Is every rational number a fraction?

Answer 1

Every fraction is a rational number, but not every rational number is a fraction.

A fraction is a number that expresses part of a whole as a quotient of integers (where the denominator is not zero).*

A rational number is a number that can be expressed as a quotient of integers (where the denominator is not zero), or as a repeating or terminating decimal. Every fraction fits the first part of that definition. Therefore, every fraction is a rational number.

Both 22/7 and 1/3 are fractions, therefore they are both rational numbers. They also are repeating decimals, as 22/7 = 3.142857142857142857... (notice that the 142857 repeats) and as 1/3 = .333...

An irrational number, on the other hand, neither terminates nor repeats.

(The confusion about 22/7 may come because that fraction is often used to represent the number pi. It is not the number pi, just an approximation. The number pi is a decimal that begins 3.1415... and continues on without terminating or repeating. )

But even though every fraction is a rational number, not every rational number is a fraction. Basically because rational numbers do not have to express a part of a whole. It can express a whole, as in an integer. And an integer is not a fraction.

Answer 2

Every rational number can be written as a vulgar or an improper fraction - even whole numbers can be written as x/1 - for example, 7 is equal to 7/1. The definition of rational is "can be rewritten as a ratio" - or, in most cases, a fraction.

New answer:

A fraction is a number that expresses part of a whole as a quotient of integers (where the denominator is not zero).

A rational number is a number that can be expressed as a quotient of integers (where the denominator is not zero), or as a repeating or terminating decimal. Every fraction fits the first part of that definition. Therefore, every fraction is a rational number.

But even though every fraction is a rational number, not every rational number is a fraction.

Why? Consider this:

Every integer (all the whole numbers, including zero, and their negatives....-3,-2,-1,0,1,2,3...) is a rational number, because it can be expressed as a quotient of integers, as in the case of 4 = 8/2 or 1 = 3/3 or -3 = 3/-1 and so on. So integers such as 4 or 1 can be expressed as the quotient of integers.

But an integer is not a fraction. 4 is an integer, but it is not a fraction. 4 is not expressed as the quotient of integers. The difference here is in the wording.

A fraction is a number that expresses part of a whole. An integer does not express a part. It only expresses a whole number.

A rational number is a number that can be expressed as a quotient of integers, or as part of a whole, but fraction is a number that is (must be) expressed as a quotient of integers, or as part of a whole - there is a difference. The difference is subtle, but it is real.

In a nutshell, the fractions are a subset of the rational numbers. The rational numbers contain the integers, and fractions don't.