# 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

The fact that Gabriel's horn has infinite surface area, but finite volume, you can "fill it with paint, but you can never cover the whole surface".

Gabriel's horn is the graph of $y = 1\frac{1}{x}$ for $x\geq 1$ rotated around the $x$ axis.

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For me, it was the beauty of the number 1, how it can be multiplied with anything , and it won't change the number it is being multiplied with, also how it can be represented as any number divided by itself such as 4/4=1 I would also love to share this beautiful poem by Dave Feinberg that is titled "the square root of 3" and was also featured in a Harold and Kumar Movie, it renewed my love for math and is and always has been one of my favorite poems! :

I’m sure that I will always be A lonely number like root three

The three is all that’s good and right, Why must my three keep out of sight Beneath the vicious square root sign, I wish instead I were a nine

For nine could thwart this evil trick, with just some quick arithmetic

I know I’ll never see the sun, as 1.7321 Such is my reality, a sad irrationality

When hark! What is this I see, Another square root of a three

As quietly co-waltzing by, Together now we multiply To form a number we prefer, Rejoicing as an integer

We break free from our mortal bonds With the wave of magic wands

Our square root signs become unglued Your love for me has been renewed

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Here is my favorite classic illustrating the power and beauty of mathematical argument. Consider the question:

Question: Can an irrational number raised to an irrational number be rational?

Answer: One of the classic answer goes as follows. Consider the number $x=\sqrt{2}^\sqrt{2}$. If $x$ is rational, we are done. If $x$ is irrational, then consider $x^{\sqrt2}$, which is $2$ and now we are done.

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I love those "anyway" proofs. –  Joe Z. Mar 8 at 1:00
The really amusing thing is that after the proof is given, we still don't know whether $\sqrt{2}^{\sqrt{2}}$ is rational or not! –  Jesse Madnick Mar 8 at 9:29
@ted $(a^{b})^c = a^{bc}$. Hence, $(\sqrt2^{\sqrt2})^{\sqrt2} = \sqrt{2}^{\sqrt2 \cdot \sqrt2} = \sqrt2^2 = 2$. –  user17762 Mar 8 at 20:49

When I was a kid I realized that $$0^2 + 1\ (\text{the first odd number}) = 1^2$$ $$1^2 + 3\ (\text{the second odd number}) = 2^2$$ $$2^2 + 5\ (\text{the third odd number}) = 3^2$$ and so on...

I checked it for A LOT of numbers :D

Years passed before someone taught me the basics of multiplication of polynomial and hence that $$(x + 1)^2 = x^2 + 2x + 1.$$ I know that this may sound stupid, but I was very young, and I had a great time filling pages with numbers to check my conjecture!!!

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I started math reasonably late, so I'm not sure this is a perfect example for a children book. I was totally amazed that the number $\pi$ is encountered in completely unrelated situation. Of course, I knew it's a ratio of circumference of the circle to the diameter, but then I learnt of the Normal probability density function.

So if you stretch a bell-shaped curve (Gaussian function) from $-\infty$ to $\infty$ the area under it $converges$ to $\sqrt{\pi}$?? How is it possible that this area has something to do with the square root of the ratio of circumference of the circle to the diameter? To be honest, even now, after learning the related proofs and derivations I still find it quite baffling.

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When I was a kid, I remember that I used to make 'little discoveries', although after I got all excited I realised other people already knew this and I didn't discover anything new, I was always really proud when I made one of those discoveries, Here are a few that I made and approximately how old I was when I did make the discovery (just saying I am 13 years old right now).

• An odd number plus an odd number is even, an even number plus an even number is even, and odd plus even is odd (approximately age 4-5)
• The rule for the Fibonacci sequence (I remember I saw it on the board or something in my classroom and then I just found the pattern) (age 6, I remember this one accurately bc it was during the first year of my primary education)
• If you rip a piece of paper in half, and then both in half again, and again etc, the number of pieces of paper you have when you ripped it $n$ times is $2^n$ (age 10)
• The fact that you can split a triangle into two right triangles and apply trig ratios to them (I later learned the Sine and Cosine Laws so yeah) (age 11)
• I was learning differential calculus, and I didn't know integral calculus yet. My dad told me that a integral is the inverse of the derivative, Then I figured out the rule that offsets the power rule and I figured out that the integral of nothing is an arbitrary constant (When i officially learnt integral calculus I found the proper formulas and I was just using the wrong symbols) (age 12)
• I proved to myself that the cardinality of the set of Reals and the set of Complex numbers are the same (age 13)
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1) Modular arithmetic fascinated me. I could not believe that with just a few tools, I could find the remainder left when $3^{100}$ is divided by 8. ($3^2\equiv 1$ mod $8$ and hence the result.)

2) Euclid's proof of the infinitude of primes. (Let the number of primes be finite. Let them be $P=\{p_1,p_2,\dots,p_r\}$. Take $k=p_1p_2\dots p_r+1$. None of the primes in $P$ divides $k$, hence $k$ is a prime or divisible by a prime not in $P$, and so we have a contradiction.)

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I think one of my early favourite mathematical things was simply "proof by contradiction" -- any of them.

I think its appeal is that you nearly have proof by example, except that you're proving a negative.

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This might seem very elementary: but amazed me when I was a child.

The fact that $a\times b = b \times a$.

I would keep drawing boxes on the number line of different lengths and then discovering that they fit snugly into one another. Still seems amazing.

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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'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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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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For me it was when I realized that with sine and cosine I could draw a circle!

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And then experimenting with those functions to draw cool parametric shapes on the calculator ^^ –  Thomas Mar 8 at 3:32

My first think of infinity was square diagonal vs. orthogonal stepping. Start with route (0,0) -- (0,1) -- (1,1), then take (0,0) -- (0,½) -- (½,½) -- (1,½) -- (1,1) and so on. Form of later will come closer to diagonal, but lenght will not.

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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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For me it was Monty Hall problem:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

I saw this problem when I was 15 year old. I answered correctly (I probably used some kind of math intuition), but I thought that probability in the second case is $1/2$. Actually it is $2/3$. The proof is beautiful, as well as the answer. This fact amazed me. Even now, at 18, I suppose it is quite a beautiful problem.

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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... - add comment I was always good at maths as a child, and took to reading extension maths books for fun (other kids thought I was weird). When I was about 10 I was completely hooked when I saw Euclid's proof for an infinity of primes. I had been given it as a question in one of the books I was reading. I spent about an hour desperately trying to prove it . . . then I looked at the solution - I was stunned by its elegant simplicity. Another thing I really enjoyed was finding cool facts about numbers in kids maths cartoon books and proving them. I loved to show WHY things always worked, that is perhaps my favorite thing about maths. - add comment I was hooked on math by a small side note in a kid's book of mathematics about perfect numbers, numbers that are twice the sum of their factors. For example, 6 is the smallest perfect number because 1 + 2 + 3 + 6 = 2 × 6 and 28 is the next one because 1 + 2 + 4 + 7 + 14 + 28 = 2 × 28. The next perfect numbers are 496, 8,128, 33,550,336, and 8,589,869,056. I was so fascinated by the idea that I proved that those numbers were perfect by listing out all their factors and adding them together. And to this day I wonder: somewhere up there in the vast expanses of integers, could there be an odd perfect number? I think that James Sylvester stated it eloquently: "...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle." Marcel Danesi, in his book The Liar Paradox and the Towers of Hanoi, stated it significantly less eloquently: "No odd perfect numbers have ever been found. They probably do not exist." - add comment The most wonderful thing I've recently seen is this (sorry it's in French) form of the sieve of Eratosthenes and of course your question too. - add comment 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 ! - add comment I was in elementary school, drawing 3D shapes in class while bored. I drew cubes by drawing two overlapping squares and connecting the vertices, like the top row in this image: Then I thought, what if I did the same procedure, but to a cube? So I drew four squares and connected the vertices, like this: I was struck by the beauty of the resulting image, with its intricate structure of star-like patterns. Here's a static version: It was years later that I discovered, to much fascination, that this was in fact the four-dimensional analogue of the cube: the hypercube. Hence my username. Edit: Another thing I remember thinking about when I was younger was that I could not always draw a straight line through three points, but was surprised to find that it would always work for two points. - This is really amazing. This is my idea of what is math all about. – Adam Sep 25 at 20:45 show 1 more comment 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 - add comment I don't find it beautiful, but I still find the idea expressed by the following something of a psychological curiosity: How can it be that when some algebraists say "AND" and "OR" they mean exactly the same thing? OR means this that "false or false" is false, "false or true", "true or false" as well as "true or true" are true, or more compactly:  F T F F T T T T  AND means this:  F T F F F T F T  But, since NOT(x OR y)=(NOT x AND NOT y) and NOT(T)=F and NOT(F)=T, OR and AND, to an algebraist, mean exactly the same thing! - show 2 more comments Thanks to @FacebookAnswers for suggesting Conway's Game of Life, a cellular automaton devised by John Conway in 1970. ## A Gosper Glider Gun With its patterns, oscillators, spaceships, glider guns (the minimalist Gosper Gun is shown above), breeders, Turing Machines, and the many derivatives, this "game" has spawnd much thinking and imagining. ## A generation$\approx 10^{28}$Turing Machine in Golly Of course it's a challenge to replicate the wonders in a static book, but there's great potential for the CD, ebook, or website. - add comment As others have mentioned, kids love$\pi$. Prime numbers are also good, if they have a good handle on division. I think the fundamental theorem of arithmetic is intuitively true once you understand it (at east it was to me). It would be great to mention some unsolved problems, like the twin prime conjecture or the Collatz conjecture. For me, one thing that I remember being fascinated about at an early age was the fact that multiplication is commutative. That$3+3+3+3+3=5+5+5$(or if you want, five baskets with three apples each is the same as three baskets with five apples each) was not immediately obvious to me, and the fact that it worked for any two numbers amazed me. Once you understand the geometric "square of dots" proof it makes sense, but I think that before that it doesn't. Knuth up arrow notation is worth mentioning. Kids love that multiplication is repeated addition and that powers are repeated multiplication, and would be interested to see that idea taken further. - add comment This was my favourite equation. I was 16 or so, when my father showed it to me. I was amazed and I programmed a application which drew this: The interval should be <-6;6> maybe. I made it looong time ago after all ;) - add comment For me, it was the discovery that the sum of the digits in all multiples of three are themselves multiples of three, and you can recursively sum them to get to 3, 6, or 9 (i.e. an 'easy' multiple of three) E.g. The sum of the digits in$13845$is$21$, The sum of the digits in$21$is$3$Edit: Should probably add that what made this useful to me was that numbers that are not multiples of three do not have this pattern. - Or, more generally, that the ultimate result of this digital summing is the same as the remainder when dividing by 9. – cobaltduck Mar 7 at 17:11 This generalises via arithmetic modulo 10 to produce tests for divisibility by other numbers based on similar properties of their decimal digits. – user3490 Mar 7 at 20:21 show 4 more comments As a child, the Fibonacci numbers$F_n$defined by the recurrence $$F_1 = 1, \quad F_2 = 1, \quad F_{n+2} = F_{n+1} + F_n$$ were very fascinating to me. It was fun to compute them $$F_3 = F_2 + F_1 = 1 + 1 = 2\\F_4 = F_3 + F_2 = 2 + 1 = 3\\F_5 = F_4 + F_3 = 3 + 2 = 5\\F_6 = F_5 + F_4 = 5 + 3 = 8\\\vdots$$ and to collect the results in tables $$\begin{array}{c|cccccccccc}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \cdots \\ \hline F_n & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & \cdots\end{array}$$ At some point, I asked myself the question: To compute$F_{10}$, do I really have to compute all the Fibonacci numbers up to$F_9$beforehand? So I tried to figure out some formula where you can plug in$n$, do some basic arithmetics, and get$F_n$as a result. I've spent a lot of time on this. However no matter how hard I tried, I didn't succeed. After a while I found the closed form $$F_n = \frac{1}{\sqrt{5}} \left(\left(\frac{1 + \sqrt{5}}2\right)^{\!n} - \left(\frac{1 - \sqrt{5}}{2}\right)^{\!n}\right)$$ in some book. I was paralysed. How can it happen that an easy recurrence formula can only be described by such a complex expression? Where do the square roots come from, and why does the expression still always evaluate to an integer in the end? And, most importantly: How on earth can one find such a formula?? - I first discovered the Fibonacci sequence by playing with an 8-digit calculator. There's a bug in most standard 8-digit calculators (the kind you get as promotional gifts, for example) so that if you press "1 + = + = + = ...", it will give you the Fibonacci sequence. – Joe Z. Mar 7 at 21:16 In order to find this formula, you usually will use linear algebra, and the property that linear recurrences can always be represented in matrix form. In particular, the two exponential terms correspond to the eigenvalues of the matrix$\left [ \begin{matrix}1 & 1 \\ 1 & 0\end{matrix} \right ]$, which, as you could probably guess, has a characteristic polynomial of$x^2 - x - 1\$. –  Joe Z. Mar 7 at 21:23
JoeZeng: Wow, that's a nice bug-abuse :) Nowadays I know, of course, how to derive the formula. Besides the linear algebra method you indicated (my favorite), there is also a standard way to do it by generating functions. –  azimut Mar 7 at 21:32
There's a way to derive this using only plain algebra, which would be nicer for a children's book. Let a = (1+sqrt(5))/2 and b = (1-sqrt(5))/2. Show that a^2 = a + 1 (and similarly for b). Multiply by a and simplify to get a^3 = 2a + 1, and show that you can repeat this to get a^n = F(n)*a + F(n-1) (and similarly for b). Take a^n - b^n, simplify and rearrange and you're done. –  Michael Shaw Mar 8 at 5:55