Decimals are pretty basic stuff. You should be able to easily breeze through this (especially with a calculator! :)).
Decimals are another way of expressing a part of a whole. Instead, they are using the ordinary number system. Each additional decimal digit represents significantly less, so for example pi is a number which goes on forever yet even engineers only really need to know it to 10 digits.
| $$10^0$$ | $$10^{-1}$$ | $$10^{-2}$$ | $$10^{-3}$$ | $$10^{-4}$$ |
|---|---|---|---|---|
| $$1$$ | $$\dfrac{1}{10}$$ | $$\dfrac{1}{100}$$ | $$\dfrac{1}{1000}$$ | $$\dfrac{1}{10000}$$ |
| $$1.$$ | $$2$$ | $$3$$ |
Just like place values for the tens and hundreds column, these get smaller (instead of increase) by a power of 10 each time. Each decimal digit represents a fraction, for example the 2 in 1.23 represents 2 out of 10 or:
$$ \dfrac{2}{10} $$
Place values allow you to order decimals very easily, all you have to do is compare the highest (different) digit. For example, let’s try and order these decimals in ascending order of size:
$$ 5.4, 3.45, 6.75, 8.99, 5.41 $$
We look at the highest digit, if they are the same then we look at the next digit and the next until we see a difference. Now we can rearrange them really easily:
$$ 3.45, 5.4, 5.41, 6.75, 8.99 $$
Another way to convert a decimal to a fraction is by taking the decimal and dividing it by it’s place value. So for example we can find the fraction of 0.65:
$$ 0.65 = \dfrac{6}{10} + \dfrac{5}{100} = \dfrac{60 + 5}{100} = \dfrac{65 \color{red}{\div 5}}{100 \color{red}{\div 5}} = \dfrac{13}{20} $$
We can convert recurring decimals, let’s try converting 0.277… (the three dots means it repeats forever, or a dot above the number can mean the same thing!) into a fraction. To start with let’s make the number equal to a letter:
$$ x = 0.27\dot{7} $$
Now we need to get all of the non-repeating part of the number over to the right of the decimal and we can do this by multiplying both sides (if you know algebra you know you must do the same to both sides of the equation!):
$$ \color{red}{10 \times} x = 0.27\dot{7} \color{red}{\times 10} = 2.7\dot{7} $$
We now need to find another multiple of the original number that we can subtract our new number to get a whole number! We can do this simply by making a new multiple of the number where all the non-repeating part is on the left but also some repeating part:
$$ \color{red}{100 \times} x = 0.27\dot{7} \color{red}{\times 100} = 27.7\dot{7} $$
We can now subtract the new multiples of the original number we made to leave us with a whole number and a multiple of the original number. We are able to ignore all of the repeating bit because we are subtracting them so they will be nothing in value or 0. Instead we subtract 2 from 27 :
$$ (100 \times x) - (10 \times x) = 27.7\dot{7} - 2.7\dot{7} $$ $$ \dfrac{(90 \times x)}{\color{red}{90}} = \dfrac{25 \color{blue}{\div 5}}{\color{red}{90 \color{blue}{\div 5}}} = \dfrac{5}{18} $$
More being added soon…