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I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found that I, too, lament the uninspiring quality of my elementary math education.

I want to make a book that discredits the notion that math is merely a series of calculations, and inspires a sense of awe and genuine curiosity in young readers.

However, I myself am mathematically unsophisticated.

What was the first bit of mathematics that made you realize that math is beautiful?

For the purposes of this children's book, accessible answers would be appreciated.

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For me Euclid's proof of the infinitude of primes was the first thing that made me realize the beauty of mathematics. –  Manjil P. Saikia Mar 7 '13 at 7:02
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Wow. Just last night I had a fierce argument with one of the bartenders of my usual watering hole who is a mechanical engineering student. He insisted that he has a better idea than me of what is mathematics. I am so going to print him a copy of Lockhart's text. Thank you for that link! –  Asaf Karagila Mar 7 '13 at 7:59
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I can’t remember a time when I didn’t think that mathematics was beautiful and fascinating. –  Brian M. Scott Mar 7 '13 at 15:06
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Although I don't know if it's what you are looking for, try looking up "vihart" on youtube--Even if it's not helpful, I guarantee you will appreciate it. –  Bill K Mar 8 '13 at 2:57
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I think it's a shame that this question was voted closed... –  Will Mar 10 '13 at 19:50

151 Answers 151

I was completely baffled when I learned the approach of C.F. Gauß for summing 1+2+3+...+100. Of course I would have gone for the hard way as well and I was deeply impressed when I learned that this equates to 1+100 + 2+99 + 3+98 + ... = 50*101 = 5050.

The next big thing for me was when I discovered that you can reduce multiplication to looking up squares by the identity

a*b = ((a+b)(a+b)-(a-b)(a-b))/4

However by that time I was already hooked.

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Though a lot have been said (I too worked out Pascals triangle as a kid) no one has (yet) mentioned Gauss' method for adding sequential numbers.

It may be apocryphal but the story I heard was that a teacher wanted busy work, so she told the class to add the numbers 1-100, thinking that would take forever. Gauss was smart, he knew that the pair 100+1 was the same as the pair 99+2, the same as the pair 98+3... and now that he paired these numbers off, he now had 100/2 or 50 pairs of them. 50 pairs of 101 was 5050. He told the teacher the answer way before it was expected, and shocked them.

The coolness of the story is that it's probably at the level of your audience, something they can do and experiment with. and the guy's a legend too.

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There had been a mentioning of Gauss, more than one person actually. Look closer. –  Asaf Karagila Mar 7 '13 at 18:59

I would have to say that it was the square root. There was (ans still is) something very fascinating about being able to recover the number that was multiplied by itself. If I know that $x^2 = 9$ then I knew that $x$ could be $3$ (just thinking about positive numbers here). And I thought that it was crazy how one could also take square roots of numbers that aren't actually squares themselves.

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A Fibonacci spiral and the way that at large enough scales it converges on the golden ratio.

Also the golden ratio.

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From an interview with Vladimir Arnold (NOTICES OF THE AMS, Vol. 44, No. 4):

Please tell us a little bit about your early education. Were you already interested in mathematics as a child?

...

The first real mathematical experience I had was when our schoolteacher I. V. Morozkin gave us the following problem: Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on this day?

I spent a whole day thinking on this oldie, and the solution (based on what is now called scaling arguments, dimensional analysis, or toricvariety theory, depending on your taste) came as a revelation.

The feeling of discovery that I had then (1949) was exactly the same as in all the subsequent much more serious problems—be it the discovery of the relation between algebraic geometry of real plane curves and four-dimensional topology (1970) or between singularities of caustics and of wave fronts and simple Lie algebra and Coxeter groups (1972). It is the greed to experience such a wonderful feeling more and more times that was, and still is, my main mo- tivation in mathematics.

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Not an example of my own youth I've followed a small seminaryseminar on how to teach math a few years ago, and one of the things the teacher mentioned was that counter-intuitive results were more likely to mark the kids in a way they would start to try to understand why the results wasn't what they expected.

The example he gave us was fairly simple:

Imagine you ran a rope around Earth's diameter, lying on the ground. Then, add 1 meter to the length of the rope, keeping its shape as a circle (let's forget mountains and pretend Earth is just a ball for a while) - at what distance of the ground will the rope be?

For most people, adding one meter to such a long rope is negligible so that there's simply no way it would be far from the ground. Convincing them that it's actually nearly 16cm above the ground is fun to do.

As far as I remember, that example was extracted from a book, full of such examples and historical references which are also useful to show math isn't just a boring school obligation; but I can't find the name of the book right now.

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Seminary? There's a religion of mathematical pedagogy? –  Joe Z. Mar 13 '13 at 20:41
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Not that I know, but now that you mention it, maybe it isn't such a bad idea. –  Joubarc Mar 14 '13 at 13:36

When I was still pretty young (I don't remember my exact age) I was very proud that I could already compute with decimal fractions which nobody I knew in my age could at the time. Around that time my aunt had a student for a visit in her home, and he talked to me about math, and asked me to compute $1/3+2/3$. I asked to how many digits and he said as many as you like. So, I sat down and computed it to 10 digits or something: \begin{align} 0.3333333333\\ \underline{+0.6666666666}\\ 0.9999999999 \end{align} Proudly, I presented my result. He said well done, but it's way easier \begin{align} \frac13+ \frac23= \frac{1+2}3= \frac{3}3=1. \end{align} The beauty in this impressed me a lot and kind of got me started in math.

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When I was 10, I read a math booklet, that talked about Euler characteristic. There were drawings of all Plato's polyhedrons, and I counted, and realize that their Euler characteristic was always 2. I was amazed, asked my mom, math teacher, if she knew anything about it, and she told me she didn't. Now I'm 21, and I am just starting to be math-mature enough to understand this theorem. Maths are beautiful :D !

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I'm not sure if this is suitable, but for me, the power of Mathematics lies in the absoluteness of its proofs. This is the only discipline where you can prove something to be true and it will stand up to the test of time, where no textbooks need replacing and facts are always right. (I'm assuming we don't make fundamental changes in axioms and what not!) This cannot be found in any other human endeavour and I find this to be very reassuring!

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I found the formula $(a+b)^2=a^2 + 2ab + b^2$ that my father told me at a young age fascinating. (And also that $(a+b)(a-b)=a^2-b^2$.)

Overall, it seems that a parents duty is to teach his children two of the following: (a) to ride bicycles, (b) To play chess, (c) The formula for $(a+b)^2$, and my father took (b) and (c).

My mother let me read her high-school calculus book (incidentally one of the authors had the same last name as mine) and there what I found really fascinating (but I could not understand) is that you can add a variable to a triangle. (This was a misunderstanding of what $f(x+\Delta)$ means.)

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I have to admit that although I'd frequently been told that mathematics was "beautiful", I didn't really get that while I was in school - even high school. I enjoyed mathematics, and saw plenty of things that were fun, and even cute, but I never really understood any ideas with sufficient depth to think of them as beautiful.

When I did encounter ideas that I found beautiful, it was in my first year at college. In fact, there were two closely related ideas in quick succession. We were just being introduced to vector spaces. This was the first time I'd seen an abstract space, but it didn't really seem to mean much except as a fancy way to talk about high dimensional Euclidean spaces.

But then I saw my first example of a vector space that didn't just look like the vectors I'd seen in high school. It was the space of infinitely differentiable functions: $C^{\infty}$. We were shown the linear operators associated with two common differential equations (exponential growth and simple harmonic motion): \begin{eqnarray} &\frac{\textrm{d}\phantom{y}}{\textrm{d}t} - kI \\ &\frac{\textrm{d}^{2}\phantom{y}}{\textrm{dt}^2} + kI. \end{eqnarray} We saw the fairly routine proofs that these were linear operators on $C^{\infty}$, but then came the magical part: The solution sets to these differential equations were subspaces of $C^{\infty}$, the canonical solutions I was familiar with were basis sets for these solution spaces, and the solution spaces were actually the nullspaces of these operators!

Later (maybe even in that same lecture) we saw how linear regression - the hitherto tedious process of finding the "line of best fit" - could be understood as a linear projection $P$ operator onto a two dimensional subspace of the data space. Given a data vector $\mathbf{x}$, the projected vector $P\mathbf{x}$ represented the line that was closest to the data - the line of best fit - and the difference term $\mathbf{x}-P\mathbf{x}$ represented the error term. I was astonished at how much more elegant this was than the clunky formulas I'd had to memorize in high school.

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The realization that you can go on counting forever.

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I think the first thing that amazed me in this way was $\pi$. An irrationnal number, which means it has an infinite number of digits, which involves humans can't manage it, we can't know it on the whole, but already the Greeks discovered it. They knew it has something to do in the circumference or the area of a circle, that is, they could manipulate it, and I find this unbelievable.

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The first interesting mathematics problem I remember in my limited memory is 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... It never totals to TWO :-)

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Sounds like Zeno's Dichotomy paradox. –  BobStein-VisiBone Mar 7 '13 at 15:28

This was probably the very first mathematic riddle which absolutely got me. It is called Algebrogram in my language, but I couldn't find a reference in English.
I was attending mathematic group after normal school (at age 11-14) and then I made few of my own for my classmates. I loved it ^^

You use characters instead of numbers and you construct some words. You then let others solve it.

F O R T Y
    T E N
    T E N
---------
S I X T Y

Solution:

2 9 7 8 6
    8 5 0
    8 5 0
---------
3 1 4 8 6

It was common to construct sentences as well, but it is kind of hard. This is only an example, which is unsolvable ;)

You could specify if there were some other operations or you could let your solvers find it out by themselves.

        O U R
    H O U S E
        H A S
      - T E N
-------------
W I N D O W S
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In English these are sometimes known as cryptarithmetic puzzles. –  Will Mar 7 '13 at 16:41

I felt like an Einstein and was really interested in mathematics when I myself discovered the truth behind a^0 =1. That is, a^0 = (a)^(1-1) = a^1/a^1 = 1

Yeah, I know this is simple.. But generally it is taught as a formula. Instead this one can be used to change the way of thinking...

Also, multiplication is repeated addition... This used to fascinate me a lot...

2 * 3 = 6 that is, 2 + 2 +2

4 * 3 = 4 + 4 + 4

5 * 8 = 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5

And then in the end you can say that, for very big numbers, you cant sit adding all of them and hence, multiplication is the shortcut to add all of them :)

I am not a writer... But probably you can take some god examples to explain what I am trying to say here... I think this will be really interesting approach to teach multiplication! All the best for your book. Do let us know the name of the book. We will also cherish it... :)

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Mine was the discovery of sets in higher order math classes, and how all the lower math classes including physics theories were strictly derived from higher order calculus, and all of the formulas I had ever learned became such simple child's toys.

I don't think those belong in a children's text, however.

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When you realize that taking derivatives is so simple, you look back and realize, "I can't believe people use this as an example of difficult mathematics!" –  Joe Z. Mar 13 '13 at 20:25
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I had a similar moment of realization for reducing polynomials, back in middle school when my Sunday School teacher used a really long rational polynomial expression as an example of a "problem that's too hard for you to solve" (it was part of a teaching package). She had to resort to using trigonometry and asking me how I would calculate $\tan 35^\circ$, which I didn't know at the time. The polynomial ended up being something contrived, but it did actually reduce quite a bit. –  Joe Z. Mar 13 '13 at 20:32
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Of course, now when I look back at it, I think, that wasn't actually hard! –  Joe Z. Mar 14 '13 at 14:20

The commutative law doesn't hold for some series. I think this is an amazing fact to teach.

http://www.math.tamu.edu/~tvogel/gallery/node10.html

The example in the link amazed me.

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The one that I was particularly intrigued in my late years was the execution of the proof of Gambler's Ruin. However, it might be too deep for small children.

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Compared to most answers this is certainly not going to blow anyone away, but at the time it did amaze me. Our maths teacher asked us how long it would take us to get home if, we only walked half the way home, and then half the way of what was left, and then half the way of what was left, etc, etc. The realisation that if you kept dividing something by two (no matter how many times), you would never get to zero.

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Well, you'd take the time it takes you to go halfway home, plus half of that, plus half of this next amount, and so on. –  Joe Z. Mar 13 '13 at 20:40
  1. Take your age, and reverse it.
  2. Subtract smaller number from bigger.
  3. Add the digits of subtraction.
  4. You get 9.
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What if I'm 22? –  Kobi Mar 7 '13 at 12:06
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What if I am 4? –  Jack Aidley Mar 7 '13 at 14:04
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What if I am 100? –  Trevor Wilson Mar 7 '13 at 16:32
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What if I'm 27.9? –  Asaf Karagila Mar 9 '13 at 20:28
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^ Then your answer will be expressed in terms of the p-adic numbers. –  Joe Z. Mar 12 '13 at 13:14

I like cars and automotive racing and such. What got me real interested in it were two things:

The first, to a great extent, in Calculus:

  • $\displaystyle \frac{d}{dt}\ \text{Displacement} = \text{Speed}$
  • $\displaystyle \frac{d}{dt}\ \text{Speed} = \text{Acceleration}$
  • $\displaystyle \frac{d}{dt}\ \text{Acceleration} = \text{Jerk}$

It all made sense to me after that!

Then there was a problem in my Cal. book about calculating the force of a piston in an engine. I can't quite remember it, but it was basically:

$\text{Force} = \text{RPM}^3$

or something similarly extreme. It relates to the automotive aphorism: Power doesn't kill motors, RPM does.

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So whenever I commented that someone is a jerk, I was deriving this from their acceleration? :-) –  Asaf Karagila Mar 7 '13 at 17:31
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@AsafKaragila Sir, I'd like to present to you your well deserved award, 'worst joke ever'! –  OmnipresentAbsence Mar 7 '13 at 18:10
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@OmnipresentAbsence: For this? Nah, I have had infinitely worse jokes before, and in probability $1$ I will have infinitely many more. –  Asaf Karagila Mar 7 '13 at 18:11
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@AsafKaragila I know, I'm just kidding. I actually chuckled because of how cheesy the joke was –  OmnipresentAbsence Mar 7 '13 at 18:15
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d/dt jerk is called "jounce", I believe. –  Joe Z. Mar 12 '13 at 13:11

The fact that you can't divide by zero always amazed me. I once read the following analogy:

Imagine you go to a shop with 100 dollars in your pocket, and imagine that everything in the shop costs 1 dollar. How many things can you buy? 100. What if instead of 1 dollar, each thing costed $0.5? How many things can you buy? 200. Now imagine that everything is free. How many things can you buy? Obviously, this question doesn't make sense anymore, because things are free, so you can take 0, or 1, or 2, or...

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If you're writing a children's book on mathematics, please start by reading some excellent children's books dealing with mathematics. Here are some books I have fond memories of:

  • The Man Who Counted
  • The Phantom Tollbooth
  • Flatland
  • Alice in Wonderland / Through the Looking Glass
  • Everything by Martin Gardner
  • Godel, Escher, Bach: An Eternal Golden Thread
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The first time I heard that 3 times 5 is the same as 5 times 3, I was really intrigued, and I've been hooked ever since. It is pretty weird when you think that five groups of three people is as many as three groups of five.

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I first discovered that math was beautiful upon learning the divisibility rules. At that point I was just like "IT WORKS IT WORKS! HOW DID PEOPLE KNOW THAT?!" I remember I once stayed up to test the divisibility rule of dividing by $8$ (if the last three numbers in the dividend are divisible by $8$ then the whole number is divisible by $8$).

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Not an experience of mine, but I'm currently reading The Greeks by H. D. F. Kitto and I think this page deserves to be here:

But let us not be too superior to those Greeks who "shut their eyes." They kept something else wide open, namely their minds, and although the eye-shutting retarded the growth of science, the mind-opening led to things perhaps equally important, metaphysics and mathematics.

Mathematics are perhaps the most characteristic of all the Greek discoveries, and the one that excited them most. We shall be more understanding of those who shut their eyes to facts if first of all we keep in mind the Greek conviction that the Universe is a logical whole, and therefore simple (despite appearances) and probably symmetrical, and then try to imagine the impact of their minds on elementary mathematics.

It happens that I myself—if I may be personal for a moment—was enabled to do this by an insomnia-beguiling piece of mathematical research that I once did myself. (Mathematical readers are permitted to smile.) It occurred to me to wonder what was the difference between the square of a number and the product of its next-door neighbors. $10 \times 10$ proved to be $100$, and $11 \times 9 = 99$—one less. It was interesting to find that the difference between $6 \times 6$ and $7 \times 5$ was just the same, and with growing excitement I discovered, and algebraically proved, the law that this product must always be one less than the square. The next step was to consider the behavior of next-door neighbors but one, and it was with great delight that I disclosed to myself a whole system of numerical behavior of which my mathematical teachers had left me (I am glad to say) in complete ignorance. With increasing wonder I worked out the series to $10 \times 10 = 100$; $9 \times 11 = 99$; $8 \times 12 = 96$; $7 \times 13 = 91$… and found that the differences were, successively, $1, 3, 5, 7, \ldots$, the odd-number series. Even more marvelous was the discovery that if each successive product is subtracted from the original $100$, there is produced the series $1, 4, 9, 16, \ldots$. They had never told me, and I had never suspected, that Numbers play these grave and beautiful games with each other, from everlasting to everlasting, independently (apparently) of time, space, and the human mind. It was an impressive peep into a new and perfect universe.

(original source image)

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At about 10 or 11 I discovered that the area of a circle was half the circumference multiplied by the radius.

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When I was 11, my math teacher asked us whether we thought that any three points in the plane that do not lie on a single line would lie on a single circle, and I remember being amazed to see that this was true.

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  1. 17 + 20 = 8
  2. 17 − 20 = 26
  3. 17 · 20 = 21
  4. 17^(−1) = 12 (inverse of 17)

I got really upset when I saw this. The professor explained, to do network communication you will need to understand this.

I found maths awesome after dealing with these. What we are normally learning can not always help (it's real numbers mathematics). But the best things deal with fields. Therefore, the below is the explanation of the above meaningless things.

(i) Addition: 17 + 20 = 8 since 37 mod 29 = 8

(ii) Subtraction: 17 − 20 = 26 since −3 mod 29 = 26

(iii) Multiplication: 17 · 20 = 21 since 340 mod 29 = 21

(iv) Inversion: 17^(−1) = 12 since 17 · 12 mod 29 = 1

The elements of F29 are {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27 28}

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