Rational numbers

A rational number is a number that can be written in the form of a common fraction of two integers, where the denominator is not 0. Formally, a rational number is a number that can be expressed in the form

where p and q are integers, and q ≠ 0.

In other words, a rational number is one that can be expressed as one integer divided by another non-zero integer. The following are some examples of rational numbers.

What are rational numbers?

As can be seen from the examples provided above, rational numbers take on a number of different forms. Recall that a rational number is one that can be written in the form of a fraction involving two integers; this does not mean that it can only be written in the form of a fraction. As long as a given number can be converted to a fraction involving two integers, it is a rational number. The following sections provide more details on what exactly can constitute a rational number.

Integers

All integers are rational numbers. This is because any integer can be written as that integer over 1 (also an integer). This includes all negative numbers. For example, the integers 5, 31214, and -9952 are all rational numbers because they can be written as:

Fractions made up of integers

Based on the definition of rational numbers above, all fractions that are made up of integers are rational numbers, as long as the denominator is not 0. It is also worth noting that since integers include negative numbers, a rational number will be positive if both numerator and denominator are positive or negative integers. Similarly, a rational number will be negative if either the numerator or the denominator is negative. Also, while 0 cannot be in the denominator of the fraction, 0 divided by any integer is also a rational number, since the result will be 0. Thus, the following are all rational numbers:

Terminating decimals

All terminating decimals are rational numbers. The longer the terminating decimal, the more difficult it typically is to convert to a fraction made up of integers. However, it is always possible. One example of a terminating decimal is 0.125, which we can write in the form of a fraction made up of integers as follows:

Non-terminating decimals that repeat

Non-terminating decimals that have an infinitely repeating pattern are rational numbers. A common example is the fraction ⅓, which can be written in decimal form as 0.3, where the bar over the 3 indicates that the "3" is repeated infinitely.

Another example is 0.126236472113562. It does not matter how complicated or long the pattern is. If there is a pattern that repeats infinitely, the non-terminating decimal is still a rational number. Although it may be difficult to express the infinitely-repeating non-terminating decimal as a fraction made up of integers, it is always possible.

Is 0 a rational number?

Zero is a rational number. It is an integer that can be written in the form of a fraction made up of integers (without 0 in the denominator), so it meets the requirements of being a rational number.

Are negative numbers rational?

All negative numbers, as long as they are not irrational, are rational numbers. This is because any negative number can be expressed in the form of a fraction of integers as long as the number is not irrational.

What numbers are not rational?

Any number that does not meet the definition of a rational number is referred to as an irrational number. Formally, irrational numbers are non-terminating decimals that do not have an infinitely repeating pattern. Common examples include:

The symbols above from left to right are the square root of 2, pi (π), Euler's number (e), and the golden ratio (φ). The table below shows some of the decimal places of the above irrational numbers. The ellipsis (...) indicates that the numbers do not terminate.

Since the above numbers do not terminate and do not have any discernible pattern, they are not rational numbers; there is no way to express the above numbers as a fraction made up of integers.

Rational vs irrational numbers

The key difference between rational and irrational numbers is that a rational number can be expressed in the form of a fraction made up of integers while an irrational number cannot; in fact, an irrational number cannot be written in the form of a fraction. More simply, an irrational number is any number that is not a rational number.

The subset of numbers that fit the criteria of being an irrational number are all numbers that are non-terminating decimals that do not have an infinitely repeating pattern.

Rational numbers include perfect squares such as 4, 9, 16, 25, 36, and so on. Irrational numbers include surds (numbers that cannot be simplified in a manner that removes the square root symbol) such as , and so on.

Properties of rational numbers

Rational numbers, as a subset of the set of real numbers, shares all the properties of real numbers. Below are some specific properties of rational numbers, some of which differentiate them from irrational numbers.

Closure

One of the properties of rational numbers that separates them from their irrational counterpart is the property of closure. Rational numbers are closed under the operations of addition, subtraction, multiplication, and division. This means that performing any of these operations using two rational numbers will always result in another rational number:

2 + 2 = 4

2 - 2 = 0

2 × 2 = 4

2 ÷ 2 = 1

All of the results are rational numbers, and the result of these operations will always be rational given that the initial two values are rational numbers. This is not true of irrational numbers, which can either result in rational or irrational numbers depending on the original values.

Additive inverses

All rational numbers have an additive inverse. Given a rational number a/b, its additive inverse is:

Also, given a non-zero rational number, a/b, its multiplicative inverse is:

The multiplicative inverse is also known as the reciprocal.

Below are some other general things to note about rational numbers.

• Rational numbers can be written in the form of a terminating decimal (the decimal ends) or a repeating decimal (the decimal does not end but has repeating digits).
• Non-terminating decimals are not rational numbers because they cannot be expressed in the form of a common fraction.
• The denominator of the common fraction used to express a rational number cannot be 0.
• All integers are rational numbers since the denominator of the common fraction can be 1.

Rational numbers and other number sets

There are many different sets of numbers that are commonly used throughout mathematics. Many of them overlap, and it can be helpful to know the various differences between number sets and how they relate to each other.

The set of rational numbers is typically denoted as Q. It is a subset of the set of real numbers (R), which is made up of the sets of rational and irrational numbers.

The set of rational numbers also includes two other commonly used subsets: the sets of integers (Z) and natural numbers (N). Rational numbers include all of the integers as well as all the values between each integer, while integers include all of the natural numbers in addition to their negative values.

The following image depicts the relationships described above (excluding irrational numbers):