What is the difference between Fractions and Rational Numbers?
Fractions and Integers
A fraction (common, or vulgar fraction) is a number that expresses part of a whole as a quotient of integers (in the form of m/n) where the denominator (divisor) is not zero.*
OK, what’s a quotient of integers? Funny you should ask that. That term is used all over the web and in math texts, but is almost never explained.
A quotient is the result of a division problem.
An integer is all the whole numbers, including zero, or their negatives. (… -3, -2, -1, 0, 1 ,2 ,3…)
A quotient of integers is simply two integers that are being divided to obtain a quotient. In the example 6/3, 2 would be the quotient, and 6/3 is the quotient of integers.
So a quotient of integers is a number expressed like 3/5, 8/4, etc. In mathematics, a fraction must be expressed like that.
Another way to say it is, a fraction (common fraction) is a division expression (in the form of m/n) where both dividend and divisor are integers, and the divisor is not zero.
There are different forms fractions can take. These are some basic descriptions:
- A proper fraction is a common fraction in which the numerator (also referred to as the dividend, or the “top number,” although the term denominator is generally preferred when referring to fractions) is greater than the denominator (also referred to as the divisor, or “bottom number'” although the term denominator is generally preferred when referring to fractions.) Example: 3/4 (three fourths).
- An improper fractions is a common fraction in which the numerator is greater than the denominator Example: 4/3 (four thirds). This can be reduced to 1 1/3 to make a mixed fraction (see below.)
- A mixed fraction is a common fraction that is combined with an integer. Example: 2 5/6 (two and five sixths).
All of the above fractions are considered simple fractions, as opposed to complex fractions (see below).
- A complex fraction (or compound fraction) is a fraction in which the numerator or denominator contains a fraction. For example, 1/3/3/5 (one third over three fifths)
It is important to know that the term “fraction” is not a term most mathematicians use. It is a tool used by math educators and teachers.
Rational Numbers
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.
Consider this:
Every integer 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, can integers such as 4 or 1 be expressed as the quotient of integers? Yes.
But an integer is not a fraction. 4 is an integer, but it is not a fraction. Is 4 expressed as the quotient of integers? No. 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.
I’ve seen the definition of fraction described as many things, including, “A fraction is the ratio of two whole numbers, or to put it simply, one whole number divided by another whole number.” This 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.
This kind of logic can be a little hard to understand, so let me make an analogy:
A person can be a chemist. But the definition of a person is not “a chemist.” A person is only a chemist when s/he is fulfilling a set of requirements that are beyond the definition of what a “person” is. So a person is not automatically a chemist. A person can be a chemist, but there is a difference between the Idea of “can be,” and the Idea of “is.”
Marie Curie was a chemist. Marie Curie was a person. Joe the plumber is a person (allegedly) but that does not make him a chemist. Therefore, the definition of a person is not necessarily “a chemist.”
The same is true of an integer not necessarily being a fraction, or a rational number not necessarily being a fraction.
So in a nutshell, the fractions are a subset of the rational numbers. The rational numbers contain the integers, and fractions don’t. Please remember that that is just the “nutshell” and not the definition, or the explanation of the difference, it’s just a rule of thumb. And keep in mind that “fraction” is not a term generally used by mathematicians. They don’t need it. They can speak in terms of rational numbers and integers.
* According to Wikipedia http://en.wikipedia.org/wiki/Fraction_(mathematics“A fraction (from the Latin fractus, broken) is a number that can represent part of a whole.” This definition is a little ambiguous. I believe the word “can” was used indiscriminantly – the word should have been “does.”*) I know this sounds like I’m splitting hairs, here, but when you use the word “can” in a definition, you are not really defining the word. A good definition implies definite – at least as definite as you can be – (hence the name), not just possible.
The reason “can” is used in the definition of rational numbers, is because that is as definite as you can get. It is an inherent part of it’s definition, just as in the definition of “bendable” meaning “can be bent.” Something that is bendable doesn’t have to be bent, but it can be. In its unbent state, it is still bendable, just like an integer is rational number because it can be expressed as a ratio of two integers.
There are more definitions for most terms than any one person could ever know. Here’s an interesting one from Bertrand Russell
We shall define the fraction m/n as being that relation which holds between two inductive numbers x, y when xn=ym. This definition enables us to prove that m/n is a one-one relation, provided neither m or n is zero. And of course n/m is the converse relation to m/n.
– Introduction to Mathematical Philosophy Page 64 Bertrand Russell (1919)
Remember to take all definitions with a grain of salt. 
The more you learn about math, the more you’ll find that some of the things you “know” have different meanings in different contexts. It can be very confusing. Understanding comes with maturity and keeping and open mind. Unfortunately those are things that schools do not value (there are no standardized tests for them).
Related posts:
More on Math Terminology Mis-Explained
Afterthoughts:
Some confusion can exist about such numbers as 22/7 and 1/3, as their decimals go on forever. Sometimes people misunderstand this to mean that those numbers are irrational. But irrational does not mean never-ending.
Both 22/7 and 1/3 are fractions, therefore they are both rational numbers. They can also be expressed as 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. If you’d like to know more about the irrationals, check out The Math Mojo Monthly Newsletter Issue #11. It will be published soon after this post is published. The only way to get it is to sign up for it. You can sign up for the Math Mojo Monthly on the top left navigation bar of this page.
(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. )
Post-afterthoughts:
I just consulted John H. Conway and Richard K. Guy’s wonder-filled The Book of Numbers. On page 25 he implies that fractions and rational numbers can be used interchangeably. I’ve been asserting that fractions are a subset of rational numbers, because, although integers can be expressed as fractions, they themselves are not fractions.
This leads me to believe –
- a) I am utterly mistaken and presumptuous, or
- b) hell has frozen over and I may be right on some minor point concerning anything at all.
So, I may have shot myself in the butt with this whole article. I don’t think so, though. Even the big guns can miss the target occasionally. (See More on Math Terminology Mis-Explained ).
I’ve been trying vainly (in both senses of the word I suppose) to reach Dr. Conway to clear up this discrepancy. If any readers should have a conduit to Conway (a conduitway?) please let me know. If my post is to be ripped apart, it should at least be by the best.
I’m still betting on Professor Homunculus, though.
Hotcha!
Brian (a.k.a. Professor Homunculus)
A challenge for you
Using the logic above, and your own reasoning and calculating skills, can you prove that every complex fraction must be a rational number? If you can, leave your proof in a comment below.
“Don’t fall into the trap of standard education – keep an open mind, and keep exploring. The more you understand, the less you know, but at least you know that.”
