My class is mostly are in business and politics and most have made the choice never to look at math again. Is there still any chance I can motivate exact notions like $\sqrt{2}$ in a manner that is relevant to them?

One counterargument might be that we can use the decimal system and we usually only need 2 points after the decimal anyway...

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    $\begingroup$ two points after the decimal? Ask your students to calculate the difference in earning a money manager will make charging 1% vs 1.0001% when she has $2B under management. Resulting numbers are to the penny of course, but interim calculations have far more significant digits than 2 beyond the decimal. $\endgroup$ Jun 26, 2014 at 14:58
  • $\begingroup$ @Joe "My calculator says..." $\endgroup$
    – Ryan Reich
    Jun 26, 2014 at 15:15
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    $\begingroup$ @JoeTaxpayer, that's not a persuasive example. People don't actually charge 100.01 bps in management fees, and for good reason. If someone tried to charge me that amount, I would wonder what other non-standard features they had inserted in the contract, and I wouldn't trust them. $\endgroup$
    – user173
    Jun 26, 2014 at 18:09
  • $\begingroup$ Ok. This comment is an exaggeration. To show that with large numbers, it's important to keep the decimals well beyond 2. Forget fees, how about 2 year's returns? Do you want your manager to drop beyond the second decimal each year? $\endgroup$ Jun 26, 2014 at 20:01
  • $\begingroup$ @MattF.: I didn't get it. Your "good reason" for not doing it is that noone is doing it? $\endgroup$
    – Taladris
    Jun 27, 2014 at 6:10

6 Answers 6


There seem to be two different issues mixed together in your question, and I think they have different answers.

The first issue is:

Is there a way to motivate the concept of square roots, for example, $\sqrt{2}$?

As the other answerers have pointed out, square roots do arise quite naturally in many financial applications. If some quantity (say, an investment account balance) grows in successive periods of time by $r_1\%, r_2\%, \ldots r_n\%$ then the average rate of growth can be computed by defining the growth factors $k_i=1+\frac{r_i}{100}$, computing their geometric mean $k=\sqrt[n]{k_1 k_2 k_3 \cdots k_n}$, and interpreting $k-1$ as a percent. (This also works if some of the $r_i$ are negative, i.e. in the case of loss rather than growth.)

The second issue that I think seems to be implied by your question is:

Is there a way to motivate the concept of irrationality, and in particular the virtue of using a symbolic representation of an exact quantity like $\sqrt{2}$ rather than a rational approximation?

Here I think you are on shakier ground. I think the idea of irrationality -- and in particular, proving that $\sqrt{2}$ cannot be expressed as a ratio of two integers -- is one of the greatest and most beautiful discoveries of pure logic in human history, and I think it's a crying shame that it is not known to all educated people. But even I have to concede that for practical purposes, there is no real harm in using (let's say) a 12-digit approximation to $\sqrt{2}$ rather than its true value. I do think JoeTaxpayer's comment above regarding precision beyond the 2nd decimal place is definitely on-target: Students should see that truncating everything to the nearest penny at each stage of a calculation results in an error that accumulates dramatically over time. (See also: the plots of the movies Superman III and Office Space.) And this could, I suppose, be used to justify as a heuristic that it is better to work with exact expressions like $\sqrt{2}$ throughout a calculation and only produce an approximate value at the very last stage. But to be honest this is a little bit untrue, because (as I noted) using a 12-digit decimal approximation is going to produce a final result almost certainly every bit as reliable as what you would get if you resisted using approximations until the very last moment.

Edited to add: Can't believe that neither I nor anybody else has mentioned it yet, but one can hardly discuss compound interest without talking about continuous compounding, which naturally introduces the irrational number $e$ into the conversation. But here again the issues raised above apply: While the relevance of the irrational number is undeniable, the fact that it is irrational seems more like a curiosity, and it is hard to argue with a straight face that there is some real harm in using a rational approximation, as long as that approximation is "good enough". Here too there is probably some good substantive discussion to be had about how many digits is "enough" to get reliable results; 2 digits is definitely not enough, 8 is probably more than enough, and rounding off too much at intermediate stages of a calculation produces an error that can accumulate substantially by the time you reach the end.

  • $\begingroup$ The second issue is very shaky. Let's assume, your executives have launched a campaign "With us, you can double your money in two years! Yearly access to your money!" and you as a business mathematician have to model it in your computer, so that the computer can compute the yearly profit. You can philosphy about the irrationality of $\sqrt{2}$ as much as you want. The software you have to use might even support a sqrt function. I'll bet my yearly profit, that it doesn't compute symbolically. It will compute $\sqrt{2}$ to it's internal exactness and be no different than your students. $\endgroup$
    – Toscho
    Aug 25, 2014 at 13:10
  • $\begingroup$ Additonal note: The construction of irrational numbers from rationals need some infinite approximation. For example one can represent $\sqrt 2$ by an interval sequence where each interval approximates $\sqrt 2$. $\endgroup$ May 28, 2015 at 21:09

I don't know about the square root of 2, per se, but the nth root is an important concept. e.g. If a stock goes up 40% one year, and down 40% the next, the average gain is 0%, yet the compound growth is 1.4 * .6 = .86 and the square root of this number, about .927 shows an average compound loss of 7.3% per year.

  • $\begingroup$ Is this sort of (as you illustrate) meaningless addition of percentages really how average gain is defined? Ugh. No wonder people think that a stock that fell 50% and then went up by 60% would be a good buy. $\endgroup$
    – LSpice
    Aug 29, 2014 at 23:40
  • $\begingroup$ @LSpice - this phenomenon is the subject of much discussion at Money.SE. Long term, the US S&P 500 index may show an average prior 100 year return of 11.93%, but CAGR (compound annualized growth rate) of 10.05%. As I say, "you can't spend average." $\endgroup$ Apr 19, 2017 at 16:12

First I have to make an assumption: that when you say "they are into business" you do not mean "professionals" in general -because if they are, say, engineers building bridges or spacecrafts, they should and most probably do care a lot about whatever mathematical toll can give them increased precision. So my assumption is that "into business" means "into business administration" (I won't bother discussing the "politics" part).

Then, from my experience as a "business person", I suggest (for precision's shake) that you separate "business people" (let's call them BP's), into those that are in the Financial Services Sector, and those that are not.
The first group has as a usual level of precision four decimal digits, the "basis points": interest rates are usually quoted as percentages with two decimal digits: $6.23$% $= 0.0623$. Of course, in very large contracts this can go further, but I am stating the usual everyday practice.
The second group, well, they haven't decided yet: business reports, charts, "contribution margins" etc are quite often expressed also as above... but almost nobody will talk, communicate, think, negotiate or decide at that level of precision. 2 decimal digits (i.e. whole percentages and nothing more, $6% = 0.06$), is the usual level here...

...for percentages. because when you go into levels of magnitudes, then you loose at least 3 zeros before the dot: The amount $100.450$ USD is "a hundred thousand", while if the business is in the tens of millions, well, we talk using tens of millions (only accountants resist this, and this is why BP's call them derogatorily "bean counters").

But this apparent avoidance of precision, perhaps surprisingly, means that BP's should adore things like $\sqrt 2$. Why?

Well, why are they not after precision in the first place? Because, there is always the trade-off between the gains from precision and the cost of attaining it.This cost is measured in time units: time required to think precisely, time required to compute precisely...

...and BP's are obsessed (rightfully) with efficient time management. So they will take anything that cuts the time to do anything: $\sqrt 2$ requires less button pushes compared to its rational approximation -and so it increases precision at a time gain: that's a dream come true (I am not joking). BP's will still think of $\sqrt 2$ as "about $1.5$" -but when it comes to actual computing, $1.5$ takes three button hits, while $\sqrt 2$ takes only two.

So it appears that if you want to push this agenda, your chance is to make the BP's see the reality that, with all these computers and calculators around, they can think in whatever level of precision they like -but that it is in their time-management interest to compute using these strange looking mathematical concepts and tools...while avoiding also the possibility of fatal compounding approximation error: a dream come true.


Our money is growing such that it is compounded once per year.

If we double our money in two years, what is the yearly interest rate?

$(1+r)^2 = 2$

$r = \sqrt2 - 1$ which is about 41%. It's going well!


It (the square root of 2) is the one and only aspect ratio that results in the same aspect ratio of the two halves when cut in half (width-wise). This is the basis of "metric-size" paper, in particular, its most common form: A4. Establishing this fact about aspect ratio is a trivial algebraic exercise.


Unless the concept of square root (or any other operation that gives irrational numbers, what I'm guessing you are really after) comes up naturally, it is probably hopeless.

Areas where such come up are when you compute e.g. interest rates (given a present value and a schedule of payments, what is the interest rate). Sure, you can compute approximate values, but the exact value will most of the time turn out irrational.

Sorry, I don't know enough of this area (even less what your class has seen) to come up with more examples.

  • $\begingroup$ this is a legitimate answer... that it's not possible $\endgroup$ Jun 26, 2014 at 12:23

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