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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

153 Answers 153

Complex numbers was awesome to me

        i² = - 1
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$$\lim_{n\to\infty}\left(1+\frac in\right)^n=\cos1+i\sin1$$which can be proven geometrically. Expanding it out with the binomial theorem gives us series representations of the cosine and some of 1. I thought I'd share it with you, because it's fairly amazing. –  columbus8myhw Nov 13 '14 at 2:57

Pi has always fascinated me. The notion that perimeter of every possible circle imaginable divided by its diameter always results Pi is astonishing.

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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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As a child, I liked drawing.

When I realized that there was an easy way of telling whether it is possible to draw a given figure in a single stroke, I was intrigued.

I read this in a popular mathematics book and it can be easily explained to a child.

(if there is 0 or 2 intersection with odd degree, the figure can be drawn in a single stroke)

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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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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

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

(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

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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I must have been very small, around three of four, when I suddenly dashed out of my room, full of excitement, wanting to show my dad something that had made a great impression on me.

I held a book, it's front cover facing me.

enter image description here

In a flash, I gave it two half-turns. One upside-down, the other left to right. This is what came out:

enter image description here

I held my breath, as the trick wasn't over yet. Sure enough, two same quick moves and -lo and behold- the front cover was facing me properly again, just as in the beginning.

"Look! Dad!" :)

That surely must have been my first conscious encounter with symmetry.

I held the memory dearly close for a number of years but then forgot about it completely. It came back to me, only very recently, after going through the first pages of Nathan Carter's Visual Group Theory and seeing this image:

enter image description here

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Parallel lines. I was amazed to find out that they would never, ever meet.

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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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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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To me, it was probably an old animated book "Och ta geometria" (eng. "Oh that geometry") written and illustrated by Zlatko Šporer, Nedejko Dragić. In the form of funny comix (check the link for samples), this book introduces basics of geometry from points, segments (not sure if this is the correct name), lines, flat figures and their area to cartesian coordinate system. This was probably the catalyst for my interest in 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
  • 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

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

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2  
And then experimenting with those functions to draw cool parametric shapes on the calculator ^^ –  Thomas Mar 8 '13 at 3:32

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 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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Tupper's self-referential formula is a self-referential formula defined by Jeff Tupper that, when graphed in two dimensions, can visually reproduce the formula itself. It is used in various math and computer science courses as an exercise in graphing formulae.

The formula was first published in his 2001 SIGGRAPH paper that discusses methods related to the GrafEq formula-graphing program he developed.

The formula is an inequality defined by: $${1\over 2} < \left\lfloor \mathrm{mod}\left(\left\lfloor {y \over 17} \right\rfloor 2^{-17 \lfloor x \rfloor - \mathrm{mod}(\lfloor y\rfloor, 17)},2\right)\right\rfloor$$ where $\lfloor \cdot \rfloor$ denotes the floor function and $\mathrm{mod}$ is the modulo operation.

Let k equal the following 543-digit integer:

960 939 379 918 958 884 971 672 962 127 852 754 715 004 339 660 129 306 651 505 519 271 702 802 395 266 424 689 642 842 174 350 718 121 267 153 782 770 623 355 993 237 280 874 144 307 891 325 963 941 337 723 487 857 735 749 823 926 629 715 517 173 716 995 165 232 890 538 221 612 403 238 855 866 184 013 235 585 136 048 828 693 337 902 491 454 229 288 667 081 096 184 496 091 705 183 454 067 827 731 551 705 405 381 627 380 967 602 565 625 016 981 482 083 418 783 163 849 115 590 225 610 003 652 351 370 343 874 461 848 378 737 238 198 224 849 863 465 033 159 410 054 974 700 593 138 339 226 497 249 461 751 545 728 366 702 369 745 461 014 655 997 933 798 537 483 143 786 841 806 593 422 227 898 388 722 980 000 748 404 719

If one graphs the set of points $(x, y)$ in $0 \le x < 106$ and $k \le y < k + 17$ satisfying the inequality given above, the resulting graph looks like this (note that the axes in this plot have been reversed, otherwise the picture comes out upside-down):

enter image description here

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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.

I was granted immunity from this idea at a very young age.

When I was in 3rd grade (possibly 4th, my memory is a little fuzzy) my father showed my sister and I knotplot (http://knotplot.com/), a piece of software that one of his friends was working with. He explained that his friend was a knot theorist, a kind of mathematician who studied knots.

I had no idea what that really meant, of course. All I knew was that studying knots was apparently a thing that mathematicians did. It wasn't a lot, but it was enough. Whenever the North American math curriculum tried to trick me into thinking that math was about numbers, my brain would reply "You say that... but what about knots?" It was my vaccine.

It also helped that my physicist father insisted on teaching us the interesting bits of math whenever it seemed appropriate. He told us about negative numbers the moment we learned about subtraction (I then obnoxiously quizzed my classmates on the playground. "What's $5-6$?", "Zero?", "Nope! Minus one." It's a wonder I didn't have more friends...). He showed us imaginary numbers when we learned about squares (the mysterious $i$ became another mental vaccine.) When I mentioned $\pi$, he countered by telling me about $e$.

The thing about math is, once you have a general curiosity, you'll start discovering interesting things all over the place. Once that starts happening, you're doomed. Math will never be boring again.

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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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7  
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, and 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 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, and things like that. You might want to see if you can find that book to get some inspiration.

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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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