Irrational number

Irrational numbers can also be expressed as non-terminating continued fractions and many other ways.

Set of real numbers (R), which include the rationals (Q), which include the integers (Z), which include the natural numbers (N). The real numbers also include the irrationals (R\Q).

It is also suggested that Aryabhata (5th century AD), in calculating a value of pi to 5 significant figures, used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational).

"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc."

"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it."

Conversely, suppose we are faced with a repeating decimal, we can prove that it is a fraction of two integers. For example, consider:

Here the repetend is 162 and the length of the repetend is 3. First, we multiply by an appropriate power of 10 to move the decimal point to the right so that it is just in front of a repetend. In this example we would multiply by 10 to obtain: