More than just maths: “1089 and All That”

[dropcap]T[/dropcap]ake a three-digit number where the first and last digits vary by more than two. Now flip it, and subtract the smaller number from the bigger number. You should end up with another three-digit number. Now, add that number to its reverse. You should always end up with 1089.

[pullquote style=”right” quote=”dark”]This book is a nugget of pure mathematical gold.[/pullquote] This is how David Acheson begins 1089 and All That, his 177-page journey into the world of mathematics. It provides an exciting, accessible and enjoyable tour through theorems, beautiful proofs, and great applications.

Acheson starts simply, with the basics of geometry like Pythagoras’ theorem that many people will remember from school. He takes us through algebra, Kepler’s laws and calculus, all the way through to imaginary numbers. If the mere mention of maths makes you want to steer as far away as possible from this book, I promise you that this is a world away from maths lessons at school. This book made me go from despising mathematics to studying both maths and further maths at A-level, and eventually going on to study physics. Suffice to say, it’s pretty good.

[pullquote style=”right” quote=”dark”]Real-life examples are used to show that the abstract mathematics from school does have purpose.[/pullquote] Particularly interesting is chapter five, which discusses the movement of the heavens, and how Newton pinned down the ellipsoidal motion of the planets. Acheson talks about how Edmund Haley (of comet fame) visited Newton in order to discuss planetary motion, but he also manages to fit in a bit of background on Newton. Far from the collected scientist we think of today, we discover that Newton’s students almost never turned up to his lectures because no one understood him, and that even when he did manage to make it to the dining hall at Cambridge, he never managed to brush his hair.  It’s the personality that Acheson conveys in his books, with the cartoons throughout and quirky stories like Newton’s that make 1089 and All That such a compelling read.

Having read many books on maths and physics (in my fresh-faced, enthusiastic A-level days), Acheson’s is by far the most accessible. Complex ideas are broken down into understandable steps, but you won’t ever feel patronised. There are even animations available on Acheson’s website to support some sections, and he points you in the right direction at the appropriate moments. At the rear of the book, there are further reading suggestions included, with miniature reviews so that you’re sure to pick up the right book for you.

Acheson proves the 1089 trick about a quarter of the way through the book, but the whole book leads up to proving something far more amazing (I’ll leave that for you to read). However, I’d like to imitate my favourite book, so I’ll finish by proving the trick I started with (and show you that algebra can be quite exciting):

If we write each digit of your number as a, b and c, then your number is:

100a + 10b + c

and the reverse of this number is:

100c + 10b +a

If we subtract these two, we get:

(100a + 10b +c)-(100c+10b+a) = 99a-99c = 99(a-c)

Since a and c are both whole numbers, we will always get a multiple of 99. The three digit multiples of 99 are:

198, 297, 396, 495, 594, 693, 792, and 891

Now, we can see that the first and third digits of each of these numbers add up to 9, and the second digit is always 9, so:

a + c = 9, b=9

We can therefore write the sum of the number and it’s reverse as:

(100a + 10b + c)+(100c + 10b + a) =

100(a+c) + 20b + (a+c) =

(100*9+20*9+9) =

1089

Throughout the book, he works up to proving what is considered to be the most beautiful result in all of mathematics. At that moment, where all of the pieces fit together, you realize just how clever this book is. I encourage everyone, mathematicians and math-haters alike to pick up this fantastic book and follow Acheson on his journey into mathematics.