# What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

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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Nice question, but should probably be community wiki? –  mrf Mar 7 at 7:02
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 at 7:02
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 at 7:59
I can’t remember a time when I didn’t think that mathematics was beautiful and fascinating. –  Brian M. Scott Mar 7 at 15:06
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 at 2:57

Arithmetic series might be interesting: straightforward to explain and amenable to pictorial representation ...and the child might love the fact that they've learnt how to do huge sums that might stump many (non-mathematical) adults.

You could show how $1 + 2 + 3 + \cdots + 100$ could be worked out by pairing numbers from opposite ends of the sum together $(1 + 100) + (2 + 99) + \cdots + (50 + 51) = \underbrace{(101 + 101 + \cdots )}_{\text{50 terms}} = 5050$.

or by adding the series to itself with terms running in ascending and descending order $1 + 2 + \cdots + 99 + 100$

$100 + 99 + \cdots + 2 + 1$

to get $101 + 101 + ... = 101 \times 100$ which is twice the sum.

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The story could even be about a little boy named Carl Freidrich... –  Will Mar 7 at 16:31
I also remember a teacher (must have been around 10yo) asking us to calculate the sum of all numbers from 1 to 100, letting us sweat uselessly, then showing us how to pair the numbers. He liked to play tricks on us, but the lessons were always useful. Well, sometimes he played us for fools by being wrong on purpose - once he "calculated" that there were about half of the days in the years that were holidays to show we shouldn't complain about school. Took me years to realize he had counted a lot of days twice (like, 2 months holiday & 52 week-ends.) –  Joubarc Mar 8 at 8:55
@Joubarc: That reminds me of a joke. A clerk asks his boss for a raise, and the boss calculates how many days the clerk works there. There were 366 days in that year, and he worked for 8 hours a day, so it became 122. Then, he had to subtract 52 Saturdays, two weeks' vacation, four bank holidays, and 52 Sundays, for a grand total of no days at all. "And you have the nerve to ask me for a raise!" –  Joe Z. Mar 8 at 20:10
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Many years ago, before I knew multiplication, I wrote numbers 1 to 10 in a row:

1  2  3  4  5  6  7  8  9 10


Then I wrote a second row, just for the fun of it, starting with 2, increasing each number by 2:

2  4  6  8 10 12 14 16 18 20


And then a next row, starting with 3, with an increment of 3, and so on, until I got:

 1  2  3  4  5  6  7  8  9  10
2  4  6  8 10 12 14 16 18  20
3  6  9 12 15 18 21 24 27  30
...
9 18 27 36 45 54 63 72 81  90
10 20 30 40 50 60 70 80 90 100


I showed this to my parents, and they told me it was this thing called the "multiplication table" and explained how it worked. I was amazed.

Still today I'm very proud that I reinvented the multiplication table :)

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For me, the result that really captured my imagination was the divergence of the harmonic series:

$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\ldots=\infty$$

It combines some wonderful ideas about the infinite and the infinitesimal, and it seemed (at the time) completely absurd to me that adding infinitely small numbers could result in an infinitely large one.

As an illustration of this idea, say we have a big pile of 1-foot square boards. We stack the first board on the second, hanging half-way (6 inches) over the edge. Then we stack the third on top the second, hanging 1/3 of the way (4 inches) over the edge. The forth is stacked on, hanging 1/4 of the way (3 inches) over the edge. The fifth...you get the idea. At first glance, one might think that our pile can only extend horizontally a finite distance - we might take bets that it gets at most 2 feet, or maybe 5 or 10 feet horizontally. But it turns out that if we have enough boards (negligibly thin, say), we could build a bridge across any river, any ocean, in fact we could build a bridge across the entire universe this way.

Here is a Wolfram demonstration of this, although their stack is upside-down from how I have described it: http://demonstrations.wolfram.com/OverhangingCards/

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Sure, okay. Hey, I'm a mathematician, not an engineer =) –  icurays1 Mar 9 at 6:42
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Euclidean geometry was the first thing that got me (about grade 9 or 10). That's where I first found out that

1) There is such a thing as mathematical proof (rather than just calculation).

2) Mathematics is not a closed subject: new and interesting results can still be found.

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