Who's afraid of the seven times table?
September 27, 2011 9:17 AM   Subscribe

Who's Afraid of the Seven Times Table? Ernst Kummer, one of the great mathematicians of the late 1800s, was hopeless at arithmetic. He was giving an advanced maths lecture and in the middle of a complicated calculation he needed to know what six times seven was. “Um ... six times seven is ... six times seven . . .” A student put up his hand: “41, Professor.” Kummer chalked 41 on the blackboard. “No, no, Professor!” shouted another. “It’s 44!” Kummer gave the students a quizzical look. “Come, come, gentlemen. It can’t be both. It must be either one or the other!”

Who can blame kids for multiplcation anxiety when the rules for learning the multiplication table seem more difficult than doing it by rote?

Abbott & Costello's take on the seven times table.
posted by storybored (168 comments total) 30 users marked this as a favorite
 
One of my rituals if I am having difficulty getting to sleep (typically due to some unwanted thoughts racing and racing in my brain) is to countdown from 100 by 7's, wrapping around when I pass zero. 100, 93, 86, ... , 9, 2, 95, 88, 81, ..., and so on. Sevens require just the right amount of attention. It is not a lucky number as far as the decimal system is concerned. (But the simple fractions 1/7, 2/7, 3/7, 4/7, 5/7, and 6/7 do have a special charm in decimal).
posted by Wolfdog at 9:21 AM on September 27, 2011 [7 favorites]


Rote is good in this regard. Doesn't waste time pulling up the requisite number when you need it. I had to memorize upto 20 x 20 back in the day but can only regularly remember uptil 12x12 now. The 7 and the 17 times table were stumbling blocks.
posted by infini at 9:23 AM on September 27, 2011 [2 favorites]


I learned it rote. Eight years old. My mom and flashcards in the living room after dinner every night for a month. Sevens were trickier than 8's and 9's I recall very well. Teaching me my multiplication tables was the single biggest teacher aid thing my mom ever did in my life.
posted by bukvich at 9:24 AM on September 27, 2011


Sevens are easy; it's the score you would have if you scored a certain number of touchdowns.

I can never multiply eights in my head, though - I always have to add them individually.
posted by Curious Artificer at 9:24 AM on September 27, 2011 [4 favorites]


Yeah, the sevens were the worst. The twos and threes are easy, with the threes being even easier when you find out that within a number divisible by three, all the digits have to add up to a number divisible by three. The fours are like the twos but with skipping. Five? Only two numbers to worry about! Six is like the fours but with threes.

Then comes seven. 14? 21? 28? It's like pulling numbers out of your ass! To this day and after years of having to do quick, simple math in my head thanks to Dungeons and Dragons, I still have to take a second to think what 7 times 6 is.

Fucking sevens, man.
posted by griphus at 9:24 AM on September 27, 2011 [8 favorites]


Metafilter: Fucking sevens, man.
posted by localroger at 9:26 AM on September 27, 2011 [6 favorites]


Sevens are easy; it's the score you would have if you scored a certain number of touchdowns.

This doesn't help me because I'm a Bears fan, but sometimes I need to count higher than 14.
posted by Bulgaroktonos at 9:27 AM on September 27, 2011 [71 favorites]


If only our number system were in base seven! Then the sevens table would be easy!
posted by kaibutsu at 9:27 AM on September 27, 2011 [2 favorites]


The only seven that is hard is 7x7. If you know the twos, threes, fours, etc, you know the rest of the sevens...
posted by Philosopher Dirtbike at 9:28 AM on September 27, 2011 [4 favorites]




I never had trouble with the basic multiplication table except for 7 x 8. For some reason that one took me way longer to finally get down than anything else.
posted by kmz at 9:28 AM on September 27, 2011 [10 favorites]


I had a math teacher in high school who pointed out that all of the decimal fractions for sevenths were just the same six numbers repeated in the same order, and the only difference was the starting number. Ergo

1/7 = .142857 142857 142857...
2/7 = .285714 285714 285714...
3/7 = .428571 428571 428571...

and so on. That still completely blows my mind.
posted by theodolite at 9:28 AM on September 27, 2011 [110 favorites]


If only our number system were in base seven! Then the sevens table would be easy!

And not just that, but we'd have incontrovertible proof that there is no god, to boot!
posted by griphus at 9:28 AM on September 27, 2011 [3 favorites]


'Some of you who have small children may have found yourself in the embarrassing position of not being able to do your child's mathematics homework...'
posted by kaibutsu at 9:29 AM on September 27, 2011 [6 favorites]


Rote here as well, and sevens were definitely the stumbling blocks.

Mother Renault would ask me "what's 7x9", and I wouldn't know, and then try to trip me up with "what's 9x7?" But I was onto her crafty game...
posted by Capt. Renault at 9:29 AM on September 27, 2011


I never had trouble with the basic multiplication table except for 7 x 8

7 x 8 is 56, 5678. This only helps if you remember that 5 x 6 is not 78.
posted by Bulgaroktonos at 9:29 AM on September 27, 2011 [3 favorites]


I learned the times tables rote and I was the BOSS of my 4th grade class. We played Conductor and I circled the room 3+ times, NOBODY COULD BEAT ME. We need to have a Conductor meetup, I would like to taste that glory again.
posted by ThePinkSuperhero at 9:30 AM on September 27, 2011 [21 favorites]


This guy (MisterNumbers) has written a method for learning the times tables that depends on recognizing the patterns in the numbers. Kids work it out through creating a series of tables for different multiples. He also liked to teach kids tricks. I would have loved his stuff as a kid but my boys have not been very interested in it.

I finally figured out some of my trickier times tables in the last few years when I started to recognize some of the underlying tricks. For instance, 7x6 is easier for me if I think of it as 7x5 plus 7: 35+7 is an easy bit of addition, and 7x5 is easy because it's half of 7x10. Or quickly multiplying by 12 or 13 by realizing it's the number times 10 plus the number times 2 or 3; again, this is usually a trivial bit of mental addition because multiplying by 10 is always a nice round number.
posted by not that girl at 9:31 AM on September 27, 2011 [2 favorites]


I realized once that I think in eights. Like the way counting in fives and tens seems "natural" to people, counting in fours and eights seems natural to me. I think it's an early-musical-education thing. I tend to count things off in fours and eights.

But I don't know, I never found sevens to be especially tricky. Learned the tables by rote; seven seemed about like all the others.
posted by penduluum at 9:31 AM on September 27, 2011


I am convinced that being forced to learn multiplication tables is what led me to have a life-long terror of all things mathmatical. I avoid math like the plague, even down to keeping score when playing card games. If it requires even simply addition or subtraction on-the-fly, I walk away. I'm terrorized by math.
posted by Thorzdad at 9:31 AM on September 27, 2011 [9 favorites]


This doesn't help me because I'm a Bears fan, but sometimes I need to count higher than 14.

Seven is the number of times Jay Cutler gets clobbered per drive.
posted by Modus Pwnens at 9:31 AM on September 27, 2011 [4 favorites]


KENKEN will get you in shape!
posted by dabug at 9:32 AM on September 27, 2011 [1 favorite]


7 is like the chess knight of the times tables.
posted by steef at 9:32 AM on September 27, 2011 [10 favorites]


I firmly believe that my dislike of math can be directly traced to my failure to ever memorize the 7s.

To this day I have to stop and think about it when it comes up.

Which is why I hate it when people tell me that "seven is their lucky number." Who the hell chooses such a pain in the ass as a lucky number? It's absurd!

Not like 13. Now there's a lucky number you can count on...

posted by quin at 9:32 AM on September 27, 2011 [1 favorite]


Does anybody remember Mathnet? There was an episode where they figured out a record company was jacking up sales figures because all their record sales were multiples of 7. It was awesome.
posted by kmz at 9:32 AM on September 27, 2011 [17 favorites]


W,H,A,T,D,O,Y,O,U,G,E,T,I,F,Y,O,U,M,U,L,T,I,P,L,Y,S,I,X,B,Y,N,I,N,E
posted by dragstroke at 9:33 AM on September 27, 2011 [15 favorites]


The times tables were my hell for years. I have a learning disability called dyscalculia (amoung others), and for many years in school they had me do them over and over and over to try and memorize them. The thing is my brain just can't. There are some numbers that i can easily do in my head, but others just can't. I can do them with charts, slide rules, but even if i repeat them a million times, it can't stick. The funny thing is that after 8th grade (and being tested by professionals), over summer i taught myself algebra. I went from times tables to near the top of my class in math, as long as i could use something to help me with the multiplication and division.
posted by usagizero at 9:34 AM on September 27, 2011 [3 favorites]


I dunno. I did this by memorization back in the day. On the other hand, I am really good with patterns, so maybe that's not a great way for everyone. These days I do a mix of memorization and calculation. 7x7=49 is memorized, 6X7 for some reason I need to think of 49-7=42. Why? I am not your neuroscientist!

Years ago, I worked at a store in a state with 5% sales tax. We did not have a register but used a receipt box and a calculator. I just did the math in my head. People were occasionally amazed.

Me: "That will be $10.50."
Customer: "What? How did you do that?"
Me: "What?"
Customer: "You did that in your head! It's amazing!"
Me: "This was $7 and that was $3. That's $10 total. 5% x $10 is fifty pennies. $10.50..."
Customer: "How is that even possible!?!"
Me: "..."
Me: "Have... a nice day?"
Customer: "Wow!"

Me (years later): "..."
posted by GenjiandProust at 9:34 AM on September 27, 2011 [10 favorites]


I learned it rote, but there are tricks that can be used.

The 5s table is fairly easy to remember. 6x7 is 5x7 + 7. To add by 7 is the same as adding by 10 and subtracting 3. 35 + 7 -> 35 + 10 - 3 -> 45-3 -> 42.

That +10-X trick is very useful in multiplying low numbers by 8 or 9, and can make counting by those numbers barely harder than counting by 2s or 1s.
posted by JHarris at 9:34 AM on September 27, 2011


I love the seven times tables! I love that multiples of seven end in any number. Unlike those bigotted twos, and that asshole 10 and boring five.
posted by frecklefaerie at 9:35 AM on September 27, 2011 [4 favorites]


I'm 35 and I was, up until just now, unaware that there were rules for the times tables. I was also the boss of my class, although we did the tables up to 12x12 in 3rd grade, rote. I admit, I did it all for the HoHos awarded to the first person to memorize all the tables with 100% accuracy.
posted by peep at 9:36 AM on September 27, 2011 [1 favorite]


Obviously the answer is 42. Any additional maths are simply a distraction from that universality.
posted by nickrussell at 9:36 AM on September 27, 2011 [2 favorites]


My favorite quote from the OP's link for learning the times table: "Multiplying by 2 is easy, you just double the number."
posted by dfan at 9:36 AM on September 27, 2011 [5 favorites]


Fuck the times tables.
posted by longsleeves at 9:37 AM on September 27, 2011 [2 favorites]


I was never the boss of my class at multiplication, but long division? I was the long division king. I remembering spending a whole afternoon just wowing the class with my long division skills. In retrospect, this was probably not as much fun for everyone as I thought.
posted by Bulgaroktonos at 9:39 AM on September 27, 2011 [11 favorites]


I remember learning the multiplication tables in 2nd grade. I recall the teacher taught us a few techniques for doing multiplication and essentially said "go figure it out for yourselves." We had self-directed study after that. When we had figured out one column of the table, we'd get tested on that—we probably had to test perfectly, and it was probably under a time limit (forcing us to memorize our work), but I don't really recall.

I do recall making little matrices, like 6 × 7 dots, and counting all the dots to work out the product.
posted by adamrice at 9:39 AM on September 27, 2011


I had a math teacher in high school who pointed out that all of the decimal fractions for sevenths were just the same six numbers repeated in the same order, and the only difference was the starting number. Ergo

1/7 = .142857 142857 142857...
2/7 = .285714 285714 285714...
3/7 = .428571 428571 428571...

and so on. That still completely blows my mind.


Yeah, this is because 7s interact with powers of 10 in a nice predictable way.

For example, 2/7 = 100/7 - 98/7 = 14.2857142857... - 14 = .2857142857...
posted by dfan at 9:39 AM on September 27, 2011 [4 favorites]


All I know is that 57 is a prime number.
posted by benito.strauss at 9:42 AM on September 27, 2011 [2 favorites]


One of my rituals if I am having difficulty getting to sleep (typically due to some unwanted thoughts racing and racing in my brain) is to countdown from 100 by 7's, wrapping around when I pass zero.

Funny I use that same method for a totally different purpose, also in bed (usually).
posted by The Bellman at 9:43 AM on September 27, 2011 [3 favorites]


18, 27, 36, 45, 54, 63, 72, 81.

I used to love that.
posted by box at 9:47 AM on September 27, 2011 [13 favorites]


Sorry for my outburst above, they just wouldn't to by brain and my mother was obsessed and I mean OBSESSED with me memorizing them and had me sit after school with her hand typed tables forcing me to go over them again and again. For days. I never did learn them by rote, I have since been told that I use the wrong side of my brain to do math.
posted by longsleeves at 9:48 AM on September 27, 2011




'Some of you who have small children may have found yourself in the embarrassing position yt of not being able to do your child's mathematics homework...'
Ugh, reactionary idiocy. First of all "New Math" came out in the 1960s, before this guy was even born. Or maybe he's doing a cover of something that came out in the 1960s, I don't know.

But whatever, there is a huge emphasis some people place on teaching math "the same way I learned it" even though the way they learned it was horrible and made generations of people "hate math". They think math should be hard and if it's not difficult and frustrating then you're not learning. It's really making people dumber.

I never really studied the times table, but I think I know most of it by now. The thing is, technology these days means that, unless you're stuck on a desert island and you didn't bring along a solar powered calculator it's really an unnecessary skill.

If the time spent teaching kids arithmetic was spent teaching them algebra instead they'd be a lot smarter.
posted by delmoi at 9:49 AM on September 27, 2011 [1 favorite]


Math... memorization. I've heard other people mention that rote instruction is somehow out of vogue in certain circles. If this is true and not some urban legend, I find this silly. There are some things you just have to drill into your head, whether it's multiplication tables, methods of integration, language vocabulary, programming syntax, on and on. Yes, you must also learn theory, and understanding the theory can help you learn faster, but simply having something you know absolutely cold right there in your brain enables you to move forward much faster when figuring out a problem than if you have to work from first principles.
posted by scelerat at 9:51 AM on September 27, 2011 [3 favorites]


> If the time spent teaching kids arithmetic was spent teaching them algebra instead they'd be a lot smarter.

I strongly doubt anybody has any good idea whether or not this statement is true or false or how we could conceivably observe it falsified in our lifetimes with ethical psycho research.
posted by bukvich at 9:52 AM on September 27, 2011


Well, they would be better at advanced math, rather then better at doing arithmetic which they will never, ever need to do.
posted by delmoi at 9:55 AM on September 27, 2011


Ugh, reactionary idiocy. First of all "New Math" came out in the 1960s, before this guy was even born. Or maybe he's doing a cover of something that came out in the 1960s, I don't know.

Dude. You don't know Tom fucking Lehrer? For shame.

If the time spent teaching kids arithmetic was spent teaching them algebra instead they'd be a lot smarter.

You need to know at least some basic arithmetic to do algebra. I mean, I guess you can start kids on abstract groups and rings and fields, but I don't know that it would really work very well.
posted by kmz at 9:57 AM on September 27, 2011 [2 favorites]


delmoi, he is lip syncing to a clip of Tom Lehrer, who sang funny songs on a TV show in the early 60s called "That Was The Week That Was."
posted by gagoumot at 9:57 AM on September 27, 2011 [1 favorite]


My mom attempted to teach me what she'd learnt. I know some of it but these things make her come across as a math savant.
posted by infini at 9:58 AM on September 27, 2011 [4 favorites]


The information on Babylonian arithmetic is partially incorrect. Although it's true that the Babylonians did have times tables, there's no evidence that children were ever required to memorize them, and quite a bit of information that nobody was.

Babylonian mathematics was done in base 60, so the basic times table had 3481 entries, which is a few more entries to memories than our 81. Babylonians also had lots of other tables: tables of reciprocals (for division), tables of pythagorean triples, etc. It's quite likely that all these tables were used as lookup tables, not as something to memorize. There's no evidence on any of these tablets that they were every used as something to memorize.

Memorizing the times table comes from the hindu-arabic tradition, which used based 10 numbers, and digit-by-digit algorithms. It is in this format that memorizing the times table is both possible and advantageous. The first documented instance of someone outright saying that the base 10 times table needed to be memorized (that I know of) comes from the Treviso Arithmetic, one of the first printed books, which instructs merchants in pen and paper algorithms for calculation in base 10. It gives a number of different base 10 algorithms, including the standard one you probably learned in school, and the lattice method --- which people are currently pretending is new. The anonymous author of the Treviso Arithmetic starts by listing the times tables and telling people to memorize them. There's a fully PDF copy of the Treviso in its original latin floating around the internet, but I'm not finding it right now.

Over time, different cultures with different technologies have had to memorize different things to do math. The memorized motions for doing multiplication on an abacus or a counting table are very different from the memorized results for doing multiplication on pen and paper. The Babylonians, working with clay tablets, likely did not memorize much of anything at all. In modern times when everyone has a calculator in their cell phone and google calculate in their lap, it might be that we begin moving to memorizing different things, or the base 10 times table might continue to serve a need and stand the test of time. It's very hard to say. But if it does disappear, it's not a tragedy, it's an adaptation of a culture to the new needs of new technologies.
posted by yeolcoatl at 9:58 AM on September 27, 2011 [15 favorites]


Now here's another problem
You've gotta be so quick
What do you make of four times six?
Turn them around, and figure, 'what is six times four?'
You'll learn the solution to both is 24.
posted by box at 9:58 AM on September 27, 2011 [1 favorite]


Ugh, reactionary idiocy. First of all "New Math" came out in the 1960s, before this guy was even born. Or maybe he's doing a cover of something that came out in the 1960s, I don't know.

Tom Lehrer! Honestly!

People today just don't know about comedy music. They should learn about it the same way I did.
posted by Faint of Butt at 9:58 AM on September 27, 2011 [5 favorites]


18, 27, 36, 45, 54, 63, 72, 81.

I used to love that.


The various properties of arithmetic with 9s has been always very beautiful to me. Preservation of digits mod 9, etc. Of course it all falls out very easily from 9 = -1 mod 10.
posted by kmz at 9:59 AM on September 27, 2011 [1 favorite]


Not only was Tom Lehrer performing in the 60s, but he was an actual mathematician as well.
posted by tdismukes at 10:00 AM on September 27, 2011 [4 favorites]


We were only supposed to learn up to 10x10, because you didn't need anything higher in a fully metric world. My mum, a teacher, taught us up to 12x12, and I - annoying little shit that I was - would pipe out "... eleven sevens are seventy seven and twelve sevens are eighty four" after the class had stopped. Mrs Hughes, wherever you are, I'm sorry.

Older Scottish schools used to teach up to 14x14, 'cos you never knew when you had to multiply in stones and pounds.

I'm still waiting for that fully metric world, incidentally.
posted by scruss at 10:02 AM on September 27, 2011 [2 favorites]


I never really studied the times table, but I think I know most of it by now. The thing is, technology these days means that, unless you're stuck on a desert island and you didn't bring along a solar powered calculator it's really an unnecessary skill.

If the time spent teaching kids arithmetic was spent teaching them algebra instead they'd be a lot smarter.


I've never understood this line of reasoning; do you genuinely not need to do basic math during your day? Do you reach for a calculator to divide up a bill or figure out a tip? I use my grasp of multiplication facts all the damn time.

The other thing is when you teach arithmetic well, you're not only teaching them a list of mathematical facts, you're teaching children to think with and use numbers, a skill you absolutely need to do advanced math. That's why although we don't do much rote memorization anymore (my wife was forbidden from having her students do time tests), we do a lot of arithmetic, it just takes the form of using physical manipulative and number lines as a way of increasing the children's numeracy. It's also invaluable at making them "smarter."
posted by Bulgaroktonos at 10:03 AM on September 27, 2011 [3 favorites]


benito.strauss: "All I know is that 57 is a prime number."


Is this a joke I don't understand? Or a math troll? Because 3 * 19 = 57. Says so right in your wiki link.
posted by pwnguin at 10:04 AM on September 27, 2011


Although 57 is not prime, it is jokingly known as the "Grothendieck prime" after a story in which Grothendieck supposedly gave it as an example of a particular prime number.[1]
posted by kmz at 10:06 AM on September 27, 2011 [3 favorites]


I intend to finally learn them all when my son has to learn them. It's something we can do together.
posted by pracowity at 10:09 AM on September 27, 2011 [2 favorites]


Also, if you want to know if a number is divisible by seven, you can figure it with something that admittedly takes a bit of mental arithmetic, but it otherwise really simple. Repeatedly separate the last digit like this:

XYZ|D

Then calculate XYZ-2*D to get a new number. Repeat until you are left with something that you can easily tell either is or is not divisible by seven. For example, to work out that 56738 is divisible by seven, the procedure goes as follows:

5672 | 8 gives us 5673 - 2*8 = 5656
565 | 6 gives us 565 - 2*6 = 553
55 | 3 gives 55 - 2*3 = 49 which is 7 * 7
though you can go further and calculate 4 | 9 giving us 4-2*9 = - 14, which is obviously divisible by seven too. This also goes to show you don't always hit exactly zero with this method. It also doesn't easily show you that 56728 is 8104 * 7, but if you look closely, you can probably tell that it is.

Now, this can really help with your sevens table, if you go through all the small multiples and see that it does indeed hold for them. Numbers where the first digit is twice the second all must by this rule be divisible by seven, and it does indeed hold: 3*7 = 21, 6*7 = 42, 9*7 = 63, and 12*7 = 84. In fact, 105 being 7*15 seems to fit it too, despite 10 not being a single digit. Remembering these anchors might help one memorize the whole table.

It is possible to prove this statement with some modular arithmetic. You can also find similar methods for other divisors. My apologies to any math-phobes reading this, but numbers are cool and I like knowing things about them.
posted by tykky at 10:11 AM on September 27, 2011 [3 favorites]


All multiplying for me came down to a very simple two step process:

1. Take the smaller number and square it.
2. Add the smaller number into the total as necessary.

Still, to this day, what is 6x8? 6x6 is 36, 42, 48. Done. I had to memorize 12 cells in the chart (1x1, 2x2, etc) and then just add.

49 would always stick in my head as the largest square number less than a deck of cards. I have no idea why that stuck that way, but it did. I never had a problem with 7x7. The strangest thing.
posted by andreaazure at 10:12 AM on September 27, 2011 [1 favorite]


delmoi: Well, they would be better at advanced math, rather then better at doing arithmetic which they will never, ever need to do.

Have you never been broke? Never been in the grocery store with just $10 in your pocket and needed to know how many packages of 79 cent kim-chi ramen you could buy?
posted by 256 at 10:15 AM on September 27, 2011 [1 favorite]


"My, Professor, that is a very large herd of cattle!"
"Yes, there are 1127 of them."
"Good heavens, you didn't count them all, did you?"
"Don't be silly; I counted their legs and divided by 4."
posted by GenjiandProust at 10:15 AM on September 27, 2011 [8 favorites]


Hmmm... I'm not finding the pdfs of the Bablyonian math tablets or the Teviso with a quick search, but I know they're out there. Being thousands of years old, they're not under copyright, though, so if anybody wants a copy, they can memail me.
posted by yeolcoatl at 10:16 AM on September 27, 2011


6 X 7 is easy. It's the Meaning of Life.




Fuck the times tables.

I think you'll find the experience singularly unsatisfactory.
posted by Kirth Gerson at 10:18 AM on September 27, 2011 [2 favorites]


When I was in school, I ended up getting a pretty bad concussion that caused me to lose my memories for the entire morning, as well as several hours after the fact. The only memory I have before coming to in an ambulance, was the teacher asking me to count down from 100 by 7's. I remember concentrating really hard, and I am pretty sure I even remember how far I got before I gave up, which was 65.

I guess it was the mental difficulty of the task that created so many neural connections that allowed me to remember it, while completely and totally forgetting everything else that happened during a 6 hour period...
posted by ryanfou at 10:20 AM on September 27, 2011


I've never understood this line of reasoning; do you genuinely not need to do basic math during your day?

I said I didn't need to do math in my head very often and I never need to do it on paper. If I'm sitting at a computer I can use Google or the built in calculator.

I sometimes do arithmetic in my head for fun, and I've gotten OK at it. But other then "entertainment" have zero use for basic arithmetic on a day to day basis.

A lot of examples people use of 'math you need to do every day' are things where you could use simpler estimates, like adding up the cost of food on a shopping trip or something like that.

The other thing is that most of the times tables are made from non-prime numbers. So 4*6 is the same as 2*2*2*3, so if you know parts of the times table you can figure out the rest pretty easily without memorizing it. That's why numbers like 7 are difficult for people, it's the largest prime number and 5 is easy to remember because it's half of 10, 3 is easy to remember because it's small.
Never been in the grocery store with just $10 in your pocket and needed to know how many packages of 79 cent kim-chi ramen you could buy?
Yeah I have been broke but this was never really a problem. You can estimate it pretty easily. No one would take out a piece of paper and actually calculate 1,000/79 using long division. It's easy to figure out because .79 is close to .80, .80 * 10 = 8.00 and that leaves you with $2. And you figure you can get two more for a total of 12.

But the thing is, that's estimation not rote arithmetic. If you used arithmetic the way it is taught in school you would take out a piece of paper and do long division. Which I guarantee you you didn't do.

I think teaching kids estimation techniques is another example of something that they should be doing instead of teaching pen and paper algorithms.
posted by delmoi at 10:21 AM on September 27, 2011


N x 5 + N + N. Easy. (Well, easier than trying to remember the sevens table.)
posted by Slap*Happy at 10:24 AM on September 27, 2011


You realize you're doing arithmetic in your estimations, right?
posted by kmz at 10:25 AM on September 27, 2011 [3 favorites]


You can't take three from two, two is less than three, so you look at the four in the eights place, now that's really four eights ...

People who don't know Tom Lehrer has nothing to do in math or chemistry threads.

I'll call it ... the Tropic of Calculus!
posted by brokkr at 10:26 AM on September 27, 2011


benito.strauss: "All I know is that 57 is a prime number."


Is this a joke I don't understand? Or a math troll? Because 3 * 19 = 57. Says so right in your wiki link.


I thought it was joke about steak. Get it, PRIME? GET IT?
posted by Panjandrum at 10:26 AM on September 27, 2011


I think teaching kids estimation techniques is another example of something that they should be doing instead of teaching pen and paper algorithms.

This is an empirical question. You've made several pretty strong claims here; are you familiar with the research literature on this?
posted by Philosopher Dirtbike at 10:26 AM on September 27, 2011


For some reason my older sister forced me to learn my multiplication tables one summer holiday; I was around 8 or 9. I didn't hate or love it, but the way in which she kept emphasising "These are very, very important" got me pretty psyched up so I was motivated to learn them.

The actual immediate goal of surviving primary school was easy enough; you're forced to use the actual tables, or perform mental arithmetic, OK. In secondary school we were actually forbidden from using calculators until around grade 10, and at that stage there was a particularly gung-ho Maths teacher who kept goading us along to not rely on calculators. The one excusable reason to use calculators, trigonometry, wasn't an issue because we were allowed (even encouraged) to provide the exact "root" form of answers, in terms of square roots etc. So I have distinct memories of passing my GCSEs and A-Levels without always needing a calculator.

But to be honest it's university that my multiplication skills really came to light. At that stage there're no nagging teachers, no conceivable reason to use mental arithmetic. But I became a veritable God of estimation. I could look at a circuit diagram (I studied EE) and say "Mmmm yes, that feels like around 30 + j50 ohms of impedance there" and people would go "What the fuck did you just say?" The heart of estimation is a solid appreciation of multiplication tables.

And so, I tip my hat to my older sister, for her foresight and dedication. *hat tip*.
posted by asymptotic at 10:29 AM on September 27, 2011 [2 favorites]


This doesn't help me because I'm a Bears fan, but sometimes I need to count higher than 14.

Seven is the number of times Jay Cutler gets clobbered per drive.


I was going to come in here and complain that seems unlikely, since a Bears drive lately is usually 3 and punt. But then I realized if anybody could allow their QB to get sacked 2 1/3 times in one down, it would be the Bears offense.

Like someone above said, I too always had trouble with 8x7 (and 7x8), though never with 7x7, which I loved -- and which, if I'd been given the proper tools at the time, should have been easy for me
posted by MCMikeNamara at 10:31 AM on September 27, 2011


I never used tricks. (Other than the nines, of course. Thank you, Square One!) However, I did have a bit of trouble with some of the 7s. I don't know why, but 8x7 always got me discombobulated. So my mother used the same technique she used with pork chops: sniveling repetition to the point of abject hatred.

"Oh, look! It's your friend!" she'd trill.

"IT IS 56, AND IT IS NOT MY FRIEND."

I never did forget that. (And I still hate pork chops.)
posted by Madamina at 10:37 AM on September 27, 2011 [4 favorites]


I was going to come in here and complain that seems unlikely, since a Bears drive lately is usually 3 and punt. But then I realized if anybody could allow their QB to get sacked 2 1/3 times in one down, it would be the Bears offense.

Based on Sunday's game it look like their plan is to take dumb penalties so there are extra plays on which he can be sacked. If you don't think it's possible to false start AND take a sack on the same play, just wait till the Bears play the Lions.
posted by Bulgaroktonos at 10:38 AM on September 27, 2011 [1 favorite]


The easiest and best way to learn basic arithmetic and multiplication tables is to drink heavily and play darts all day. Darts players are amazing at putting numbers together instantly.
posted by dvdgee at 10:39 AM on September 27, 2011 [1 favorite]


seven times table is seven tables
posted by Anything at 10:39 AM on September 27, 2011 [10 favorites]


We can manage without times tables, though I admit I found it disturbing when the man at the Tesco checkout entered ten identical items at £1 each, one by one.

He gets paid by the hour. Shortcuts aren't to his advantage.
posted by Obscure Reference at 10:42 AM on September 27, 2011


I am not a big fan of memorization, but my brother, who normally cared nothing for my educational struggles, sat down with me and helped me drill flash cards for multiplication, over and over, and by Jove, it worked.

Possibly because numbers were never "intuitive" to me, so that all the little backwards/forwards/pattern tricks didn't work so much. Just brute memorizing, but now they're in my head and come in useful every day.

I must also pause to thank my mom for constantly taking me sale-shopping, which gave me a really solid grounding in percentages. "What's 80% off a $44.99 pair of shoes that's already 30% off, and then what if we have a coupon for $5?" is a pretty good math exercise.
posted by emjaybee at 10:42 AM on September 27, 2011 [3 favorites]


For example, to work out that 56738 is divisible by seven,

Less work is to subtract multiples of 1001: In this case, start with 38038 (and divide by 100) to get 187, clearly not divisible by 7, but it is by 11 (so the original number is.) Also works for 13 since 1001 is 7*11*13.
posted by Obscure Reference at 10:48 AM on September 27, 2011 [3 favorites]


The advantage of learning arithmetic in an intuitive way (as opposed to memorization) is that you can add/multiply/divide any numbers if need be by extending the rules, and at the very least you can quickly estimate the answer (17 x 6: "well it has to be at least 60 (10 x 6) and less than 120 (20 x 6), and it has to be closer to the latter so it's more than 90...") Many people not only can't answer a problem they haven't encountered before, they can't even begin to think about the answer, a la GenjiandProust's "wow!"

And yeah, the idea that you "have zero use for basic arithmetic on a day to day basis" because "you could use simpler estimates" is ridiculous. Estimation involves basic arithmetic.
posted by vorfeed at 10:51 AM on September 27, 2011 [1 favorite]


"IT IS 56, AND IT IS NOT MY FRIEND."

Somewhere in the Platonic World of Numbers, 56 is crying its eyes out or would be, if it had eyes, which it doesn't, but never mind that 56 is really, really sad.

And it's all your fault. After all 56 did for you.
posted by GenjiandProust at 11:00 AM on September 27, 2011 [2 favorites]


Well, I like it NOW. Just like I like pork, when it's not drained of everything that makes it delicious. It took me years to admit that.
posted by Madamina at 11:03 AM on September 27, 2011 [1 favorite]


7 times tables without any math.

Writing out the 7 times table doesn't require any math. Draw a 3 x 3 grid. starting from the top right, working your way down to the bottom left by columns, write 1 - 9

7|4|1    starting top right with 1, continue by column to bottom left
-------
8|5|2
-------
9|6|3

These are your ones.

Add the tens to the grid with a pattern that repeats

0 - 1 - 2
2 - 3 - 4
4 - 5 - 6

notice that the last number in the line is repeated at the start of the next line.

Separate each box with the times 10's and you can continue this pattern

-----------------
07   14   21
28   35   42
49   56   63

70

77   84   91
108 115 122
129 136 143

140

147 154 161
168 175 182
189 196 203

210

etc...
posted by joelf at 11:03 AM on September 27, 2011 [16 favorites]


"What's 80% off a $44.99 pair of shoes that's already 30% off, and then what if we have a coupon for $5?" is a pretty good math exercise

Not to mention going shopping with mum when touristing around "and then divide by 2.13 to see what a real bargain this is!"

Good training, moms. Do we ever thank them instead of writing it up in a thread here?
posted by infini at 11:07 AM on September 27, 2011 [3 favorites]


I hated sevens as a child, and after having many, many seizures and taking lots of medicine as an adult, I remembered why I hated them. Turns out, seizures can actually damage your brain. I used Brain Age to re-learn math, but I still don't know sevens... and I can't reliably do math homework with my elementary-aged kids.

I fucking rock at sudoku, though. No brain damage to that part of my brain, I guess.
posted by doyouknowwhoIam? at 11:09 AM on September 27, 2011


My mother was outraged that we only had to learn up to the nines. I'm not entirely sure she believed I knew the times tables because I couldn't recite them, even though I was the sort of kid who did timed arithmetic quizzes for fun. (There was a shop nearby that sold supplies for teachers where my mother got them. I imagine the internet suffices for such things now.)

I think, though, that the Kummer anecdote encapsulates the rule that at a certain point, you just make the students do the arithmetic. That or the fact that it's totally impossible to do simple math while lecturing.
posted by hoyland at 11:14 AM on September 27, 2011


"IT IS 56, AND IT IS NOT MY FRIEND."

A couple hundred folks in Arkansas are looking to kick your ass right now.
posted by aught at 11:28 AM on September 27, 2011 [5 favorites]


I had a math teacher in high school who pointed out that all of the decimal fractions for sevenths were just the same six numbers repeated in the same order, and the only difference was the starting number.

Not only that, but you can remember the digits by starting with 7 and doubling, though the last digit is a little special.

14 ... 28 ... 56+1

I remember the last digit by thinking of the 1 in the hundreds place of 2*56 overlapping with the one's digit. Then for something like 5/7th, you take the 5th highest digit, 7, and start from there, 0.714285714, without having to memorize any arbitrary number sequence.

I also used to do subtraction backwards. I'd look at something like this:
 47
-18
----

I'd try to add from bottom to top. "What plus 8 gives you 7?" 9 ... 9+8 is seventeen, so you carry the one, and I'd write the 1 at the bottom under the tens digit, and then think "What plus 2 gives you 4?" and the answer is obviously 2.. It was a lot easier to get 29 as an answer than with that borrowing bullshit.

Never really understood the problem with the seven times table. You can always think of 7 as (6+1) if you're stuck, so what's 7*9? Well, it's (6+1)*9 = 54+9 = 63. I'd do that while I was learning it instead of rote memorization. It's also a good trick for doing larger numbers in your head... 17*48 may look like something you want to grab pencil and paper for, but 17*(50-2) you can probably do in your head.
posted by fleacircus at 11:31 AM on September 27, 2011 [1 favorite]


77 84 91
108 115 122
129 136 143


You borked that one. Forgot to repeat the 9.
posted by Sys Rq at 11:36 AM on September 27, 2011


Damnit
posted by joelf at 11:37 AM on September 27, 2011


and at the very least you can quickly estimate the answer (17 x 6: "well it has to be at least 60 (10 x 6) and less than 120 (20 x 6), and it has to be closer to the latter so it's more than 90...")

That's "quickly"? Isn't it faster just to get the actual answer? I mean, how long does it take to do (6x10 + 6x7) = 102
posted by Justinian at 11:38 AM on September 27, 2011


I learned my times tables by listening to these songs, on cassette tapes, natch. Sooo cheesy... but highly effective.
posted by The Winsome Parker Lewis at 11:40 AM on September 27, 2011


You realize you're doing arithmetic in your estimations, right?

I'm not using arithmetic the way it's taught in school. And the whole point is that you're substituting 'hard' numbers (like 7) with 'easy' numbers (like 8 or six). You use 'easy' operations (like multiplying by 10). All we did in that example was multiply by 10 and divide 2 by itself.

So rather then knowing the whole times table, you know most of the important entries and figuring out which substitutions will get you close.
This is an empirical question. You've made several pretty strong claims here; are you familiar with the research literature on this?
I'm talking about what people do in real life, whether it's estimating stuff at the grocery store, or using computers while doing science and math. I would argue that learning a skill that will never be used, instead of learning skills that will be used is obvious, and the opposite is the extraordinary claim that needs lots of evidence. My 'empirical data' here is everyday life.
He gets paid by the hour. Shortcuts aren't to his advantage.
Cashers get paid whether they are checking out or not. And by the way, when you actually do jobs like that you can't wait to get off work, even though you're being paid by the hour.
Writing out the 7 times table doesn't require any math. Draw a 3 x 3 grid. starting from the top right, working your way down to the bottom left by columns, write 1 – 9
Sorry, that counts as math. Not arithmetic (in terms of what you learn in elementary school) but still math.
posted by delmoi at 11:42 AM on September 27, 2011


I always found the 7's to be easy, because of American Football
posted by holdkris99 at 11:45 AM on September 27, 2011


All I know is that my 18-month-old counts as such:

ONE! TWO! FREE! FIVE! SEVEN! EIGHT! TEN!

She's adamant that there's no four, in particular. At least there's a seven in there.
posted by jimmythefish at 12:01 PM on September 27, 2011 [1 favorite]


That's "quickly"? Isn't it faster just to get the actual answer? I mean, how long does it take to do (6x10 + 6x7) = 102

You don't really work out such an estimate, though. They kind of come naturally once you have a good sense of the number line, and you can still do (6x10 + 6x7) = 102 if you need to.

Besides, I've seen people do things like (6x10 + 6x7) = 122 based on mis-remembered times tables and/or bad arithmetic. Estimates also give you a sense of whether the answer is reasonable.
posted by vorfeed at 12:01 PM on September 27, 2011


I'm not using arithmetic the way it's taught in school. And the whole point is that you're substituting 'hard' numbers (like 7) with 'easy' numbers (like 8 or six). You use 'easy' operations (like multiplying by 10). All we did in that example was multiply by 10 and divide 2 by itself.

It really sounds like the problem here is that you don't know how arithmetic is taught in schools these days. My wife teaches second grade, she has kids don't know their basic addition facts well at all, but she forbidden from teaching them using rote memorization; forbidden as in, she tried and the school told her not to.
posted by Bulgaroktonos at 12:01 PM on September 27, 2011


I'm talking about what people do in real life, whether it's estimating stuff at the grocery store, or using computers while doing science and math. I would argue that learning a skill that will never be used, instead of learning skills that will be used is obvious, and the opposite is the extraordinary claim that needs lots of evidence. My 'empirical data' here is everyday life.

I find knowing the times table eminently useful in everyday life. I come across odd numbers exactly as often as you would expect from Benford's law and it seems a great deal more convenient to be able to just manipulate them with the same speed as any other number than it does to suddenly switch to an alternative mode of reckoning, which is not to say that don't use estimating techniques when dealing with unknown quantities.

I'm horrified to read that some people were introduced to multiplication in grade 2 or even grade 4. It's a hell of a lot easier to just learn this stuff by rote, in kindergarten. Learning some things by rote does not in any way impair your ability to reason about other things. You almost certainly learned the alphabet by rote, and that works out just fine for most people.
posted by anigbrowl at 12:02 PM on September 27, 2011 [5 favorites]


Wait, wait, stop, there's a number "seven"?! Damn...
posted by fallingbadgers at 12:04 PM on September 27, 2011


I'm horrified to read that some people were introduced to multiplication in grade 2 or even grade 4. It's a hell of a lot easier to just learn this stuff by rote, in kindergarten.

What????

In kindergarten I was learning to count to ten. That was 25 years ago. I doubt much has changed.
posted by Sys Rq at 12:05 PM on September 27, 2011


Depends on the country, really. That's really something I would find unsurprising in the USSR educational system at the time, for instance.
posted by griphus at 12:06 PM on September 27, 2011 [1 favorite]


Or the Indian.
posted by infini at 12:07 PM on September 27, 2011


I'm horrified to read that some people were introduced to multiplication in grade 2 or even grade 4. It's a hell of a lot easier to just learn this stuff by rote, in kindergarten.

I did that. Then everyday, they separated me from the rest of the kids for math and reading. So lonely.
posted by ego at 12:08 PM on September 27, 2011 [3 favorites]


My wife teaches second grade, she has kids don't know their basic addition facts well at all, but she forbidden from teaching them using rote memorization; forbidden as in, she tried and the school told her not to.

wtf

Who is it that thinks it's a bad idea to drill kids in the basics as young as possible? Because those are the people that need to be kicked out of the education system ASAP.
posted by anigbrowl at 12:08 PM on September 27, 2011


6 X 7 is easy. It's the Meaning of Life.

Of course, Douglas Adams fucked up his arithmetic.
posted by twirlip at 12:11 PM on September 27, 2011


What????

In kindergarten I was learning to count to ten. That was 25 years ago. I doubt much has changed.


We used to count up to 100 and did addition and subtraction and the alphabet and basic reading. Then in what would be grade 1 we did multiplication and short division. This was all chanting by rote from a book of tables several times a day. We kept using the tables up through grade 2, though less and less frequently. I don't think we used them after we started grade 3. This was in Ireland.
posted by anigbrowl at 12:13 PM on September 27, 2011


My oldest daughter, then about four years old, noted that we had bought a dozen muffins at the grocery store. Counting her younger sister, mom, myself, and herself, she announced that there were three muffins for each of us.

Mom, busy unpacking the groceries, mumbled a distracted "That's right, honey". I had to stop her and point out that she had divided twelve by four and gotten three.

I have found that I think similarly. If you asked me what 12 divided by 4 is, I'd have to stop and think a second. But if you asked me how many muffins get divided evenly, I'd have it instantly. I have always found word problems to be much easier than the symbolic notations used traditionally. I don't know why this is, and I have always been a little puzzled that so many people find word problems so difficult.

Happily, I do a fair amount of basic math in my work, so this stuff doesn't get much chance to get rusty, either.
posted by Xoebe at 12:16 PM on September 27, 2011


Oops, grade 4 I meant to say. In Ireland the school system goes (junior infants, senior infants, 1st - 6th class) = primary school, then you go to secondary school which is like high school. You do your first year, then study 2 years for your Intermediate Certificate (big scary state exam). some people leave school after that, most go on to do 2 more years and then a Leaving Certificate (another big scary state exam). So you've got 13 years in total, same as K-12. Some schools offer a transition year in the middle of secondary to explore subjects off the beaten path.

So by 3rd class you've had 4 years of school, which was when we stopped using the times tables (unless the teacher was feeling bored or sadistic).
posted by anigbrowl at 12:20 PM on September 27, 2011



I was always pretty good at math. In high school, I took pre-calc and honors geometry and all that noise. I really liked math.

I took the asvab one Saturday my senior year. It went well enough, piece of cake, really.

Then I got to the math portion. Back then, there were 120 problems of the 6x7, 4+3 variety. You had 2 minutes to do as many as you could. Here I was doing integrals and partial fractions and power series in school and now, what is this ?

4+3 WHAT ? WHERE THE FUCK IS X ? WHAT AM I SOLVING FOR ? JESUS HOPSCOTCH CHRIST, I DON"T KNOW HOW TO DO THIS!!!

I must have stared at that paper for a full minute before it occurred to me that I could just pretend the x was invisible. 4x + 3x = 7x! YES. I HAVE SOLVED IT!

Worst test ever.
posted by Pogo_Fuzzybutt at 12:27 PM on September 27, 2011 [8 favorites]


About a decade back, I spent some time one summer helping the friend of a friend with her GED studies in math. She was particularly stuck on fractions and decimals and converting between the two systems. She eventually fell back on "I don't get this, and I can't learn it, and it's no use." And I said "Yeah, you can. You know this; you just don't know that you know it." And she said "Nuh uh." And I threw a handful of change on the table and said "How much is that?" She immediately replied "65 cents." And I said "See? You converted 1/4, 1/4, 1/10, and 1/20 into a decimal without even thinking."

Technically, I might have been lying. She could have been thinking about it as .25, .24, .1, and .05. But it kind of made a light go on for her, and she kept studying.
posted by GenjiandProust at 12:54 PM on September 27, 2011


About 15 years ago, on a bus in Vancouver, I saw a father and daughter practicing the multiplication tables. He would say, "6 times 6", and she'd reply, "36!", and he'd nod. "5 times 8." "40!" *nod* "7 times 7." "... 49?" *nod* It became clear that the daughter had some trouble around the sevens, so the father came back to "7 times 7" a few times. The daughter caught on and started playing with it. He'd say, "7 times 7", and she'd say, "Oh... [looking up as if trying to remember] is it... is it... 49? [big smile]" And he'd nod calmly, same as ever.

After a particularly hammy iteration of this, the father asked "7 times 6", and there was a pause. She doesn't know, I thought. And then she said, slowly, "49 minus 7... which is... 42!" Big smile. Calm nod.

I wanted to leap up and shout, "Do you know what you just did?! If first-year university students could do what you just did, ..." Well, I wanted to rant a bit. I didn't, because I didn't want to freak them out, but it would have been a good rant. See, that kid got a question and didn't know the answer. Many students give up at that point. This kid invented a method of deducing the answer by combining what she knew about the meaning of multiplication with a freshly remembered fact, and she did it on the fly.

Bad student: "I don't know." Good student: "I don't know, but let's see if I can figure it out..."

I wish I knew how to resurrect that spirit in university students who gave up on understanding math a decade before I ever laid eyes on them.
posted by stebulus at 1:01 PM on September 27, 2011 [13 favorites]


The thing is, you don't actually have to learn the seven times table. You can just learn all the other times tables and the sequence of squares.

(The sevens are kind of hard, though. I keep wanting to look at the multiplication quizzes on Sporcle and see if there are patterns in the results, but I never actually do it.)
posted by madcaptenor at 1:06 PM on September 27, 2011


The times tables were my hell for years. I have a learning disability called dyscalculia (amoung others), and for many years in school they had me do them over and over and over to try and memorize them. The thing is my brain just can't. There are some numbers that i can easily do in my head, but others just can't. I can do them with charts, slide rules, but even if i repeat them a million times, it can't stick. The funny thing is that after 8th grade (and being tested by professionals), over summer i taught myself algebra. I went from times tables to near the top of my class in math, as long as i could use something to help me with the multiplication and division.

I have the exact opposite problem as you. I didn't have any trouble learning the multiplication tables and still remember them easily... well, through 11's anyway. I have to multiply two digit numbers on paper.

But I cannot seem to learn algebra to save my ass. Only my husband's most patient and persistent tutoring got me through business math with a C, the first and only C I've ever gotten in college, and I sweated drops of blood on the final exam. And I barely remember a single thing I learned in the class.

I am amazed that my husband can just look at an algebra problem and solve it in seconds. I rarely have the faintest clue where to start. But his arithmetic is much slower than mine.
posted by Serene Empress Dork at 1:08 PM on September 27, 2011


When my third grade teacher attempted to teach me rote memorization of times tables in the mid-1980s I more or less replied that I refused to do the busywork of a calculator or a machine.

She responded with "what are you going to do if you don't have a calculator?" to which I replied with something like "They're everywhere and they're dirt cheap. I have one on my wrist. And if I can't find a calculator within a short walk that means something went terribly wrong with society and I probably have bigger things to worry about like how to make fire."

She scoffed at this and said, completely seriously - "Computers are just a fad! They're going to go away soon!" and things went rapidly downhill from there.

I don't think she appreciated or enjoyed learning that computers were already everywhere - in her car, refrigerator, phone, and TV - everywhere. Much less from a disgruntled, screaming, frothy nerdchild 'spergin' like Kim Peek with a new phone book. It was probably deeply unsettling for her to suddenly realize her every day life was already deeply reliant on computers, that she was surrounded by them. She also probably didn't appreciate being told that a computer signs her paycheck.

Fast forward a few months and I'm at her house showing her how to use her Texas Instruments home computer. I remember she complimented me on my touch-typing skills. Which now that I think about it, it was probably pretty freaky and weird looking that I could touch type when I was 9 in the early/mid 1980s. It's probably/hopefully less freaky now.

Anyway, never had a problem with multiplying 7s, but I've always relied on addition or subtraction workarounds instead of rote memorization - though at this point I think I have the single digit tables memorized whether I wanted to or not. If I have to multiply more than a few numbers I still reach for a calculator or a piece of scrap paper. There's a reason why we invented those, and it's because rote memorization of data arrays is a terrible waste of a human mind. We've got better things to do.

And it leaves room for more important things like trivia and beer.
posted by loquacious at 1:18 PM on September 27, 2011 [4 favorites]


 1/19 = .052631578947368421...
10/19 = .526315789473684210...
 5/19 = .263157894736842105...
12/19 = .631578947368421052...
 6/19 = .315789473684210526...
 3/19 = .157894736842105263...
11/19 = .578947368421052631...
15/19 = .789473684210526315...
17/19 = .894736842105263157...
18/19 = .947368421052631578...
 9/19 = .473684210526315789...
14/19 = .736842105263157894...
 7/19 = .368421052631578947...
13/19 = .684210526315789473...
16/19 = .842105263157894736...
 8/19 = .421052631578947368...
 4/19 = .210526315789473684...
 2/19 = .105263157894736842...
 1/19 = .052631578947368421...

Alright, I'm just going to stare at that all afternoon instead of getting work done. The full sequence of primes for which this works is at the online encyclopedia of integer sequences.
posted by madcaptenor at 1:23 PM on September 27, 2011 [5 favorites]


When I was young I hated rote learning and resisted learning the multiplication tables. Then, as an adult, I studied mathematics. I still have some holes in my knowledge of the tables, but I am on much more friendly terms with numbers themselves, which unfortunately is not something that is taught in grade school. For example, whenever I need 6x7, I remember that 3 sevens is 21 (that's easy, or else just 14+7), then double that to get 42. It doesn't take more than a millisecond. Similarly 7*8 = double 7*4, which I can remember (or get by doubling 14). I think math teachers should show students this kind of fun number-play, rather than just drilling!
posted by crazy_yeti at 1:26 PM on September 27, 2011 [1 favorite]


Are there seriously that many people with problems multiplying sevens?

Jesus. WTF. I cannot understand that. It's just another number.
posted by grubi at 1:32 PM on September 27, 2011 [1 favorite]


Bulgaroktonos wrote: I've never understood this line of reasoning; do you genuinely not need to do basic math during your day? Do you reach for a calculator to divide up a bill or figure out a tip? I use my grasp of multiplication facts all the damn time.

I'm not going to make a big argument against people learning the effing multiplication tables. Flash cards make it stupidly simple, after all, and even partial memorization helps you with the rest, but tipping is stupidly easy even if your arithmetic skills are incredibly poor.

If you can't move a decimal place one digit left, take half the result and add it back in to calculate the amount you want to tip, you have big problems. These are, after all, some of the most basic operations you can possibly do. I guess if you're too dumb for that, you can always tip 20% and then you just move the decimal place and double the result.

To be fair, and to my eternal shame it took me about 5 seconds to come up with 6x7, but I was also checking my work as I went, verifying the answer from both directions, so as not to make a fool of myself like the poor schlubs in the post.
posted by wierdo at 1:39 PM on September 27, 2011


Jesus. WTF. I cannot understand that. It's just another number.

It's the only number less then 10 that doesn't decompose to multiples of 1,2,3 or 5.
posted by delmoi at 1:44 PM on September 27, 2011 [1 favorite]


Gasp - a prime number that doesn't decompose to other prime numbers? Ban it immediately, how dare you make me learn to manipulate such a monstrosity.
posted by anigbrowl at 1:46 PM on September 27, 2011 [1 favorite]


Are there seriously that many people with problems multiplying sevens?

Jesus. WTF. I cannot understand that. It's just another number.


Sure, but not everyone was drilled in math, and people almost never need to do mental arithemetic - there are always calculators around - so what ability they had as school children atrophies. Someone once commented on how I must be really smart/good at math when I did some fairly basic mental arithmetic at the checkout. I can't remember exactly what it was, but let me tell you, I'm not very good at math.

I shall never forget a thread from several years ago on AskMe in which blantant innumeracy reigned in an Ask thread - to the point that people didn't realize a correct answer when it was staring them in the face. It was immensely frustrating to watch, even more so because it turned out the poster didn't even realize he was trying to ask an unanswerable question, though I'm sure I remember it today simply because I got in early with the correct freaking answer to the question actually asked.

And that's when it comes to math, which people don't need to use every day. People do, in fact, use words every day, and there are millions of people who don't have what I would consider a fairly basic grasp of grammar and vocabulary, so go figure.
posted by Dasein at 1:51 PM on September 27, 2011 [3 favorites]


Jesus. WTF. I cannot understand that. It's just another number.

Yeah, and as implied previously, 7 is an awkward number to manipulate in base 10. The problem isn't the number itself, it's the mismatch between the base used to count and the number.
posted by Philosopher Dirtbike at 1:55 PM on September 27, 2011


Every time Khan Academy prompts me to revisit the multiplication section I mutter a silent curse. Doing them on paper and checking my answer with a calculator seems to be the only way to reliably pass that section.
posted by tmt at 1:58 PM on September 27, 2011


See, to me, it was just another number. The little tricks are great, but simply memorizing it worked best.

This isn't me insulting or putting anyone down; it's more like I can't believe a human being can't ride a bike or swim. I'm conceptually aware there are people like this; it's just a tough thing to understand.
posted by grubi at 1:59 PM on September 27, 2011


We learned our times tables by rote, and I was absent the day they drilled the sixes and sevens. To this day I have trouble with 6 times 7 and 7 times 7, and will have to pause for a second while I add. But I thought it was just me!
posted by Soliloquy at 2:56 PM on September 27, 2011


Dasein, reading that probability thread made me cringe. I sympathize with your frustration.
posted by pemberkins at 3:34 PM on September 27, 2011


Our teacher taught us the multiples of seven to the tune of Happy Birthday, and nines to Take Me Out To The Ballgame.

I've never forgotten my sevens, though I am sometimes caught humming while on a multiplication problem.
posted by NMcCoy at 3:35 PM on September 27, 2011 [1 favorite]


I always had trouble with the sevens... until I became a Michael Apted fan.
posted by thejoshu at 3:37 PM on September 27, 2011 [1 favorite]


I learned 1-10 in primary school, mostly by rote. About halfway through the semester, the entire class was sat down for umpteen consecutive classes with a (then) massive sheet of 80 questions like "7x8=?" (that one was the hardest for me to get down; 8x8 and 7x7 were my favourites. I found them beautiful).

We did these tests until every single one of us got everything right. The students who managed this were excempted from class for the 15 minutes we were given to finish the test, and could sit in a smaller room doing whatever. I was a strong incentive at the time.

If I can't sleep, I'll multiply numbers in my head. Like 3x3x3x3.... until it starts taking too long. I mentally split the numbers up, multiply them separately and then add them back together. I can usually keep going for quite a while.
posted by flippant at 3:44 PM on September 27, 2011


I don't understand. Why is this form of memorization any harder than remembering your phone number? Do it enough and you will remember.
posted by mrgrimm at 4:03 PM on September 27, 2011 [1 favorite]


I don't understand. Why is this form of memorization any harder than remembering your phone number? Do it enough and you will remember.

Not everyone can remember their phone number. I have actually had my mother ask me what her phone number is. This is not because my mother can't remember things (she can), but because I call her a lot more often than she calls herself.
posted by madcaptenor at 4:17 PM on September 27, 2011 [1 favorite]


The only seven that is hard is 7x7.

Hawkwind can help with that one.
posted by ovvl at 5:18 PM on September 27, 2011


Are there seriously that many people with problems multiplying sevens?

Some people are just less comfortable with numbers than other people are.
posted by ovvl at 5:38 PM on September 27, 2011


Some people are just less comfortable with numbers than other people are.

The idea that this is an ok thing to say about times tables, fractions, and basic algebra is the reason half of my science students would fail if I could get away with failing that many. Nobody thinks it's ok to let a kid graduate being unable to read (even if it does happen in some places), but for some reason it's socially acceptable to say "oh, I'm just bad at math."

(I love watching their faces when I point out that multiplying any whole number by 1.35 in your head is trivially easy: just add a quarter and a tenth of it--both easy to do mentally--together and add them to the original)
posted by Dr.Enormous at 6:39 PM on September 27, 2011 [2 favorites]


See, that kid got a question and didn't know the answer. Many students give up at that point.

Chalk that up to an education system in need of a serious overhaul. If you're not teaching students to be thinkers, then you're teaching them to be ...............
posted by Twang at 6:40 PM on September 27, 2011


Here's the most useful math 'trick' I use when shopping for groceries to keep a running total of how much I'm spending. Round to the nearest dollar, and just keep track of that total. Unless you're only buying a few items, your average will be surprisingly accurate.

$1.79 = $2, $4.49 = $4, $2.19 = $2, $8.75=$9 and so on. The cents will tend to cancel each other out.

If you need to be really safe (only have $15 along and can't go over), round everything up.
posted by Twang at 7:00 PM on September 27, 2011 [1 favorite]


If the time spent teaching kids arithmetic was spent teaching them algebra instead they'd be a lot smarter.

I'm sorry, I just simply couldn't let this one go. I could not POSSIBLY disagree more strongly. This is absolutely untrue, but it is an idea that has gained significant traction in education and is the primary reason my job as a high school math teacher is often so incredibly frustrating.

See, here's the thing. I can teach a high school student algebra. Totally. I can get down with the whole concept of that crazy unknown x, and the Cartesian coordinate system, and all the rest of it. Easy to teach. What I can't do is start from scratch and teach a kid how to add two damn numbers together, or their entire times tables. Well, I can but then I wouldn't get too far with the curriculum I'm supposed to be covering. There is no algebra without basic number sense. NONE!!! I can teach them how to solve systems of linear equations and totally get them down with the methods, but if they can't add or multiply they will NEVER get the right answer. But hey! Why not use a calculator, you say? Well, you've clearly forgotten just how much simple math is involved in basic algebra. Pull out a calculator for every single damn calculation and you're solving one question every half hour or so. Not so fun for the kid. They feel like an idiot. Same thing with factoring--my students think it's some kind of witchcraft when I can instantly think of two numbers that add to 9 and multiply to 14. Yet they can give me ten different synonyms for the word "large" without even thinking twice.

Everyone freaks if a kid is illiterate, but nobody seems to give a damn if they have no concept of numeracy. You have to memorize your alphabet to read. You have to memorize your times tables (at some basic level--not up to 20 or anything but please, at least up to five or something) to do math. Or at least be able to add. I think many of you aren't truly aware of how dreadful the general level of numeracy is these days amongst typical North American students.

So here's my plea: elementary educators, I could give a rat's ass if you teach your kids problem solving, algebra, etc. Just teach them how to add and subtract (with integers would be nice, but I realize I'm asking a bit much), teach them their times tables (or how to figure them out--patterning is fun!), and you know, maybe a smattering of basic geometry. If you have time--don't sweat that last one. And maybe inequalities. I don't want to have any more damn ten-minute discussions in a grade 10 academic math class where I have to assure a student that yes, it is possible for something to be greater than AND equal to a given number. Both! At the same time! Craziness!!!
posted by Go Banana at 7:26 PM on September 27, 2011 [15 favorites]


No it's not.
posted by Crabby Appleton at 7:36 PM on September 27, 2011


I have long suspected that I have some form of dyscalculia as well; it certainly jibes with my frustrating experiences "not working hard enough" to remember math that I had previously "learned."

Nevertheless, I learned a few tricks, like counting back change, that keep me from totally embarrassing myself. But no, none of what anyone has described as "easy" in this thread is easy for me.

On the other hand, it's never occurred to me that spelling a word would require any sort of memorization. Huh? You just learn how to spell, and then you can do it correctly almost all the time, whether you know the word or not.
posted by desuetude at 7:51 PM on September 27, 2011


Here's the most useful math 'trick' I use when shopping for groceries to keep a running total of how much I'm spending. Round to the nearest dollar, and just keep track of that total. Unless you're only buying a few items, your average will be surprisingly accurate.

I've tried to explain this trick to my probability students, in the context of showing how big the error is. They just look at me like "wait, people can do mental arithmetic"?
posted by madcaptenor at 8:15 PM on September 27, 2011


I've tried to explain this trick to my probability students, in the context of showing how big the error is. They just look at me like "wait, people can do mental arithmetic"?

They've just never had The Feeling Of Power...
posted by vorfeed at 8:27 PM on September 27, 2011 [2 favorites]


See, here's the thing. I can teach a high school student algebra. Totally. I can get down with the whole concept of that crazy unknown x, and the Cartesian coordinate system, and all the rest of it. Easy to teach. What I can't do is start from scratch and teach a kid how to add two damn numbers together, or their entire times tables.
First of all, those kids actually did get basic mathematical education in school. It just didn't take, apparently. Apparently a lot of places teach elementary students math differently now then in the 1980s but if they just got the basic pencil/paper algorithm drills then and don't have a good number sense now then can't that be attributed to the type of math education they got. Teaching kids "number sense" is exactly what I'm talking about
But hey! Why not use a calculator, you say? Well, you've clearly forgotten just how much simple math is involved in basic algebra. Pull out a calculator for every single damn calculation and you're solving one question every half hour or so.
Well, I had a graphing calculator that could solve algebraic equations in highschool. In fact in Algebra ii they gave everyone a Ti-83. By the time I got to calculus I had '92 that could differentiate and integrate for me.
posted by delmoi at 8:30 PM on September 27, 2011


scruss: "I'm still waiting for that fully metric world, incidentally."

May 14th, 2019
"NASA announced today the discovery of a new planet, internally christened 'Metron 10'. This unique world has a solar day precisely one hundred thousand seconds long, a satellite with an orbital period of 10 Metron days, and a solar year consisting of 10 Metron months. Fans of round numbers rejoiced at the obvious perfection of such a world."

October 2nd, 2023
"Mankind was shocked today by NASA's announcement that the recently-discovered world of Metron 10 was inhabited by the first intelligent extraterrestrial species ever detected by man. High-energy video signals coming from the planet allowed scientists to gather vast amounts of information about the aliens' culture.

Fans of round numbers mourned the revelation that members of the alien race appear to possess 28, 30 or 31 fingers (29 in a rare genetic mutation) and twelve toes, and have developed a unique variable-base numeric system accordingly.

The civilization has accordingly been dubbed the 'Ironites'."
posted by Riki tiki at 9:27 PM on September 27, 2011 [1 favorite]


oh wow that Lucky Seven Sampson Schoolhouse Rock is THE MOST GORGEOUS SCHOOLHOUSE ROCK EVER. It's in the funky 70s style of the rest of the things but it's clearly animated by someone who learnt their chops in the classic theatrical era, the whole thing is animated with so much snap and verve and subtlety that I could tell was kinda lacking in the SHRs even when I was a kid in the 70s.

It is so awesome that I made a blog post full of screengrabs and me raving about how awesome it is.

WHY HAVE I NOT SEEN THIS ONE BEFORE. EVER.

(Little Twelvetoes looks to be by the same hand, but is nowhere near as beautifully-animated. I think I saw it once.)
posted by egypturnash at 12:34 AM on September 28, 2011


delmoi: well, then, why bother teaching the kids algebra instead of arithmetic, if they can get calculators to do both?
posted by alexei at 12:50 AM on September 28, 2011


Turns out that yes, Twelvetoes was drawn by the same hand as Sampson - one Rowland B. Wilson, who was a pretty awesome artist.

Anyway, back to talking about math I guess.
posted by egypturnash at 1:01 AM on September 28, 2011


Neat math grammar fact: When you say stuff like "three times five," the "times" doesn't function the same way as the operators in phrases like "three plus five" or "three minus five." You're really saying "five (three times)" -- it's just worded somewhat archaically. Like how you might say "He was three times married before the age of thirty," which doesn't mean "three multiplied by married," but "married (three times)."

I picked this up reading a passage in a book where some British kids were practicing their times tables and started with "one time one is one, two times one is two," etc. Because one (one time) is one! I've also heard the twos said as "twice three is six" and so on, as a rearrangement of "three twice is six."

egypturnash: " (Little Twelvetoes yt looks to be by the same hand, but is nowhere near as beautifully-animated. I think I saw it once.)"

It may not be as pretty, but elementary school me thought the lyrics were SO COOL. Just the idea of us having to invent two extra numbers -- dec (X) and el (ε), followed by doe ("10") -- just because we had two extra fingers, it was such an abstract, challenging, fundamental idea for something that was supposed to teach kids how to add up their twelves.
posted by Rhaomi at 1:12 AM on September 28, 2011


Neat math grammar fact: When you say stuff like "three times five," the "times" doesn't function the same way as the operators in phrases like "three plus five" or "three minus five." You're really saying "five (three times)" -- it's just worded somewhat archaically. Like how you might say "He was three times married before the age of thirty," which doesn't mean "three multiplied by married," but "married (three times)."

You could say the same thing about "plus". It means "more", so when you say "three plus five" you're really saying, "five more than three" -- it's just worded somewhat archaically. Like "times", "plus" can be used in a non-mathematical way ("...plus, I wasn't ready for it anyway..." where it means "what's more..."). But context matters. In mathematical phrases like "three times four" or "three plus four" they do take on the role of an operator.

I picked this up reading a passage in a book where some British kids were practicing their times tables and started with "one time one is one, two times one is two,"

This is not typical, which explains why it surprised you. When they are learning grammar, children often apply rules that they've learned to situations where they are not appropriate. In English (at least, American English; I can't speak for British English...) a fluent speaker of English would not say "one time one". But a child who hasn't learned the role of 'times' as an operator, and the corresponding exception to the rule about pluralizing 'times', might make that mistake.

It's also possible that along with that grammatical mistake came an understanding of what 'times' means, but that doesn't mean that "'times' doesn't function the same way as the operators in phrases like 'three plus five' or 'three minus five,'" because it does.
posted by Philosopher Dirtbike at 2:14 AM on September 28, 2011


Yeah, to be clear, I was just talking about the phrase "x times y" in basic arithmetic; there is of course a multiplication operator there that can function in much more complex ways. I just think it's interesting to point out where the "times" construction came from, especially since it makes it intuitively clear what's going on mathematically.

That's a big weakness of rote memorization of times tables -- without sufficient explanation of what multiplication is, it's easy for kids to fall into a lazier mode of thinking where you just memorize that this number combined with that number on the table produce this bigger number, for boring, dimly-understood reasons. It's simpler to drill in the idea of it meaning "this number that many times" when you realize that's what you're actually saying, in slightly more formal English. That gets obscured when you think of the word "times" solely as the name for that little "x" symbol.

(Also, I think the "one time x" thing is either a Britishism or an obsolete wording, but not a mistake. I've seen it a few times in reference to old schoolhouse-style drills; googling "one time two is two" pulls up multiple Google Books hits from the late 1800s. I guess it just fell out of general use, like the data/datum distinction.)
posted by Rhaomi at 3:19 AM on September 28, 2011


That's a big weakness of rote memorization of times tables -- without sufficient explanation of what multiplication is, it's easy for kids to fall into a lazier mode of thinking where you just memorize that this number combined with that number on the table produce this bigger number, for boring, dimly-understood reasons.

That's similar to Delmoi's objection above, but I don't know that it is a real problem; I'd have to be convinced with real data. It seems to me that memorization promotes basic fluency. I could, for instance, "understand" the rules of phonetics, and therefore read nearly anything (let's ignore the exceptions; they aren't important for my point). But if I rely on "understanding" the rules, and sound everything out every time I have to read, I'm going to be a very slow reader. In fact, I won't be able to read anything more complex than a sentence or so.

However, if I memorize the pronunciation of words then I can read much faster. The more words I memorize, the better I'll be able to read, and I can apprehend more complex texts. Imagine reading a novel or a textbook having to sound out words! Luckily, we read so much that memorization eventually happens automatically, but the same can't be said for times tables.

Without automatization, of certain things, "understanding" won't help. We are limited in our capacity to grasp multiple ideas at the same time. It seems reasonable to me that memorizing would help to make learning more complex ideas and seeing patterns easier, because recalling and putting together memorized facts is much easier than working it out every time. In my example above, imagine if it took 30 seconds to read and sound out a whole sentence. You may have "read" the sentence, but you will have trouble understanding it. By the time you got to the end, you'll have forgotten the beginning. With memorization, you can work around inherent human working memory capacity limits (see Go Banana's example above).
posted by Philosopher Dirtbike at 3:52 AM on September 28, 2011 [1 favorite]


When you get to about 3:40 in this video, there's a nice description of the bizarre number patterns seven can spin off.
posted by CaptainCaseous at 6:20 AM on September 28, 2011


Commutative property. As mentioned, you just gotta know 49.

For me, single-digit multiplication doesn't seem like memorizing facts. It's more like familiarizing yourself with a set of numbers that are used very often in our society, most importantly in employment. And single-digit multiplication opens up multiple-digit multiplication on paper (or in the head), which is also pretty critical.

I know we have portable computers, but I'm not sure how "math tables" (or "multiplier memorization," as I would call it) can be avoided. I want people to be able to calculate 127x36 when the lights go out. (Preferably in their heads, as it will be dark at night with no lights.)

the thing is, that's estimation not rote arithmetic. If you used arithmetic the way it is taught in school you would take out a piece of paper and do long division. Which I guarantee you you didn't do.

I had a pretty good (not great) math education, and we learned estimation. It wasn't a huge subject, but I remember spending time on it. e.g. 128 * 83 is 13*8*10 or

My mom attempted to teach me what she'd learnt. I know some of it but these things make her come across as a math savant.

I'm disappointed they use an estimate for Fahrenheit to Celsius. That conversion was always my favorite when I was a kid, just because it was hard to remember--(F-32) * 5/9 = C--and hard to calculate. (Quick, what's 33 * 5/9?)

When I was a teenager and applying for work, LOTS of employers would ask quick math questions, like "what's 14 x 16?"

I enjoyed all the primary school math nerd stories because I was certainly one of those. The times tables were a somewhat sensitive subject at my school, because lots of kids who weren't good at memorization had trouble with them, so we didn't have any bee-type competition, but I would have smoked all your asses, repeatedly.
posted by mrgrimm at 7:45 AM on September 28, 2011


13*8*10 or ... 1,040 (i think), lol
posted by mrgrimm at 8:16 AM on September 28, 2011


I'm disappointed they use an estimate for Fahrenheit to Celsius. That conversion was always my favorite when I was a kid, just because it was hard to remember--(F-32) * 5/9 = C--and hard to calculate. (Quick, what's 33 * 5/9?)

In practice I do Celsius to Fahrenheit and vice versa as follows:
1. I know that 0 C = 32 F, 10 C = 50 F, 20 C = 68 F, 30 C = 86 F, 40 C = 104 F.
2. from this it's easy to derive 5 C = 41 F, ..., 35 C = 95 F.
3. then basically just guess for anything in between those numbers. (I'd call it "interpolation" but that would imply I do some calculations, which I don't.) If pressed for a method I'd say that I just use 1/2 in place of 5/9, rounding up.

For the example implied in your comment: 65 F is three degrees below 68 F, so it must be 3/2 rounded up = 2 degrees below 20 C, i. e. 18 C.

To convert back, 18 C is two degrees below 20 C, so it must be twice two = four degrees below 68 F, i. e. 64 F. (The conversion isn't one-to-one, because Celsius degrees are bigger than Fahrenheit degrees.)

(Exercise for the reader: prove this always works. Before you do that, define "this".)
posted by madcaptenor at 9:07 AM on September 28, 2011


Do you also know the one value that is the same in both F and C? (I figured it out as a kid. Such a delight.)
posted by grubi at 9:26 AM on September 28, 2011


-40. (I used to live in Edmonton.)
posted by Sys Rq at 9:42 AM on September 28, 2011


You discovered it the hard way. Wow.
posted by grubi at 9:59 AM on September 28, 2011 [1 favorite]


Not really. They cancelled school when it got that cold, 'cause the buses wouldn't start.

And that's why I never learned my 12 times tables.
posted by Sys Rq at 10:26 AM on September 28, 2011 [1 favorite]


OH SEE NOW

grr
posted by grubi at 11:56 AM on September 28, 2011


I'm still waiting to learn about Go Banana's pair of numbers such that one is "greater than AND equal" to the other.
posted by Pyry at 1:20 PM on September 28, 2011 [1 favorite]


What, we have gotten this far without mentioning that the proper way to write five plus three is:
5 3 +
posted by eriko at 3:55 PM on September 28, 2011


I'm still waiting to learn about Go Banana's pair of numbers such that one is "greater than AND equal" to the other.

Me too.
posted by Crabby Appleton at 3:59 PM on September 28, 2011


No, eriko, you heretic, it's (+ 3 5). Let's have a religious war!
posted by Crabby Appleton at 7:38 PM on September 28, 2011 [1 favorite]


α α  α  α  α  α  α  α  α  α  α  α  α  α  α  α  α  α  α  α  α  α  α  α
α β  γ  δ  ε  ζ  η  θ  ι  κ  λ  μ  ν  ξ  ο  π  ρ  σ  τ  υ  φ  χ  ψ  ω
α γ  ε  η  ι  λ  ν  ο  ρ  τ  φ  ψ βα βγ βε βη βι βλ βν βο βρ βτ βφ βψ
α δ  η  κ  ν  π  τ  χ βα βδ βη βκ βν βπ βτ βχ γα γδ γη γκ γν γπ γτ γχ
α ε  ι  ν  ρ  φ βα βε βι βν βρ βφ γα γε γι γν γρ γφ δα δε δι δν δρ δφ
α ζ  λ  π  φ ββ βη βμ βρ βχ γγ γθ γν γσ γψ δδ δι δξ δτ δω εε εκ εο ευ
α η  ν  τ βα βη βν βτ γα γη γν γτ δα δη δν δτ εα εη εν ετ ζα ζη ζν ζτ
α θ  ο  χ βε βμ βτ γβ γι γπ γψ δζ δν δυ εγ εκ ερ εω ζη ζξ ζφ ηδ ηλ ησ
α ι  ρ βα βι βρ γα γι γρ δα δι δρ εα ει ερ ζα ζι ζρ ηα ηι ηρ θα θι θρ
α κ  τ βδ βν βχ γη γπ δα δκ δτ εδ εν εχ ζη ζπ ηα ηκ ητ θδ θν θχ ιη ιπ
α λ  φ βη βρ γγ γν γψ δι δτ εε εο ζα ζλ ζφ ηη ηρ θγ θν θψ ιι ιτ κε κο
α μ  ψ βκ βφ γθ γτ δζ δρ εδ εο ζβ ζν ζω ηλ ηχ θι θυ ιη ισ κε κπ λγ λξ
α ν βα βν γα γν δα δν εα εν ζα ζν ηα ην θα θν ια ιν κα κν λα λν μα μν
α ξ βγ βπ γε γσ δη δυ ει εχ ζλ ζω ην θβ θο ιδ ιρ κζ κτ λθ λφ μκ μψ νμ
α ο βε βτ γι γψ δν εγ ερ ζη ζφ ηλ θα θο ιε ιτ κι κψ λν μγ μρ νη νφ ξλ
α π βη βχ γν δδ δτ εκ ζα ζπ ηη ηχ θν ιδ ιτ κκ λα λπ μη μχ νν ξδ ξτ οκ
α ρ βι γα γρ δι εα ερ ζι ηα ηρ θι ια ιρ κι λα λρ μι να νρ ξι οα ορ πι
α σ βλ γδ γφ δξ εη εω ζρ ηκ θγ θυ ιν κζ κψ λπ μι νβ ντ ξμ οε οχ πο ρθ
α τ βν γη δα δτ εν ζη ηα ητ θν ιη κα κτ λν μη να ντ ξν οη πα πτ ρν ση
α υ βο γκ δε δω ετ ζξ ηι θδ θψ ισ κν λθ μγ μχ νρ ξμ οη πβ πφ ρπ σλ τζ
α φ βρ γν δι εε ζα ζφ ηρ θν ιι κε λα λφ μρ νν ξι οε πα πφ ρρ σν τι υε
α χ βτ γπ δν εκ ζη ηδ θα θχ ιτ κπ λν μκ νη ξδ οα οχ πτ ρπ σν τκ υη φδ
α ψ βφ γτ δρ εο ζν ηλ θι ιη κε λγ μα μψ νφ ξτ ορ πο ρν σλ τι υη φε χγ
α ω βψ γχ δφ ευ ζτ ησ θρ ιπ κο λξ μν νμ ξλ οκ πι ρθ ση τζ υε φδ χγ ψβ
What?
posted by wobh at 5:09 PM on September 30, 2011 [2 favorites]


Beautiful, wobh. I really like that you can see the hyperbolas at around x*y = ι, x*y = ια, and x*y = τα.

Though I wonder where you got this, as it seems you're sometimes assigning values 1 .. 24, and sometimes 1 .. 9, 10 .. 90, 100 .. 900.
posted by benito.strauss at 7:05 AM on October 1, 2011


I'm sorry to come late to the party!

I think it might have helped me a lot to have also been made to study at a division table somewhen during my childhood math education. Here's one:
| 9 | 9.0 | 4.5 | 3.000000 | 2.25 | 1.8 | 1.500000 | 1.285714 | 1.125 | 0.999999 |
| 8 | 8.0 | 4.0 | 2.666666 | 2.00 | 1.6 | 1.333333 | 1.142857 | 1.000 | 0.888888 |
| 7 | 7.0 | 3.5 | 2.333333 | 1.75 | 1.4 | 1.166666 | 1.000000 | 0.875 | 0.777777 |
| 6 | 6.0 | 3.0 | 2.000000 | 1.50 | 1.2 | 1.000000 | 0.857142 | 0.750 | 0.666666 |
| 5 | 5.0 | 2.5 | 1.666666 | 1.25 | 1.0 | 0.833333 | 0.714285 | 0.625 | 0.555555 |
| 4 | 4.0 | 2.0 | 1.333333 | 1.00 | 0.8 | 0.666666 | 0.571428 | 0.500 | 0.444444 |
| 3 | 3.0 | 1.5 | 1.000000 | 0.75 | 0.6 | 0.500000 | 0.428571 | 0.375 | 0.333333 |
| 2 | 2.0 | 1.0 | 0.666666 | 0.50 | 0.4 | 0.333333 | 0.285714 | 0.250 | 0.222222 |
| 1 | 1.0 | 0.5 | 0.333333 | 0.25 | 0.2 | 0.166666 | 0.142857 | 0.125 | 0.111111 |
|---+-----+-----+----------+------+-----+----------+----------+-------+----------|
| ÷ |   1 |   2 |        3 |    4 |   5 |        6 |        7 |     8 |        9 |
I find the division table simpler than the multiplication table in most respects, and it illustrates many useful patterns that I didn't figure out until much later. On the other hand, you can see that 7 is extra tricky here as well.

It wouldn't be a math party without exercises. Here's two:

1. The repeating decimal patterns of n/7 represent 6 of 720 possible permutations of those digits. Why 6? Why those 6? Where are the others?

2. Generate a similar division table corresponding to the base-24 greek-numeral multiplication table given in my earlier comment.
posted by wobh at 10:14 AM on October 1, 2011 [3 favorites]


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