Most of us go through years of school math courses and still are confused about some basic things. For example: Why can’t you divide by zero? Why is .999… equal to 1, and not a bit less?
There are loads of these kinds of questions, that wouldn’t be a cause of frustration at all, if they were taught reasonably and clearly.
Unfortunately most of these things are supposed to be covered in elementary school, and most elementary school teachers don’t have a good understanding of basic math concepts. Instead they are supposed to teach just a collection of “skills.”
One of the simplest concepts that is usually left inadequately explained is the difference between fractions and rational numbers. Let’s see if we can clear it up now.
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.
There are slightly different variations of the definition of a fraction, including, “A fraction is the ratio of two whole numbers, or to put it simply, one whole number divided by another whole number.”
That definition also shows that an integer is not a fraction, because an integer is not a ratio. It can be expressed as a ratio, but it is not a ratio in itself; it can be divided by another whole number, but it is not being divided.
In a nutshell, the fractions are a subset of the rational numbers. The rational numbers contain the integers, and fractions don’t.
