More simply, an irrational number is any number that is not a rational number. 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. 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. The ellipsis (.) indicates that the numbers do not terminate. The table below shows some of the decimal places of the above irrational numbers. The symbols above from left to right are the square root of 2, pi (π), Euler's number ( e), and the golden ratio (φ). Formally, irrational numbers are non-terminating decimals that do not have an infinitely repeating pattern. What numbers are not rational?Īny number that does not meet the definition of a rational number is referred to as an irrational number. 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. Are negative numbers rational?Īll negative numbers, as long as they are not irrational, are rational numbers.
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. Although it may be difficult to express the infinitely-repeating non-terminating decimal as a fraction made up of integers, it is always possible.
If there is a pattern that repeats infinitely, the non-terminating decimal is still a rational number. It does not matter how complicated or long the pattern is. 3, where the bar over the 3 indicates that the "3" is repeated infinitely.Īnother example is 0. A common example is the fraction ⅓, which can be written in decimal form as 0. Non-terminating decimals that have an infinitely repeating pattern are rational numbers. 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: The longer the terminating decimal, the more difficult it typically is to convert to a fraction made up of integers. Thus, the following are all rational numbers:Īll terminating decimals are rational numbers. 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. Similarly, a rational number will be negative if either the numerator or the denominator is negative. 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. For example, the integers 5, 31214, and -9952 are all rational numbers because they can be written as:īased 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. This is because any integer can be written as that integer over 1 (also an integer). IntegersĪll integers are rational numbers. The following sections provide more details on what exactly can constitute a rational number. As long as a given number can be converted to a fraction involving two integers, it is a rational number. 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. The following are some examples of rational numbers.Īs can be seen from the examples provided above, rational numbers take on a number of different forms. In other words, a rational number is one that can be expressed as one integer divided by another non-zero integer. Formally, a rational number is a number that can be expressed in the form Home / primary math / number / rational numbers Rational numbersĪ 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.
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