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The Magic of Maths

Solving for x and Figuring Out Why

By Arthur Benjamin
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  • Contains 9 key ideas
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The Magic of Maths by Arthur Benjamin

The Magic of Maths (2015) reveals the magic hidden within the fascinating world of mathematics. These blinks will show you the beautiful and often surprising patterns of mathematical observations, expand your knowledge of geometry and algebra, teach you numerical party tricks and illuminate the mysterious properties of numbers like π (pi).

Key idea 1 of 9

Numerical patterns are not only magic – they can lead to real applications too.

Mathematics is more than just tedious textbooks and laborious calculations; it is a whole world of patterns that are not only magical, they can be really useful too.

Consider the so-called numerical patterns – patterns made of numbers – and their surprising and beautiful properties.

The author first discovered these patterns as a child, when he was playing with pairs of numbers that add up to 20:10 and 10, or 9 and 11, for instance.

He wondered: what’s the largest possible product I can get by multiplying these pairs together?

Let’s find out:





So, the largest product is when both numbers are 10. Nothing extraordinary, right?

But if you look closer, there’s something interesting about these numbers. Examine how far each product is from 100, counting downwards, and you’ll notice a pattern: 0, 1, 4, 9. These are the first square numbers, that is, the numbers that follow the sequence 1², 2², 3², and so on.

This pattern applies all the way up and down the scale: if we calculate 5 x 15, we can get back to 100 by adding 5². And what’s more, the same pattern emerges no matter what number the pairs add up to!

These numerical patterns aren’t only magical, they’re also useful in the real world. If we can learn their secrets, we can use them to increase the power of our mental arithmetic, that is, the calculations we make in our head.

For example, we can use the above pattern to easily calculate the square of a number.

Say you want to square the number 13. Instead of calculating 13x13– a nightmare to do mentally – we can perform the easier calculation 10*16 where both numbers add up to 26, just like 13 and 13.

Now we’ve got 10x16=160, but we’re not quite there yet. Our previous pattern tells us that since we went up and down 3 from 13, we need to add 3² to the result. Thus we get 13²=(10*16)+3²=160+9=169.

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