Around 7pm, I was heading home in the BART with a splitting headache after a frustrating day at work ( one of those days :), so I check out my twitter feed on my Android and I see the pattern. Yeah, ok, I say to myself, "each line is half the previous", but only if the number is 2. Why begin with a 2 ? So I put in a 3, and guess what, the whole thing falls apart! You can't halve 3^4 = 81 to get 3^3 = 27. The whole thing makes no sense and works for a very specific example of 2. So I write back -

Should have been obvious upon a moment of reflection.

In fact, replace these 2's & 3s with a k, and a k-inverse will do the trick.

I went to bed thinking the whole thing was too cute, but clearly not a proof.

But then, what is a proof ?

As a math major, we have this notion that a math problem is the enemy, and a proof is a weapon. So you use the weapon to make the enemy submit to you. Sometimes I think math problems are the enemy fortresses, to be blown up by a tank, which obviously is the proof.

This sort of vivid imagery is not unique to me - Supposedly, Alfred Einstein had problems with Algebra in high school. He couldn't solve equations with x in them. You know, the kinds like 2x+3 = 5, solve for x etc. "What is x ?" Einstein wondered. x is 1, said his teacher. But he didn't see it. Finally, his teacher understood that he didn't understand the concept of an equation!

So Einstein's Algebra teacher told him "x is an animal in the forest. You can't see its face. The animal is running away and you chase after it. Finally you hunt it down & you now see the animal! That's your x !! " That's how Einstein understood what the whole deal was, and went on to win the Nobel etc...

But proofs don't have to be weapons. Proofs can be an object of persuasion. We often insist on rigor at the expense of insight. This is a mistake.

Most mathematicians don't proceed by assembling one rigorous theorem after another to arrive at the final proof - even though that's the exact impression you get when you read any math paper. A typical math paper will state the problem in say proposition 8.1, then quote a few lemmas & axioms 8.2a, 8.2b, 8.2c, then infer a few theorems 8.3, 8.4 leading up to 8.17 which then completes the proof. You read the whole thing & exclaim "Wow! How did the guy know he had to start with this particular axiom & move on to that particular theorem & from there to the next one ? The chain of reasoning is mind boggling! I could never do that!!"

However, that's just presentation for the purposes of publishing in the AMS/MAA, for tenure, etc :) How one arrives at the theorem-chain is mostly through rough heuristics, pattern matching & a feel for the territory. The pattern that Dr. Strogatz presented above is far more valuable than any rigorous proof. Its a simple magical intuitive pattern! You will remember it a decade from now. I sure will!

I recall this problem posed in a college math exam -

2.7^1.01 - 2.7*1.01 is closest to

a. 0.5

b. 0.1

c. 0

d. -0.6

e. -0.09

No calculators, of course!

I walked around with a smile the entire day because I was the only student who got that right !!!

Now, the problem seems quite opaque. The typical reasoning goes thusly - What's so special about 2.7 and 1.01 ? So if I exponentiate 2.7 with 1.01, ok I'll probably get something very close to 2.7. If I multiply 2.7 with 1.01 I will also get something very close to 2.7. But clearly they won't be equal! I mean, exponentiation is not multiplication! One will be larger than the other. So there'll be a difference. So clearly c isn't the answer. Now the question is, will the difference be positive or negative ? Well, exponentiation is clearly more "powerful" than mere multiplication. So the answer is either a or b.

So roughly 50% of the student population picks a, and the other 50% picks b and they all get it wrong!

Say you are a diligent math major.

So you spot that 2.7 is really very close to e ie. the base of the natural logarithm!

You think you've got it now, but you still don't!

e^1.01 - 1.01*e = ???

Now ofcourse, if you think Calculus, you stand a chance.

What do we know about e with regards to Calculus ?

Well, the most obvious fact is that e^x is the only function whose derivatives & integrals preserve the function.

So you differentiate e^x, you get back e^x.

Well, that's a LOT of hints, so I'll let you figure out the solution one more time. If you don't see it, scroll down the page for an Aha! moment !!!

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Ok, so you give up!

Here we go.

What's a derivative of a function a some point ? Well, you tweak the function very slightly around the point ie. just move a little bit to the right ( or the left) of the point. Evaluate the function at this new position. Take the difference and divide by the tweak amount.

So say you want the derivative of e^x around 1, you would move a little bit to the right say 1.01.

Now evaluate e^x at this new point 1.01

That's e^1.01

Now take the difference with your original point

So you get e^1.01 - e

Now divide by the tweak amount 0.01

So (e^1.01 - e)/0.01

So there's your derivative of e^x at 1 ( a very good approximation )

But what is the derivative of e^x at 1 ? Why, its simply e^x evaluated at 1 !

So (e^1.01 - e)/0.01 = e^1

ie. e^1.01 - e = 0.01*e

ie. e^1.01 = 1.01*e

Voila!

So you should expect the numbers to be very, very close.

Which means

2.7^1.01 - 2.7*1.01 is closest to

c. 0

That's your Aha! moment right there!

Indeed!

If you still don't believe me, here's some scala

scala> math.pow(2.7,1.01)-2.7*1.01

res3: Double = -4.857595395302283 E-5

So the difference is very, very small. Definitely closest to 0 than to 0.1 or any of those other choices.

I will let Patrick Honner, a math teacher extraordinaire, have the last word -