Rational numbers are those numbers that can be expressed as a quotient (the result in a regular division equation) or in the format of a simple fraction. Even if you express the resulting number not as a fraction and it repeats infinitely, it can still be a rational number. Zero is a rational number.
Rational Number Definition
A rational number is any number that satisfies the following three criteria:
- It can be expressed in the form of a simple fraction with a numerator (p) divided by a (/) a denominator (q).
- Both the numerator and the denominator must be regular integers themselves. An integer is what we would normally call a "whole number" like 3 or 15. It can be positive or negative.
- The denominator (q) cannot be zero.
Any number divided by zero (i.e., where the denominator is zero) approaches infinity (or negative infinity), but is undefined.
Zero Is a Rational Number
With this explanation in mind, you can see how zero (0) is a rational number. That's because while there is a restriction on the denominator (the "bottom" number in a fraction), there is no similar restriction on the numerator (the "top" number in a fraction).
As such, if the numerator is zero (0), and the denominator is any non-zero integer, the resulting quotient is itself zero.
- 0/5 = 0
- 0/200 = 0
- 0/(-25) = 0
Calculating Rational Numbers
Numbers only need to satisfy the three requirements listed above to qualify as rational numbers. The numerator or the denominator can be positive or negative, as long as the denominator is not zero.
The table below shows several examples of positive and negative rational numbers. It shows the relationship between the numerator (p) and denominator (q), the fraction (p/q), and the rational number.
Numerator (p) | Denominator (q) | p / q | Rational Number |
6 | 1 | 6/1 | 6.000 |
1 | 1 | 1/1 | 1 |
2 | 3 | 2/3 | 0.667 |
1 | 1000 | 1/1000 | 0.001 |
86 | 34 | 86/34 | 2.529 |
122 | 70 | 122/70 | 1.743 |
353 | 10 | 353/10 | 35.3 |
-2 | 1 | -2/1 | -2.0 |
-5 | 4 | -5/4 | -1.25 |
You'll also notice two more things about rational numbers:
- They can be expressed with any number of decimal places. When you calculate 6/1, the resulting rational number of 6 can also be written as 6.0, 6.00, 6.000, and so forth.
- Rational numbers can have an infinite number of decimal places, so long as the digits repeat following a predictable pattern. In the case of 2/3, the chart above shows the rational number of 0.667. However, the true number actually has the "6" repeating into infinity. You place a horizontal bar (called a vinculum) above the repeating number ("6" in this case) to indicate this.
With the second point, there can be more than one repeating digit, as long as it follows a repeating pattern. For instance, 123/999 is equal to 0.123123123... where the "123" repeats into infinity. That is still a rational number, since it can be expressed as 123/999, a regular fraction.
Rational and Irrational Numbers
As with so many other concepts, both within mathematics and beyond it, rational numbers also have a counterpart or opposite. Unsurprisingly, this counterpart is called the irrational number. As you might guess, an irrational number is one that cannot be expressed as a fraction or quotient of integers.
A well-known example of an irrational number is pi (π), defined as the ratio of the circumference of a circle to its diameter. It is usually approximated as 3.14, but its true value extends into infinite decimal points with no repeating pattern. Check out some examples of irrational numbers to further explore this mathematical concept.