An Irrational Post

In my posts to date, you have seen different types of numbers: Positive numbers, Negative numbers, Integers, Fractions, and Prime numbers. Some numbers can fit into more than one category. Today I want to introduce two more types: Rational and Irrational numbers.

It is easier to define what a rational number is then anything that is not rational, is an irrational number.

A rational number is any number that can be put into the form

\[\frac{a}{b}\]

where a and b are integers. So all fractions are rational numbers, but so are all  the integers like 0, -1, 3, -9,999 and so on because they can all be represented as the integer over “1”:

\[\frac{0}{1}{,}\hspace{0.33em}\frac{{-}{1}}{1}{,}\hspace{0.33em}\frac{3}{1}{,}\hspace{0.33em}\frac{{-}{9}{,}{999}}{1}\]

Now decimal numbers can be rational as well. The way to tell is if the decimal part repeats a pattern no matter how short or long the pattern is. That is because all repeating pattern decimals can be put into the a/b form. So the following are all rational numbers. Note that a line over a section of the decimal means that that section is repeated over and over: 

\[{3}{.}{0}\overline{0}{,}\hspace{0.33em}{0}{.}{111}\overline{1}{,}\hspace{0.33em}{6}{.}{25}\overline{25}{,}\hspace{0.33em}{-}{2}{.}{1275}\overline{1275}\]

That is in fact another way to test for whether a number is rational: if the decimal part (even if it is “0”) repeats eventually, it is a rational number. Any number that cannot be expressed as a repeating decimal is irrational. You have been exposed to one famous irrational number: 𝜋. This number has a non-repeating decimal that goes on forever without repeating. This has been proven to be true many times. Other numbers that are irrational are many (but not all) square roots and other roots like cube roots. For example, the following are also irrational numbers:

\[\sqrt{2}{,}\hspace{0.33em}\sqrt{3}{,}\hspace{0.33em}\sqrt[3]{7}{,}\hspace{0.33em}\sqrt[4]{2}\]

In fact, using the square root symbols or other symbols like 𝜋 are the only way we can express the number exactly. Even modern calculators with many digits of accuracy can only represent an irrational number to a limited number of decimal places. So when using an irrational number in a calculation, you use the number of decimal places required for the accuracy required in the final answer.

The set of rational numbers and irrational numbers comprise all the numbers on the number line. This complete set of numbers on the number line are called real numbers.

There are many other types of numbers. For example transcendental numbers and things called imaginary numbers. So we have real numbers and imaginary numbers. These can be combined to form another type of number, complex numbers. These are used in many branches of engineering and have physical meaning, even though imaginary numbers are used. Imaginary numbers are not on the real number line, they are plotted using another different number line. Maybe I will cover these later when I have a real, rational moment.