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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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Nice question, but should probably be community wiki? –  mrf Mar 7 '13 at 7:02
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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

138 Answers 138

Not sure if it was the first, but one very early one for me was realizing if you knew the square of an integer, you could easily step to the next one by adding the known square, the original integer and the next integer together.
Know $2^{2}$
Want $3^2$

$3^2 = 2^2 +2 +3 $
$3^2 = 4+2 +3$
$3^2 = 9$

or $(n+1)^2 = n^2 + n + (n +1)$

Another although much later point for me was when Calculus just clicked (4th time taking Calc 1). It was like cracking The Matrix.

I could see derivatives and integrals in everything around me and the relationship between the trig functions suddenly made sense.

Also the magic of Fourier and Laplace transformations.

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I discover that math is beautiful again and again, I suspect there is no end to this. Some of things that blew my mind:

  • The very first thing that hooked me to science in general was the ability to model reality. When I was a child my father showed me how differential equations (by then I had no idea what it was) could simulate reality (it was masses connected by springs and dampers with some simple graphical representation).
  • Prime numbers. The natural numbers are intrinsic to the world and so a prime numbers. If there is a sentient alien race, they are aware of prime numbers (or they will be someday).
  • The numbers $e$ and $\pi$, their interconnection, and importance in both continuous and discrete worlds.
  • Fourier transform (both continuous and discrete). There is no denying: Fourier transform is just awesome, the sheer number of applications, implications, similar transforms, etc. speaks for itself.
  • The 3D proof of Desargues' theorem. It's one of those illuminative cases where considering a harder case simplifies the problem.
  • The probabilistic method. The first time I saw it, it was very inspiring, also the person of Paul Erdős is very inspiring too (and funny, e.g. the title of "Permanent Visiting Professor").
  • Randomness, randomized algorithms, and the P vs. NP problem. There is something incredible in the fact that for some problems the only fast solutions we know are randomized, any known deterministic approach is way slower (the canonical example being checking if the symbolic determinant of a matrix containing variables is zero).
  • ... this ever-growing list continues...

Cheers!

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$$\sqrt{\sqrt{\dotsb\sqrt{x}}} = 1$$

(or its more precise version: $lim_{n \rightarrow \infty} \sqrt[n]{x}$, for x positive)

As a kid I would always type in a number in my calculator and then keep hitting the square root key until the display went to 1. I would also do this with other keys on the calculator to see what would happen (some would blow up past the capacity of the floating point storage and some would go to 0, some to 1).

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Actually $\sqrt[n]{x}\to 1$, as $n\to\infty$. –  Asaf Karagila Mar 8 '13 at 21:15

I remember reading a magazine when I was a kid that asked this question. If you have 6 pieces of spaghetti that extend as long as you please, and you cross them so that they create as many overlaps as 6 sticks allow. How many cross points do you have. Then it asked how many would it be for 17 spaghetti sticks, could you figure it out for any number of spaghetti?

And I remember concluding that $\frac {n(n-1)}2$ is the formula for finding the answer. I was excited at the time. Looking back now I see how elementary that was.

Here is a visual way to see it: enter image description here

So, I started by drawing out a strand of spaghetti assuming that they could be as long as you please and as thin as you please, then I start with one and work my way up to 5 counting how many times they cross.

enter image description here

So then for 5 sticks of spaghetti I labeled all of the crossings. I did this for 6 as well just to see what was happening. I noticed that the number of crossings on each strand of spaghetti was the number of total spaghetti - 1 because it didn't cross itself. So from now on I will refer to the number of spaghetti as n. So to count the number of crossing I knew it was $n-1$ crossings for every stick and $n$ sticks so the total number of crossings was $(n-1)n$ and I noticed that each crossing occurs on two separate sticks, because one crossing is the crossing of two sticks to the total number of crossings is half of the number in the diagram so it was $\frac 12 (n-1)n = \frac {n^2-n}2$

P.S. sorry for using the words crossing and sticks instead of points and lines. It was something that stuck because of the spaghetti analogy in my head. I didn't realize I was doing it until it was too late.

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Another different but related question is how many pieces of a round pie can you cut using $n$ straight cuts. The answer turns out to be $\displaystyle \frac{n(n+1)}{2} + 1$. –  Joe Z. Mar 13 '13 at 20:27

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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Complex numbers was awesome to me

        i² = - 1
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I didn't so much discover mathematics was beautiful as I discovered everything I found beautiful was mathematics. I would have said the dodecahedron was my favorite elementary shape when I was little, but as a teenager I was exposed to the fourth dimension. I became obsessed with symmetry and analogy-based objects, and with the Johnson solids, which gave me a then-ineffable feeling of filling out the quality of symmetry that it always creates good shapes, that there are rules you can make that name exactly the set of good shapes. That "looks right" and "looks wrong" can be made precise, and hence the feeling can be explained, you can learn what it is you're noticing about those shapes that you wouldn't feel when you look at a 3D mesh of a face or a blanket.

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Pythagorean theorem

If $(a, b, c)$ is a Pythagorean triple, then so is $(ka, kb, kc)$ for any positive integer $k$.

$3^2 + 4^2 = 5^2$

Pythagorean triples

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In my school when I learned about Cartesian coordinate system was shocking! Because that was the time I learned that was possible to make drawings with numbers.

Unlike most people here, I didn't have so much fun playing with numbers, but everything changed when I realized that I could convert numbers (more precisely, ordered pairs) in drawings over the coordinate plane.

And YES, that was a lot of fun.

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If I wrote a book, a few pages would be dedicated to visualizing square root through blocks like this:

square root blocks

A kid can put cards on a table and count the edge rectangles to figure out the approximate square root of any number. With the help of some legos you can even teach cube root!

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The possibilities of abstraction. This liberated me. Until I was about 13, I always had trouble with solving problems involving proportionality and inverse proportionality. Until I learned about variables. When I realized that you could just put a symbol instead of the number you don't know and just perform computations with it until everything simplifies in a way you can find back the number, I had a feeling of unstoppability.

The power of abstraction is so great that I'm very saddened by our current educational system in which it has nearly disappeared. All the students I get are struggling with symbols there were I have always seen them as my friends.

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The fact that you can add natural numbers successively in the order you prefer and that you can split subtraction:
enter image description here ...I remember that day when got taught this in class which made me really excited so I had to tell my Mum =D

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It's not the first one that made me love math (what made me love math isn't math itself at all, but rather someone pointing out to me that I was pretty good at math -- and then I proceeded to like math haha), but this is the most amazing discovery I made when I was 15.

$$ Pr(X = r) = \frac{(nCr)(x-1)^{n-r}}{x^{n}} $$

Which is really just a restatement of the binomial distribution:

$$ Pr(X = r) = (nCr)(p^{r})(p^{n-r}) $$

where $p = 1/x$, so it works only makes sense for integer values. For example, the chances of choosing one blue jar out of 10 differently colored ones would be $x=10$, but also $p=0.1$.

I discovered it after one week of exhaustively listing down all the permutations of the letters n, t, g, and b and figuring out what patterns they looked like when you took only 1, 2, 3, and 4 elements. Then I went ahead and added more and more letters until I arrived at that formula by inspection.

In my opinion, it isn't the math itself that makes kids dislike math. It's all the people around them who dislike math who make kids dislike math.

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If you've ever heard of $3,529,411,764,705,882$ being multiplied by $3/2$ to give $5,294,117,647,058,823$ (which is the same as the 3 being shifted to the back), you might consider including that in the book.

There are lots of other examples, like $285,714$ turning into $428,571$ (moving the 4 from back to front) when multiplied by $3/2$, or the front digit of $842,105,263,157,894,736$ moving to the back four times in a row when you divide it by $2$. (There's a leading zero in front of the last term, though.)

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Or without some good ol' pencil and paper. –  Joe Z. Mar 7 '13 at 15:51
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Here's the trick: Take the decimal expansion of 1/(any prime number). That will give you lots of ratios to work with. –  Joe Z. Mar 7 '13 at 15:54

My two favorite mathy things (not mentioned in other answers) were the powers of 11

11 ^ 0  = 1  (1)
11 ^ 1  = 11  (1, 1)
11 ^ 2  = 121  (1, (1+1), 1) 
11 ^ 3  = 1331 (1, (1+2), (2+1), 1)
11 ^ 4  = 14641 (1, 1+3, 3+3, 3+1, 1) 
11 ^ 5  = 161051  (1, 1+4, (bump 1 to left) 4+6, (bump 1 to left) 6+4, 4+1, 1)
11 ^ 6  = 1771561 (1,1+6, 6+1, 1+0, 0+5, 5+1, 1)

and the estimated relationship btwn powers of 2 and powers of 10 and how they diverge (hard drive manufacturers think this is beautiful :P )

2^10 ~= 10^3 (1024, 1000) "kilo"
2^20 ~= 10^6 (1048576, 1000000) "mega"
2^30 ~= 10^9 (1073741824, 100000000) "giga"
2^40 ~= 10^12 (1099511627776 , 1000000000000) "tera" 
2^n ~= 10^floor(n/3) (where n is a multiple of 10)
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If you look closely, the first few powers of 11 are rows of Pascal's Triangle. This is not a coincidence. –  Joe Z. Mar 8 '13 at 20:16
  • Like Trevor Wilson, I was awed by human ingenuity where just by looking at "few" given numbers, one can deduce that there are infinitely many primes from reason alone and doing basic operations. (Here is more on Euclid's proof.)

  • As freshman undergrad in college, always loved Theoni Pappas' Joy of Mathematics before being introduced to Raymond Smullyan, Charles Seife (Biography of Zero), Rudy Rucker and Hoftstadter's books.

  • As far as Math.SE's question is concerned, this was an interesting brain teaser and simplicity at best.

===================EDITED THE FOLLOWING BELOW==================== I just realized although the above have been influential, but earliest memory of the workings of mathematics came in the manner of following magic trick aged six or seven:

Effect: Performer asks someone to write down a random long number 4567829872367783456753745673456347567346534756 and he writes the another line of the matching digit, so let's say she writes 1263347567346534756378567563434543534543534545 and after that performer asks another audience member to approach the blackboard on dimly lit stage. Let's say the next random line 8636652432653465243621432436565456465456465454 and random number and as he writes 5555555555555555555555555555555555555999999990 the performer quickly writes below 4444444444444444444444444444444444444000000009 and then pauses. Then he continues his patter: Now I could not have possibly known what digits you would have chosen, right ladies and gentlemen? Well, let me gather my thoughts for a while and clear my mind....as I attempt to add this rather cumbersome mess in just the time of writing it down. Then he approaches the blackboard and without hesitation calculates the answer:

24567829872367783456753745673456347567346534754

which of course proves to be correct.

Method: There is of course no telepathy and the trick is entirely mathematical in nature. If the reader wants the audience to choose the first line make sure the last digit does not end in 0 or 1. So what the practitioner would do is matching the digits of audience write the complement of the number adding to 9. Say the line is of a 10-digit sequence of 5s then the performer should write a 10-digit sequence of 4s. To add the whole block one simply copies down the first line with 2 in front of it and subtracting 2 from the last digit. Hence the need for no 0 or 1 in the first line.

Tips: To make it realistic, make sure it is a cumbersome mess that is not too big of a block. Because say one smart aleck chooses all digits of 0000... and then when performer writes 999999... it may be a give away. Strike a balance between how big the mammoth block should be to appeal awe from students and the reality of randomness in the numbers. The rest is, of course, all up to the showmanship of mentalist.

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+1 for the first bullet point! I had the same enthralling reaction to the Euclid's proof that there are infinitely many primes. It was the first time in my life that I realized there was more to mathematics than mindless calculation, and it fundamentally changed the course of my life. –  user5501 Mar 7 '13 at 9:58

I found a book by Isaac Asimov called "Asimov On Numbers" which is a compilation of his essays related to math and numbers. It was all very fascinating - things like why Roman Numerals are inefficient, why Zero was such a groundbreaking number to invent, things like that. You might want to see if you can find that book to get some inspiration.

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In the elementary school, when I was learning about maps on the geography lessons, I was amazed by the concept of the scale of a map.

It might sound silly now but I was very happy when I understood the relation between ratios of distances and ratios of areas (and ratios of volumes:).

It somehow provoked me to thinking about what length, area and volumes really are, how to define them. And how to define what a map is.

Of course I got familiar with precise definitions much, much later :)

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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

When I was in second grade we memorized the times tables through $9$. At the very end of the year, our teacher taught us simple single-digit division. I was floored: "We can reverse multiply?!?!"

I think that got me to pay more attention in math. The first thing that really cemented my love of math was learning set theory in seventh grade (widely reviled as "the new math" by parents and politicians in the U.S.). I wasn't hooked for life until 11th grade when we were given the definition of a relation as a subset of the cartesian cross-product between two sets. I still remember getting chills when I understood that.

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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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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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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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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 '13 at 3:32

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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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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The fact that $\Bbb C$ is algebraically closed.

About 12 years old, after I just learned about quadratic equation such as $x^2=a$ may or may not have solutions, my mother told me about complex numbers: you attach the number $i=\sqrt{-1}$ to real numbers and after that $x^2=-1$ have solutions.

"Nah", I said, "that doesn't help much: although you now have solutions for $x^2=a$ for $a$ in the old number system, which are reals, you still don't have solutions of $x^2=a$ in the new number system, which are complex. You still don't have a solution of $x^2=i$, for example. And having complex solutions for some of the complex numbers is no better than having real solutions for some of the real numbers."

Then she showed me the roots of $x^2=i$ and explained that $x^2=a$ has complex solutions for any complex $a$. The ingenuity of the complex numbers impressed me a lot.

Then she told me about polynomials of degree higher than 2 and that they all have roots in the same field of complex numbers, that you don't need to "attach" $n^{th}$ root of $-1$ or any other number in order to have any polynomial of degree $n$ have roots, that $\sqrt{-1}$ is sufficient for them all. And I was impressed even further.

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These amazed me quite a lot when i first saw them:

$1.$ Prove that $|(a,b)| =|\Bbb R|$, $\forall a,b\in\Bbb R$ and $a<b$.

$2.$ Both $\Bbb Q$ and $\Bbb R\setminus \Bbb Q$ are dense in $\Bbb R$, but $\Bbb Q$ is countable set while $\Bbb R\setminus \Bbb Q$ is uncountable.

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At what age?${}$ –  mrf Mar 7 '13 at 7:42
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My brother used to tell me these kind of things from an early age. –  Aang Mar 7 '13 at 11:15
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You learned about dense, countable and uncountable sets at that age?? Its hard to believe...and its very strange that you have to get so far to see something you considered beautiful. –  Integral Mar 7 '13 at 15:23
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@tttppp … I read $(a,b)$ as an tuple and completely forgot it could also denote an open interval (which I write $(a..b)$ now) – even after searching for different meanings! It’s times like this that I wonder whether I have some serious brain condition. –  k.stm Mar 15 '13 at 15:51

I remember in geometry using direct reasoning once and another by the absurd, I manage to show that lines are parallel, intersecting at a point, a triangle is isosceles, it is inscribed in a circle ..... I was fascinated by geometry.

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I think the first think that amazed me in this way is $\pi$. 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 Greeks discovered it, they knew it has something to do in the circumference or the area of a circle, i.e. they could manipulate it, and I find this unbelievable.

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