Expressing Fractions as Decimal Numbers

The fraction \frac { 1 }{ 2 } can also be written as 0.5. On a number line 0.5 is halfway between 0 and 1.

This image has an empty alt attribute; its file name is image-9.png

The fraction \frac { 1 }{ 4 } is 0.25 and \frac { 3 }{ 4 } is 0.75.

This image has an empty alt attribute; its file name is image-10.png

All fractions can be written as a decimal number. The decimal form of a fraction will either stop after a finite number of decimal or will recur (periods); it keeps repeating.

  • \frac { 1 }{ 2 } =0.5, stops 1 place after the decimal.
  • \frac { 3 }{ 4 } =0.75, stops 2 places after the decimal.
  • \frac { 1 }{ 8 } =0.125, stops 3 places after the decimal.


But the fraction \frac { 1 }{ 3 } is 0.3333333…, the 3 never ends, but keeps going forever.
\frac { 2 }{ 3 } is also 0.6666666…, here the 6 never stops.

It is important to be able to write functions as their decimal equivalent.
Every fraction has the form \frac { a }{ b } where a and b are integers.
b\neq 0. We also call the fraction \frac { a }{ b } rational.
But numbers exist which cannot be written as fractions, their decimals go on forever without repeating (recur).

Another example of such a number is \sqrt { 2 }. There are many that exist, which we call irrational numbers.
\pi ,\sqrt { 3 } ,\sqrt { 5 } ,\sqrt { 7 } ,… are other examples of irrational numbers.


When we take all types of numbers, we simply call them real numbers. So the real numbers include integers, fractions and also the irrational numbers.

The integers are also rational numbers because they can all be written in the form a/b with b=1.

Thus 2=\frac { 2 }{ 1 } , 5=\frac { 5 }{ 1 } , -3=-\frac { 3 }{ 1 } and so on.

The fraction \frac { 2 }{ 7 } which in decimal form is 0.285714285714285714… can be written as 0.2\dot { 8 } 571\dot { 4 } …

With the dots placed on top of the group of numbers that repeat.

\frac { 4 }{ 11 } =0.36363636… will be written as 0.\dot { 3 } \dot { 6 } and \frac { 2 }{ 3 } =0.6666666… as 0.\dot { 6 } ̇