At Sixes and Sevens

How to Understand Numbers and Make Maths Easy

Rachel Riley

• Editore: HarperCollins
• Anno: 2021
• ISBN:
  9780008491086
• Argomento: Mathematik
• Categoria: Mathematik-Referate

CHAPTER 1

TOOLS OF THE TRADE

ABSOLUTELY MENTAL ARITHMETIC

When most people think of maths, the first thing they think of is numbers. So, if we wanted to go all Julie Andrews à la The Sound of Music, and start at the very beginning, with the ABCs or do-re-mi’s of maths, we begin, of course, with one, two, three.

It would have been fitting for Julie to have sung about these counting numbers, as she did in the film, surrounded by the giant fir trees, babbling brooks and female deer of the Austrian Alps, as they are known officially as the ‘natural’ numbers, precisely because they appear in nature. So, you’ll have one dog, two flowers, three bees, four bottles of wine, etc – and these are all known as natural, counting or cardinal numbers.

Numbers have many other inherently natural properties too, and display clues that tell us about themselves just by looking at them. We use these different properties to separate numbers into different groups – such as into odd and even numbers, or into primes and non-primes.

Unlike almost everything else in life, numbers are unlimited. They are not something you can ever run out of. They go on forever, since you can just keep on adding 1 to make them bigger and bigger, until you get bored or fall asleep. In fact, even the maths symbol for infinity, a theoretical number that is always bigger than any other number, looks like an 8 that has given up the will and passed out under the tedium: ∞.

THE POWER OF PRACTICE

Everyone can add 1 to a number and quickly tell you the answer. But what about adding 10? Or 100? Or 5000?

When it comes to mental arithmetic it’s all about practice. When applying for my job on Countdown, I literally had to relearn my times tables from scratch. I’d just finished four years studying maths at Oxford, in which we’d barely seen numbers in lieu of algebra and Greek notation. In fact, numbers took on such a supporting role that my college friend once asked what that funny symbol on the board was that looked like a backwards epsilon (ε)? Reader, it was a 3.

Also, whereas practice might not always make perfect, when it comes to mental arithmetic, practice certainly does improve your confidence and speed of thought. And then, once you feel confident manipulating numbers at will, you can use those skills to solve real mathematical problems – rather than the mental arithmetic itself being the mathematical problem.

You should treat your brain like a muscle, in that exercising it will keep it in shape, so use the numbers that pop up all around you as your training tools. When you’re out shopping you can try and decide how much things are going to cost in your head before it’s worked out for you. When you’re hot and sweaty, staring down at the display on a treadmill, or bored at work and are digital clock-watching, or just walking down a street past all the shiny door numbers, use those digits to practise your tables, or add them up, or work out which numbers divide into them. Spend a bit of time thinking about that number and work out what you know about it just by looking, as per some tips coming later in this chapter. Not only will it help your maths skills, but it will also keep your brain active and make you a bit sharper to boot. If you have a family, get your children involved in practising along with you too!

IT ALL ADDS UP

Let’s say you want to add a couple of numbers together, such as 28 + 37. What’s the best way to do that? A sum like this isn’t something most people can do without a bit of thought.

To find the easiest method of doing this mentally, it helps to think literally about what the numbers actually mean. The number 28 means one plus one plus one plus one … with 28 ones added up. Likewise, 37 means 37 ones added up, so 28 + 37 is whatever 28 + 37 is, number of ones all added together. And the thing about addition is that you can add numbers in any order you like, and it will still give the same result, i.e. 28 + 37 = 37 + 28 = any grouping of the 37 and 28 ones you like, added in any order. So, why not regroup the ones you’re adding up to make an equivalent addition, with numbers that are more friendly to think about in your head.

For example, you can take 2 off 37 to move it over to the 28, to leave the much more manageable sum of 30 + 35.

Working with round numbers reduces the amount of brainpower required, which suits mathematicians perfectly – we are essentially the laziest group of people I know. The aim of the game is to do the minimum amount of work possible – the less thinking, the better!

Additions such as this can be made much easier, due to the commutative nature of addition. This is the fancy way of saying that you can put the numbers in any order and you still get the same result.

This is probably something you will have done with wooden blocks when you were a toddler, without realising that you were already thinking like a mathematician. If you had 10 blocks, for example, it was easy to see that 10 could be formed by adding 5 blocks + 5 blocks, or by adding 1 block 10 times. At school, the same concept is known in terms of ‘number bonds’, so children become familiar with breaking up numbers into multiple smaller numbers that add up to the same value – so, 2 and 8 are number bonds that make 10. There’s also the idea of ‘chunking’, which is when you divide a bigger number up into smaller, easier parts, such as breaking 56 into 50 + 6.

It’s always worth thinking about what the numbers on the page actually represent, since you can shift them around however you like to make it easier in your mind. You don’t have to go straight for adding two awkward numbers together.

In fact, I’m often asked how I ‘see’ numbers myself when I’m doing calculations, with some people expecting a kind of magical answer in te...