Everyone remembers learning about decimals at school. Even if you don’t remember much about it, you will recall the long lists of numbers and the strange “moving” of the point when decimals are multiplied or divided by any multiple of 10. It can seem complicated at first to work with decimals, but if you can remember a few basic concepts and a couple of rules, it becomes a lot easier.
Where should you start?
Decimals are just fractions. A number with a decimal point and any other number after it is a whole number “and a bit”. So 23.5 is 23 and 0.5, or twenty-three and a half. 42.0 is just 42.
So the reason they are called “decimals” is that they are a way of putting fractions into the place value system. When children start school, they learn about “tens and units” – 835 is made up of eight hundreds, three tens and five units, for example. Each digit in a number “stands for” something, such as thousands, units, or millions. Decimals express fractions in exactly the same way.
As you look from left to right at a number, the quantity represented by the digit gets smaller by a factor of 10 each time. The decimal point is the dividing line between whole number and fraction: to the right of the decimal point the number being represented is less than 1.
To take an example: A police officer measures a car’s speed at 74.345mph.
At first sight this appears to be a complicated number. The whole number, to the left of the decimal point, is 74. There are 74 “and a bit” miles per hour. The fraction part of it is three tenths, four hundredths, and five thousandths. With each step to the right the digits get ten times smaller, starting with tenths, which are ten times smaller than units (0.6 x 10 = 6, for example).
How can we do calculations involving this number?
If the speed limit was 70mph, the police could very easily calculate by how much the car was breaking the limit. Write 70mph as 70.000, to match the digits in 74.345, and simply take the smaller number away from the larger, using the normal rules of subtraction but remembering where the decimal point is. The answer is 4.345mph. If the police were testing the car, and needed it to be 10.5mph faster, then they could simply add the two numbers together. As before, write 10.5 as 10.500, to make the calculation easier to perform by hand, and add normally. The answer is 84.845mph.
One thing that the use of decimals really enables (compared to “ordinary” fractions) is easy multiplication or division by multiples of 10. Since they express a quantity in base-10, multiplying by 10 moves everything a space to the left: 34.21 x 10 = 342.1 since every digit in the number is now ten times greater than it was. Division by ten is the same process but in reverse: 34.21 ÷ 10 = 3.421, since each digit in the number is now ten times smaller.
Multiplying more complex decimals is simpler than you might imagine. Say you had to multiply 4.56 by 9.31. Set it out like a normal long multiplication and do the calculation. When you have the answer, count the number of digits that are to the right of the points in the two numbers you have used – in this case, .56 and .31. This makes four digits. Go back to your answer (424536) and, starting from the right, count four places to the left. That’s where the point goes. So 4.56 x 9.31 = 42.4536 or 42.5, rounding up to one decimal place. The answer always has as many digits after its decimal points as the two numbers you start with put together.
With division, again perform the calculation in the same ways as division of whole numbers. Set it out as a “bus stop” division, with a line separating the two numbers. This time, check where the decimal point is in the larger number (the dividend). Above the decimal point, put a new point, so that your answer will be a correct decimal. Try it with 85÷4. The answer is 21.25. Remembering to put the decimal point in exactly the same place above the dividend, 8.5÷4 will be 2.125 (ten times less).
A more difficult question would be something like 86.2÷4.3. Now you have decimal points in both numbers. The important thing to do is to remove the points by multiplying both numbers by the same amount – as long as you do the same thing to both numbers, it won’t affect the answer. In this case, multiply both by 10 to create 862÷43. Divide in the usual way!
Decimals are easier to work with than many people think. They allow very precise measurements to be made and it is useful if you are doing any kind of home improvement work, building or even shopping to be able to perform basic calculations with decimals. It is a good idea to practise some of the calculations first, so that you can feel more confident before you need to use them.
