Decimals and fractions are very closely related. Both can be used to express and manipulate parts of whole numbers.
To appreciate decimals we need to thoroughly understand fractions. A fraction is a device to express how many parts we have of the whole. To do this we write down two numbers separated by a diagonal line. The top number tells us how many parts we have while the lower number tells us how many parts make up the whole. This is quite easy to understand. For example, 3/8 means that we have three parts of an eight part whole.
Provided that the same number of parts make up the whole it is very easy to compare fractions. We know that 5/8 is a bigger number than 3/8.
It is much harder to compare fractions in which a different number of parts make up the whole. This requires us to rearrange the fractions so that the same number of parts make up the whole in each case. We can do this by scaling up the fractions so that they become parts of a common denominator. The size of the fraction will not change provided we do the same to top and bottom.
For example, suppose we wish to compare 1/8 with 1/5.
We can rewrite 1/8 as 5 / (8*5) and 1/5 as 8/ (8*5) and see at a glance that 5 parts of 40 is smaller than 8 parts of 40.
Decimals contain a shorthand so that we can directly compare and manipulate fractions without doing these calculations.
Decimals are a natural extension of the way in which we write whole numbers. The decimal number system relies upon just ten symbols. We reuse the symbols in subsequent columns and know that each column to the left of the start of a column is worth ten times its neighbour in the right. Thus 321 means 3 * 10 *10 + 2 * 10 + 1.
If we put a dot in the number we can use the same method to denote fractions. Thus 0.321 can represent 3/10 + 2/100 + 1/1000.
Because the leftmost numbers are the largest parts of the decimal be can directly compare the magnitude of decimals without having to find a common denominator.
If we wish to convert a decimal into a fraction we can write the numbers to the right of the decimal point divided by 1 followed by the number of digits to the right of the decimal point. Thus 0.321 is the same as 321/1000. If there is a common factor between the top and bottom of the fraction it would be usual to factor the number out of the fraction. Thus, we would prefer to write 0.322 as 161/500 rather than 322/1000.
When comparing a fraction with a decimal it would be usual to convert the fraction into a decimal format. This can readily be done using the divide button on a calculator.