March 12, 2013 1:27 PM Subscribe

The origins of plus and minus signs - "There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made – and betokeneth lesse."

posted by spbmp (30 comments total) 31 users marked this as a favorite

posted by spbmp (30 comments total) 31 users marked this as a favorite

The etymology of "sine" is pretty interesting as well.

posted by griphus at 1:47 PM on March 12, 2013 [6 favorites]

posted by griphus at 1:47 PM on March 12, 2013 [6 favorites]

Fascinating read. It's interesting that today we use many symbols for multiplication; 'X', '*', '•', and the most confusing to future symbolic historians, simple proximity; ab.

posted by Mei's lost sandal at 1:52 PM on March 12, 2013 [2 favorites]

posted by Mei's lost sandal at 1:52 PM on March 12, 2013 [2 favorites]

Is this where I can submit my pet peeve that + and - can be used as both unary and binary operators (giving them very slightly different meanings depending on the context)?

Mathematics would be just a tiny bit simpler and less ambiguous if we used a separate set of symbols to indicate positive/negative (as opposed to addition/subtraction).

posted by schmod at 1:53 PM on March 12, 2013 [2 favorites]

Mathematics would be just a tiny bit simpler and less ambiguous if we used a separate set of symbols to indicate positive/negative (as opposed to addition/subtraction).

posted by schmod at 1:53 PM on March 12, 2013 [2 favorites]

Finding out just what the heck all those symbols mean is one of the things that drew me in to math. "∂" is a thing of beauty.

But even I get a little intimidated by all the operators that Perl 6 is contemplating. Check out the Periodic Table of Perl Operators, if you're not squeamish.

Mind you, it's not as bad as APL, whose Wikipedia page has its very own custom warning: "This article contains APL source code. Without proper rendering support, you may see question marks, boxes, or other symbols instead of APL symbols.". Here, test your browser with this snippet of valid APL code:

posted by benito.strauss at 2:11 PM on March 12, 2013 [1 favorite]

But even I get a little intimidated by all the operators that Perl 6 is contemplating. Check out the Periodic Table of Perl Operators, if you're not squeamish.

Mind you, it's not as bad as APL, whose Wikipedia page has its very own custom warning: "This article contains APL source code. Without proper rendering support, you may see question marks, boxes, or other symbols instead of APL symbols.". Here, test your browser with this snippet of valid APL code:

life←{↑1 ⍵∨.∧3 4=+/,¯1 0 1∘.⊖¯1 0 1∘.⌽⊂⍵}

posted by benito.strauss at 2:11 PM on March 12, 2013 [1 favorite]

I feel pretty safe in asserting that Perl 6 will never be a major force in computing. Perl 5 will probably exist forever, but its time in the sun has passed, and Perl 6, even if it ever actually is finished, which is quite questionable at this point, has missed the window for wide acceptance.

Python and Ruby are probably the dominant interpreted languages now, though considering the trouble Python is having with its 2.X->3.X transition, I suspect Ruby may eventually eclipse its popularity.

posted by Malor at 2:19 PM on March 12, 2013 [1 favorite]

Order of operations. Today I learned, not once but twice, that ')' is a synonym for '÷'.

posted by fantabulous timewaster at 2:21 PM on March 12, 2013 [1 favorite]

posted by fantabulous timewaster at 2:21 PM on March 12, 2013 [1 favorite]

I've got an old engineering handbook from the 1840s which still felt it necessary to provide an explanation (and justification) of the basic arithmetical operators.

posted by Segundus at 2:31 PM on March 12, 2013

posted by Segundus at 2:31 PM on March 12, 2013

It what context would it be less ambiguous?

posted by milestogo at 2:33 PM on March 12, 2013

milestogo: "*It what context would it be less ambiguous?*"

one example of the inconsistencies that arise:

a = 5

b = -6

ab = -30

5-6 = ?

posted by idiopath at 2:35 PM on March 12, 2013 [1 favorite]

one example of the inconsistencies that arise:

a = 5

b = -6

ab = -30

5-6 = ?

posted by idiopath at 2:35 PM on March 12, 2013 [1 favorite]

Oh, I see, you've "replaced" each symbol with its value. So if one thinks this is always permitted, then you can have ambiguity, got it.

Of course, this isn't always permitted, but it does take some experience to know this.

posted by milestogo at 2:38 PM on March 12, 2013

Of course, this isn't always permitted, but it does take some experience to know this.

posted by milestogo at 2:38 PM on March 12, 2013

That doesn't arise from plus/minus positive/negative, it arises from variable notation.

a = 5

b = 6

ab = a*b = 30?

or

ab = 56?

posted by muddgirl at 3:11 PM on March 12, 2013 [2 favorites]

You have to be careful, since '))' is an emoticon for mooning someone. Won't want that in your mathematics treatise.

posted by GenjiandProust at 3:18 PM on March 12, 2013

I really need to use 'betokeneth' more in my life.

posted by estuardo at 3:45 PM on March 12, 2013 [6 favorites]

posted by estuardo at 3:45 PM on March 12, 2013 [6 favorites]

I guess the ambiguity in 5-6 is resolved with 5(-6), but it is not intuitive.

posted by Mei's lost sandal at 3:48 PM on March 12, 2013

posted by Mei's lost sandal at 3:48 PM on March 12, 2013

posted by Mei's lost sandal at 3:48 PM on March 12 [+] [!]

Of course it's not intuitive, but it is standard.

Consider the case

a=5

b=x-7

then the standard substitution is

ab=5(x-7).

when substituting, always add parentheses, because you are substituting values, so the grouping must be maintained. In fact, strictly speaking, the correct method is:

ab=(5)(x-7) followed by a step that drops the redundant parentheses.

Similarly, for idiopath's problem, the correct substitution is

a=5

b=-6

ab=(5)(-6)

posted by yeolcoatl at 3:54 PM on March 12, 2013

When I taught basic math at the community college, I split negative numbers and subtraction into separate lessons. A short dash was used to make the opposite of a number (-6 is the additive opposite of 6) and subtraction was with an em dash - nice and looooong. My reason: so that I could physically demonstrate why subtracting a negative number is like adding its opposite.

I'd have the students imagine a number line stretching across the front of the classroom, positive to their right, negative to their left. Start facing positive ("Stay positive, folks!") on zero.

A positive number means, take that many steps. So "2" means, you take two steps and wind up on the 2 of the number line. I'd do the walking up front.

A negative number means, take that many steps, backwards. So "-5" means walk backwards and wind up at negative five.

Addition of two numbers means, do the first number, then do the second number. 2+3 means take two steps, so you're at 2; then take three steps, and you're at five. At this point students would start getting antsy, does this kid (I was 28 but looked 18) think we're dumb?

Ok, so 2+ (-3). Take two steps, then take three steps backwards and wind up at negative one. (-3)+(-4)? I'd wind up at -7. Mind you, I'd do three or four of each type of addition, ask questions, etc. Mutiny was nigh.

Subtraction! OK, subtraction means do the first number then turn around and do the second number. 7-2, take seven steps, don't run into the wall, turn around, walk two steps. Bam, I'm standing at 5. Three more examples. Then some examples like 1 - 4; take a step, then turn around and take four steps; winding up at -3. Ok, people would quiet down here a little; maybe they started to understand a little.

OK - subtracting a negative number. Does the class want to take two minutes to clear your head? No? OK! You already know the steps. 3- (-2). WHat do you think the answer is? One, negative one, two (?), five. Let's see what happens. Start at zero, I'm facing positive. Take three steps. I'm on the three. Turn around, since I'm subtracting. What's next? Two steps! Which way? *Backwards*! I"m at 5! So subtracting a negative number is like turning around and walking backwards!

If you're in community college but are taking Basic Math, the education system has failed you. Maybe you were a jerk that teachers passed so they wouldn't have to see you. Maybe you were just a little too timid to ask "why" in third grade, and you never caught back up; hell you never made forward progress again, for want of a single answered question. Most of the students really did want to succeed but didn't have high hopes for themselves because they'd never grokked math in any way before. To see the light come on in so many eyes - Hey that weird teacher just did a ridiculous shimmy and walk backwards to explain math, and I GOT IT! - was one of the most rewarding moments of my life.

posted by notsnot at 4:23 PM on March 12, 2013 [23 favorites]

I'd have the students imagine a number line stretching across the front of the classroom, positive to their right, negative to their left. Start facing positive ("Stay positive, folks!") on zero.

A positive number means, take that many steps. So "2" means, you take two steps and wind up on the 2 of the number line. I'd do the walking up front.

A negative number means, take that many steps, backwards. So "-5" means walk backwards and wind up at negative five.

Addition of two numbers means, do the first number, then do the second number. 2+3 means take two steps, so you're at 2; then take three steps, and you're at five. At this point students would start getting antsy, does this kid (I was 28 but looked 18) think we're dumb?

Ok, so 2+ (-3). Take two steps, then take three steps backwards and wind up at negative one. (-3)+(-4)? I'd wind up at -7. Mind you, I'd do three or four of each type of addition, ask questions, etc. Mutiny was nigh.

Subtraction! OK, subtraction means do the first number then turn around and do the second number. 7-2, take seven steps, don't run into the wall, turn around, walk two steps. Bam, I'm standing at 5. Three more examples. Then some examples like 1 - 4; take a step, then turn around and take four steps; winding up at -3. Ok, people would quiet down here a little; maybe they started to understand a little.

OK - subtracting a negative number. Does the class want to take two minutes to clear your head? No? OK! You already know the steps. 3- (-2). WHat do you think the answer is? One, negative one, two (?), five. Let's see what happens. Start at zero, I'm facing positive. Take three steps. I'm on the three. Turn around, since I'm subtracting. What's next? Two steps! Which way? *Backwards*! I"m at 5! So subtracting a negative number is like turning around and walking backwards!

If you're in community college but are taking Basic Math, the education system has failed you. Maybe you were a jerk that teachers passed so they wouldn't have to see you. Maybe you were just a little too timid to ask "why" in third grade, and you never caught back up; hell you never made forward progress again, for want of a single answered question. Most of the students really did want to succeed but didn't have high hopes for themselves because they'd never grokked math in any way before. To see the light come on in so many eyes - Hey that weird teacher just did a ridiculous shimmy and walk backwards to explain math, and I GOT IT! - was one of the most rewarding moments of my life.

posted by notsnot at 4:23 PM on March 12, 2013 [23 favorites]

Clearly the only solution is to use Reverse Polish Notation for all calculations.

posted by Elementary Penguin at 4:54 PM on March 12, 2013 [5 favorites]

posted by Elementary Penguin at 4:54 PM on March 12, 2013 [5 favorites]

I believe you were told there would be zero math.

posted by fantabulous timewaster at 4:55 PM on March 12, 2013 [2 favorites]

posted by fantabulous timewaster at 4:55 PM on March 12, 2013 [2 favorites]

- betokeneth "Favorite added!"

posted by bleep at 4:57 PM on March 12, 2013 [3 favorites]

posted by bleep at 4:57 PM on March 12, 2013 [3 favorites]

In linear algebra, × and • are two different operations. The asterisk was really only used in computer science for multiplication because there's no • button on the keyboard, and x was used for variables, etc..

posted by empath at 5:12 PM on March 12, 2013 [1 favorite]

notnot, I did something similar with explaining why multiplying a negative by a negative is a positive. I used a digital camera (I love technology) to film myself walking both forwards and backwards for positive and negative on the first number. Then I ran it forwards or backwards for positive or negative on the second number. Walking backwards in reverse looks more or less like walking forwards.

I like your method better though.

posted by Hactar at 5:40 PM on March 12, 2013

I like your method better though.

posted by Hactar at 5:40 PM on March 12, 2013

Yeah, multiplication is harder. Just getting addition and subtraction is a hell of a step. Negative numbers are like debts so if you owe four people five bucks then you owe 20. And since we already know 4x5 is the same as 5x4, the same works with negative numbers (at this level you don't have to prove things) so 4x(-5) is -20 and (-5)x4 is the same.

As for doing it on camera or in person, for my part I think my students needed to have something humorous - me running around the front of class - to cement it in my brain.

posted by notsnot at 6:34 PM on March 12, 2013

As for doing it on camera or in person, for my part I think my students needed to have something humorous - me running around the front of class - to cement it in my brain.

posted by notsnot at 6:34 PM on March 12, 2013

A friend told me that the German manuscript discussed in the blog ("The first use of the modern algebraic sign") is now thought to have been written by Regiomontanus.

posted by spbmp at 7:04 PM on March 12, 2013

posted by spbmp at 7:04 PM on March 12, 2013

It seems like constructing the rules for multiplication and addition with really basic group theory would actually not be that hard to do for a remedial class. Just kind of explaining the concept of an identity and an inverse, etc, and going from there.

posted by empath at 7:19 PM on March 12, 2013

posted by empath at 7:19 PM on March 12, 2013

I’ve long had a strange fondness for Robert Recorde’s explanation of the equals sign (“I will ſette as I doe often in woorke vſe, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicauſe noe .2. thynges, can be moare equalle”). Mainly because archaic English always looks to me like it should be pronounced with elongated and exaggerated diction like the backwards-spoken Black Lodge scenes from Twin Peaks.

posted by mubba at 6:54 PM on March 13, 2013

posted by mubba at 6:54 PM on March 13, 2013

Oddly, that was what I thought of too, and is also one of my favorite phrases: "no two things can be more equal". The oddball spelling and stuttering punctuation only adds to the charm.

posted by benito.strauss at 7:04 PM on March 13, 2013

posted by benito.strauss at 7:04 PM on March 13, 2013

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