The twelve-year-old just asked a series of three questions, the last of which was one of the Ultimate Questions:

- What is this ln - e^x button on my calculator?
- What is e=2.718281828... good for?
- If e is defined as the number for which the curve y=e^x has everywhere a slope equal to its y-axis value, why is e=2.718281828... ?

Needless to say, I could not answer the third question. Fathers really don't know very much, do they?

Posted by DeLong at August 14, 2002 12:56 PM | TrackBack

Comments

you forgot that e is the limit of the sequence (1+1/n)^n. just say that and sit back knowingly.

Posted by: Dennis on August 14, 2002 01:20 PMFor that matter, why does two plus two equal four?

Posted by: Bob Hawkins on August 14, 2002 01:30 PMWhy, because we define it so.

Posted by: Dennis on August 14, 2002 01:39 PMIt's not too hard.

(Technically speaking, you can't define e as the number for which the curve y = e^x has everywhere a slope equal to its y-axis value, because how do you know that there is such a number e? But we can let that slide.)

Observe that the curves

y = e^x

and

y = 1 + x + x^2/2! + x^3/3! + ....

are the same.

(You can show this in a non-rigorous way by pointing out that where x = 0, y = 1 for both curves, and then demonstrate by differentiating that the slope of the second curve is also everywhere equal to its y-axis value.)

Then simply plug x = 1 into the latter curve and calculate away:

e = 1 + 1 + 1/2! + 1/3! + 1/4! + ...

e = 1 + 1 + 1/2 + 1/6 + 1/24 + ...

e = 2.718 ...

Posted by: Kurt Hemr on August 14, 2002 01:56 PMthats usually nto how E is defined either. You usually define e as the limit of (1+1/n)^n, get your binomial theorem out, and prove that thats equivalent to what you just wrote.

Posted by: Dennis on August 14, 2002 02:00 PMAgreed. What I wrote is a more direct route from the property of e under discussion (namely, that De^x = e^x) to its numerical value.

Posted by: Kurt Hemr on August 14, 2002 02:11 PMWell, yes, we can certainly show that e is equal to 2.718281828.

But I, at least, interpreted the twelve-year-old to be asking a harder question, namely: Could the universe have been set up so that e had a different numerical value? (In the past, he has asked if it might have been possible for pi to be a rational number.)

Now I just have to remember Pythagoras's proof of the irrationality of root two... ah, got it...

Posted by: Brad DeLong on August 14, 2002 02:26 PM"If e is defined as the number for which the curve y=e^x has everywhere a slope equal to its y-axis value, why is e=2.718281828... ? "

A very simple way of answering this:

Because for 2^x, the derivative is too low, and for 3^x, the derivative is too high. For 2.5^x, the derivative is still too low, and for 2.75^x, the derivative is still too high. Keep going like this, and you will converge on the value of e.

Yes, I know there are more efficient ways of doing this, but this method is extremely simple conceptually. Also very simple to see in an Excel spreadsheet (with discrete approximations of derivatives):

x 2^x d(2^x)/dx 3^x d(3^x)/dx

0.1 1.072 xxxx 1.116 xxxx

0.2 1.149 0.797 1.246 1.371

0.3 1.231 0.854 1.390 1.531

0.4 1.320 0.915 1.552 1.708

0.5 1.414 0.981 1.732 1.907

0.6 1.516 1.051 1.933 2.128

0.7 1.625 1.127 2.158 2.375

0.8 1.741 1.208 2.408 2.651

0.9 1.866 1.294 2.688 2.959

1 2.000 1.387 3.000 3.302

1.1 2.144 1.487 3.348 3.686

1.2 2.297 1.594 3.737 4.114

1.3 2.462 1.708 4.171 4.592

1.4 2.639 1.831 4.656 5.125

1.5 2.828 1.962 5.196 5.720

1.6 3.031 2.103 5.800 6.384

1.7 3.249 2.254 6.473 7.126

1.8 3.482 2.416 7.225 7.953

1.9 3.732 2.589 8.064 8.877

2 4.000 2.775 9.000 9.907

...

Posted by: Curt on August 14, 2002 02:36 PM>>But I, at least, interpreted the twelve-year-old to be asking a harder question, namely: Could the universe have been set up so that e had a different numerical value? (In the past, he has asked if it might have been possible for pi to be a rational number.)<<

No. Unlike pi, e is not an intrinsically interesting number. It's the solution to an interesting calculation. If it were a different number, it would be the solution to a different calculation.

One might, given suitable metamathematical flights of fancy, imagine some bizarre geometry in which the ratio of a circle's circumference to its diameter was a rational number (I'm not sure if this is actually a coherent geometry, but let's suppose that we can vary logic as well as space-time).

However, circles are "natural kinds" in a way in which limits aren't, and e is either the limit of a specific series or (using your slope-based definition) the result of a more complicated limit argument. Taking limits is a formalism, and you change the formalism, you're naming a different entity by "e".

Posted by: Daniel Davies on August 15, 2002 01:00 AMWhat will never cease to fascinate me is that:

e^{i*pi}=-1

Not sure if it should, but it does.

Posted by: Atrios on August 15, 2002 02:53 AMI've run into the 'explaining e' problem a couple of times. The basic problem seems to be that 'e' is intimately involved with the -general- concept of analytical continuity-- so the various -specific- examples one uses to illustrate it (like continuous slope, continuous compound interest, Taylor series, e^(i*pi)=-1, etc.) all seem to miss the point.

Posted by: Matt on August 15, 2002 07:01 AM>>However, circles are "natural kinds" in a way in which limits aren't<<

Hmmm. I actually find e more of a "natural kind" than pi: I spend more of my time thinking about compound growth than drawing circles, after all...

And I have never understood why both e and pi have to be irrational numbers either...

Posted by: Brad DeLong on August 15, 2002 07:41 AMe^{pi*i}=-1 is really spooky...

Posted by: Brad DeLong on August 15, 2002 07:41 AMAnd I have never understood why both e and pi have to be irrational numbers either...

This is an interesting question. We can easily find A proof for why e is irrational (I'm sure a quick google search will show the same for pi).

Now the next question: is a proof just a meaningless manipulation of symbols, or is it a reason why e is not rational? i.e. e is irrational because if it were not, we'd reach an absurd position?

The one thing my math education taught me is that a proof isn't the same as a reason. They are often related, but they aren't the same. But I'm looking at the proof, and I can't think of anyway to point out the intuitions to a 12-year-old. About the only thing I can add is this: we can define e to be the solution of D(e^x) = e^x where D is the differential operator. Using no other information about e, we can prove e exists and show that it is equal to the sequence 1 + 1/2! + 1/3! +...

On a related note: I just finished reading a great book: "Women, fire and dangerous things" by George Lakoff. Lakoff is a cognitive scientist who is criticizing the use of logic (and proof theory) to model human intelligence.

One of Lakoff's problems is that mathematics, the way it is used in cognitive science and AI, is basically meaningless. The "objectivist" or "logical positivist" view is that meaning arises

when a mathematical expression is given some referent in the real world.

Unfortunately, this idea doesn't seem to work out: an earlier work by someone called Putnam trashed the idea of math+references=meaning. Lakoff reviews Putnam's argument (as well as talking about some experimental evidence against the objectivist view) and then goes on to propose a new basis for cognitive science and AI. But reading your question now, I am starting to wonder... does Putnam's arguments mean mathematics is fundamentally meaningless?

This is why I say this: the "reasons" we eventually come up with are based on other truths. These truths have proofs as well. Eventually, you'll get back to the "intuitive" parts: the axioms (Lakoff points out the the axioms of the various branches of mathematics are usually based on human psychology more than any kind of abstract truth). But if there are too many steps between the intuition and the reasons, does mathematics become, for all practical purposes, meaningless?

We can apply the same thinking to answering why is e=2.7yaddayaddayadda. We get an absurdity with any other value if we use the calculus definition of e. Meaningless... except for the psychologically motivated axioms.

Another way of putting it is: if, in a wierd alternate universe, we were all born with 2*e/3 (base 10) fingers, we might be asking, why is e=3 (base 2*e/3)?

I hope this made some sense...

Posted by: Amit Dubey on August 15, 2002 09:23 AM>>I spend more of my time thinking about compound growth than drawing circles, after all...<<

Oh come come .... the compound growth we think about is discrete, not continuous! Even as recently as the eighteenth century we were trying to think about differentials as ratios of infinitesimals!

Posted by: Daniel Davies on August 15, 2002 09:45 AMThe idea of a universe that exists just "because" is a terribly interestign one. Do we exist because this universe, wokring by these self consistent laws, is the onlyone we could exist in. The anthropomorphic principle is frightening.

One book that I loved is Penrose's "The Emperor's New Mind" which deals with quite a bit of mathematice, inductive logic, Turing machines, and conciousness. That and Godel, Escher, Bach form the very best of modern mathematical philosophy, at least to my knowledge.

A mroe spooky thought is considering the subtle relationships of geometry that relate e^{i \Pi} and the real exponent. It is worthwhile, every now and then, to write out the terms of the taylor expansion (real and imaginary) to see that indeed, e and pi are interestingly and intimitely intertwined.

B

Posted by: Brennan on August 15, 2002 10:44 AMMy 9 year old asks similar types of questions, but they don't involve math... e.g.

1. If a astroid impact and the resulting climate effect killed off all the dinosaurs (small, large, meat eating and not) how did mammals, lizards etc. survive?

2. If prunes are dried plumbs, how can there be prune juice?

3. Why is it that when a public square is crowded with pigeons, you never see a little baby pigeon?

Posted by: Avery Shenfeld on August 15, 2002 12:00 PMWhoops... on looking over my previous post, I realized the 2nd last paragraph makes no sense at all... please ignore it!

Brennan:

I've never read the Emporor's New Mind, but I've read reviews. Penrose's main thesis, that humans use quantum theory to think, is a radically different model than what cognitive scientists use, but is supported by no experimental evidence whatsoever, and makes no attempt to try to explain existing experiments concerning human thought. The book may be interesting, but remember that it is a "theory" in the layperson's sense of the word, rather than a theory in the scientific sense.

Posted by: Amit Dubey on August 16, 2002 01:46 AM>>>>I spend more of my time thinking about compound growth than drawing circles, after all...<<

Oh come come .... the compound growth we think about is discrete, not continuous!<<

Yeah. And the circles we draw are discrete animals too: there's not a one for which the ratio of the curve we drew to its diameter is the transcendental number pi...

Posted by: Brad DeLong on August 16, 2002 09:21 AMAmit:

"I've never read the Emporor's New Mind, but I've read reviews. Penrose's main thesis, that humans use quantum theory to think"

Well, that isn't his thesis, at all--as a matter of fact, I suspect that the arguments Penrose makes dovetail very nicely with Lakoff (who you quote), which is why I brought it up. The main issue in the book is the limits of mathematical (and by extension, computational) theory, especially as applied to consciousness. In fact, the central point is that intuitive and inductive reasoning are the critical portions of what we call consciousness, and that these are non-mathematical phenomena. Without any other explanation, quantum mechanics provides and interesting potential means of defining conscious thought.

In any case the point in reading it isn't to gain a better understanding of the human mind, but rather, to gain an appreciation of science, math, and logic from one of the better mathemeticians and physicists in current times. As a book on consciousness theory, it isn't necessarily the best (though by no means the worst--there are some very incoherent critiques of it out there). But as a book on mathematics, physics, and geometry.

I will look up the lakoff book. The idea that mathematical axioms are based on psychology sounds, well, unbelievable. And to posit that they form the basis of mathematics, especially modern mathematics....I will have to see.

B

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