The Joy of X: A guided tour of math, from one to infinity by Steven Strogatz. Copyright ©2012 Steven Strogatz. Reprinted with permission from Houghton Mifflin Harcourt." data-pin-do="buttonBookmark">
The Joy of X: A guided tour of math, from one to infinity by Steven Strogatz. Copyright ©2012 Steven Strogatz. Reprinted with permission from Houghton Mifflin Harcourt.
Image: Houghton Mifflin Harcourt
Reprinted from The Joy of X: A guided tour of math, from one to infinity by Steven Strogatz. Copyright ©2012 Steven Strogatz. Reprinted with permission from Houghton Mifflin Harcourt.
Every decade or so a new approach to teaching math comes along and creates fresh opportunities for parents to feel inadequate. Back in the 1960s, my parents were flabbergasted by their inability to help me with my second-grade homework. They’d never heard of base 3 or Venn diagrams.
Now the tables have turned. “Dad, can you show me how to do these multiplication problems?” Sure, I thought, until the headshaking began. “No, Dad, that’s not how we’re supposed to do it. That’s the old-school method. Don’t you know the lattice method? No? Well, what about partial products?”
These humbling sessions have prompted me to revisit multiplication from scratch. And it’s actually quite subtle, once you start to think about it.
Take the terminology. Does “seven times three” mean “seven added to itself three times”? Or “three added to itself seven times”? In some cultures the language is less ambiguous. A friend of mine from Belize used to recite his times tables like this: “Seven ones are seven, seven twos are fourteen, seven threes are twenty-one,” and so on. This phrasing makes it clear that the first number is the multiplier; the second number is the thing being multiplied. It’s the same convention as in Lionel Richie’s immortal lyrics “She’s once, twice, three times a lady.” (“She’s a lady times three” would never have been a hit.)
Maybe all this semantic fuss strikes you as silly, since the order in which numbers are multiplied doesn’t matter anyway: 7 × 3 = 3 × 7. Fair enough, but that begs the question I’d like to explore in some depth here: Is this commutative law of multiplication, a × b = b × a, really so obvious? I remember being surprised by it as a child; maybe you were too.
To recapture the magic, imagine not knowing what 7 × 3 equals. So you try counting by sevens: 7, 14, 21. Now turn it around and count by threes instead: 3, 6, 9, . . . Do you feel the suspense building? So far, none of the numbers match those in the sevens list, but keep going . . . 12, 15, 18, and then, bingo, 21!
My point is that if you regard multiplication as being synonymous with repeated counting by a certain number (or, in other words, with repeated addition), the commutative law isn’t transparent.
But it becomes more intuitive if you conceive of multiplication visually. Think of 7 × 3 as the number of dots in a rectangular array with seven rows and three columns.
If you turn the array on its side, it transforms into three rows and seven columns—and since rotating the picture doesn’t change the number of dots, it must be true that 7 × 3 = 3 × 7.
Yet strangely enough, in many real-world situations, especially where money is concerned, people seem to forget the commutative law, or don’t realize it applies. Let me give you two examples.
Suppose you’re shopping for a new pair of jeans. They’re on sale for 20 percent off the sticker price of $50, which sounds like a bargain, but keep in mind that you also have to pay the 8 percent sales tax. After the clerk finishes complimenting you on the flattering fit, she starts ringing up the purchase but then pauses and whispers, in a conspiratorial tone, “Hey, let me save you some money. I’ll apply the tax first, and then take twenty percent off the total, so you’ll get more money back. Okay?”
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9 Comments
Add CommentSo the second law of thermodynamics does not apply to order of operations in arithmetic. I think it is a good point that we should be clear that x * y implies y groups of x items. We certainly didn't cover it in my school.
Reply | Report Abuse | Link to thisWhile commutative property works well in some ecionomic transactions, such as applying sales tax before or after discount is applied (which Srogatz adduces as an example), in other cases commutativity doesn't work, like the second law of thermodnamics (to which Strogatz adumbrates), which itself cannot be proved "deductively," but only with statistical maximum likelihhod theory. This must be one reason why MIT's Nobel-laureate economist brought thermodynamics into economic theory in his pioneering Ph.D. thesis ["Foundations of Economic Analysis"]at Harvard. Other great economists like Solow or NYT's Paul Krugman warn us about robotic,arid mathematical explanations of economic ideas. We cannot always slavishly apply commutativity to economic constructs. Strogatz's crisp investigation gives a good insight
Reply | Report Abuse | Link to thisI dislike using 'beg' that way. To beg the question means to avoid answering a question already raised. It's 'raise' the question blah blah.
Reply | Report Abuse | Link to this(I know, it's in wide usage that way, but it's fingernails on the blackboard for me.)
The major reason why the discount before tax is superior to the tax before discount, is not related to the amount paid by the customer, but rather the amount of tax going to the government coffers. If you wish to optimise both your own out of pocket expenses and reduction in the harm that all governments do with their funding, then you should always insist on the discount before that tax. Note that this also means less overall cost for the vendor, which is why the vendor should also insist on this method.
Reply | Report Abuse | Link to thisAfter I read "1984" which I love to reference I had to go back to this very thing itself. I put the book down and thought "I can just use my fingers to add but what about multiplying?" and then I thought about 2 groups of 4. And then to add them all up. Sometimes we take our knowledge for granted!
Reply | Report Abuse | Link to thisYou didn't have to go so exotic to find a place where "seven threes are twenty one". This is how we recited it in the good ol' UK 60 years ago.
Reply | Report Abuse | Link to thisAnd you also didn't have to go quite so exotic with the example from quantum mechanics, because vector algebra is not always commutative. We also had that in pre-college maths in those days.
@phalaris
Reply | Report Abuse | Link to thisGood remembering! In Ontario Canada, I experienced the same two items that you reported also about 60 years ago. IMO, the new ways of teach arithmetic and later beginning math are not better than those old ways, which I enjoyed so much that I became a math prof for a while in my life.
In addition, the Montessori method uses physical objects to see constructs like that dot pictures above in order to understand better what is going on.
With respect to the Traditional vs Roth IRAs example, the reader should be careful. The commutative law applies in considering whether to convert a Traditional to a Roth, but not whether to invest pre-tax income. In the latter case, the investor earns an additional return on the tax that is saved [minus the tax on that return] by selecting the Traditional.
Reply | Report Abuse | Link to thisOther factors can be critical. In deciding whether to convert from a Traditional to a Roth, a [I believe it is the] major consideration is that, if the investment in the Roth turns out badly, it can be recharacterized and the tax paid on the conversion can be recouped. The option to recharacterize creates a win / win situation in many common circumstances.
While questioning the utility of the standard definition of multiplication, this article appears to accept its accuracy. As reflected by the usage in the article, the standard definition of multiplication is usually something to the effect that multiplication is the process of adding a number to itself a certain number of times, as in the references collected in Section C.2 of the Times Higher/Times Greater subpage of the Vignettes page of jpscanlan.com:
Reply | Report Abuse | Link to thishttp://www.jpscanlan.com/vignettes/timeshigherissues.html
As explained in that section, however, the definition is plainly incorrect. Multiplying a times b involves either (1) adding a to zero b times or (2) adding a to itself b-1 times.
The interesting thing reflected by the questions posed in this article – “Does ‘seven times three’ mean ‘seven added to itself three times’? Or ‘three added to itself seven times’?” – is that the two processes in fact yield different results. The former yields 28; the latter yields 24. Neither, however, is the correct result of multiplying either 7 by 3 or 3 by 7.
Put another way, with respect to the observation “Is this commutative law of multiplication, a × b = b × a, really so obvious?,” it could be said that, whether obvious or not, according to the standard definition of multiplication it is incorrect. According to that definition, a x b = ab+a, while b x a equals ab+b.