# Commuting: Steven Strogatz Explains One of the Laws of Multiplication [Excerpt]

Steven Strogatz discusses the mathematical difference between IRA contributions and footwear in this excerpt from his new book

But something about that sounds fishy to you. “No thanks,” you say. “Could you please take the twenty percent off first, then apply the tax to the sale price? That way, I’ll pay less tax.”

Which way is a better deal for you? (Assume both are legal.) When confronted with a question like this, many people approach it additively. They work out the tax and the discount under both scenarios, and then do whatever additions or subtractions are necessary to find the final price. Doing things the clerk’s way, you reason, would cost you \$4 in tax (8 percent of the sticker price of \$50). That would bring your total to \$54. Then applying the 20 percent discount to \$54 gives you \$10.80 back, so you’d end up paying \$54 minus \$10.80, which equals \$43.20. Whereas under your scenario, the 20 percent discount would be applied first, saving you \$10 off the \$50 sticker price. Then the 8 percent tax on that reduced price of \$40 would be \$3.20, so you’d still end up paying \$43.20. Amazing!

But it’s merely the commutative law in action. To see why, think multiplicatively, not additively. Applying an 8 percent tax followed by a 20 percent discount amounts to multiplying the sticker price by 1.08 and then multiplying that result by 0.80. Switching the order of tax and discount reverses the multiplication, but since 1.08 × 0.80 = 0.80 × 1.08, the final price comes out the same.

Considerations like these also arise in larger financial decisions. Is a Roth 401(k) better or worse than a traditional retirement plan? More generally, if you have a pile of money to invest and you have to pay taxes on it at some point, is it better to take the tax bite at the beginning of the investment period, or at the end?

Once again, the commutative law shows it’s a wash, all other things being equal (which, sadly, they often aren’t). If, for both scenarios, your money grows by the same factor and gets taxed at the same rate, it doesn’t matter whether you pay the taxes up front or at the end.

Please don’t mistake this mathematical remark for financial advice. Anyone facing these decisions in real life needs to be aware of many complications that muddy the waters: Do you expect to be in a higher or lower tax bracket when you retire? Will you max out your contribution limits? Do you think the government will change its policies about the tax-exempt status of withdrawals by the time you’re ready to take the money out? Leaving all this aside (and don’t get me wrong, it’s all important; I’m just trying to focus here on a simpler mathematical issue), my basic point is that the commutative law is relevant to the analysis of such decisions.

You can find heated debates about this on personal finance sites on the Internet. Even after the relevance of the commutative law has been pointed out, some bloggers don’t accept it. It’s that counterintuitive.

Maybe we’re wired to doubt the commutative law because in daily life, it usually matters what you do first. You can’t have your cake and eat it too. And when taking off your shoes and socks, you’ve got to get the sequencing right.

The physicist Murray Gell-Mann came to a similar realization one day when he was worrying about his future. As an undergraduate at Yale, he desperately wanted to stay in the Ivy League for graduate school. Unfortunately Princeton rejected his application. Harvard said yes but seemed to be dragging its feet about providing the financial support he needed. His best option, though he found it depressing, was MIT. In Gell-Mann’s eyes, MIT was a grubby technological institute, beneath his rarefied taste. Nevertheless, he accepted the offer. Years later he would explain that he had contemplated suicide at the time but decided against it once he realized that attending MIT and killing himself didn’t commute. He could always go to MIT and commit suicide later if he had to, but not the other way around.

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1. 1. RSchmidt 10:32 AM 10/5/12

So 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.

2. 2. eklimur.raza 04:31 PM 10/5/12

While 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

3. 3. g-rowley 05:54 PM 10/5/12

I dislike using 'beg' that way. To beg the question means to avoid answering a question already raised. It's 'raise' the question blah blah.

(I know, it's in wide usage that way, but it's fingernails on the blackboard for me.)

4. 4. paulwakfer 06:23 PM 10/5/12

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.

5. 5. MBarksdale 05:57 PM 10/7/12

After 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!

6. 6. phalaris 05:50 AM 10/8/12

You 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.

And 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.

7. 7. paulwakfer 11:48 AM 10/8/12

@phalaris

Good 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.

8. 8. shorewood 04:41 PM 10/13/12

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.

Other 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.

9. 9. jscanlan 03:30 PM 3/7/13

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:
http://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.

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