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Irrational Numbers

There is a type of number that does not fall into any of our four categories. An irrational number is a number with a decimal that neither terminates or repeats. An irrational number cannot be written as a fraction a/b where a and b are integers. Plug in (the square root of 2) on a calculator and the screen will display a decimal that does not repeat itself, but that continues infinitely. This is because the square root of 2 is an irrational number.

There is no number which is both an irrational number and a natural number, whole number, integer, or rational number. If a number is irrational, it cannot fall into one of the four categories we previously outlined; and if a number falls into one of the four categories, it cannot be irrational.

Real Numbers

All the rational numbers and all the irrational numbers together form the real numbers. Every rational number is real, and every irrational number is real. For our purposes at this time, the real numbers constitute all the numbers. 0.45, 5/2, -0.726495..., 18, and -65 1/4 are all real numbers.

Taking into account the irrational numbers and the real numbers, our new classification might look like this:

Figure %: Classification of Numbers
If a number falls into a category, it also falls into all the categories below that category to which it is connected by a line. For example, -7 is an integer, so it is also a rational and a real number. The square root of 2 is an irrational number, so it is also a real number.

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