Posted 6 November 2008
Definition: A rational number is a number that can be represented as a fraction , where m and n are integers and .
¨ The numbers 3/4 and 11/5 are rational.
¨ 6 is rational because 6 = 6/1.
¨ 0.33 is rational because . (See Approximations).
Rational numbers may be referred to as “rationals”. The name comes from the fact that they represent ratios and is not related to the meaning “able to reason” or “sane”.
Rational numbers have two familiar representations, as fractions and as decimals. Decimals are discussed in the section on real numbers.
The definition of rational number says that it must be a number that can be represented as a fraction of integers.
For example if you see the expression you cannot conclude that it denotes a rational number, because and e are not integers. (I do not know if is rational.)
Let be the representation of a rational number with and . The representation is in lowest terms (or is reduced) if there is no integer for which d divides m and d divides n. (See parenthetic assertion.)
The symbols 3/4 and 6/8 are two representations of the same number. One of the representations, 3/4, is in lowest terms and the other is not. So when someone says “3/4 is in lowest terms”, the symbol “3/4” refers to the representation, not the rational number. See context sensitive interpretation.
not in lowest terms because 74 and
To calculate the lowest terms representation you divide the numerator and denominator by the largest integer that divides both of them. For example, the largest number dividing both 74 and 111 is 37. 74/37 = 2 and 111/37 = 3. So 74/111 = 2/3 and “2/3” is in lowest terms.
Every nonzero rational number has a representation in lowest terms.
A proof of this will appear in the number theory section if I ever get around to writing it.
Rational numbers are closed under addition, subtraction, multiplication, as well as division by a nonzero rational.
These operations are carried out according to the familiar rules for operating with fractions. Thus for rational numbers and , we have
The expressions , and denote rational numbers because integers are closed under addition and multiplication, so that ac, bd, ad, bc and ad+bc are integers.
In other words, “being in lowest terms” is not closed under addition and multiplication.
Suppose we have a line segment L that is 5/8 of a unit long. Then we can take a line segment M that is 1 unit long and divide it into exactly 8 segments of 1/8 unit each, and we can divide L into exactly 5 segments of the same length. That is because the ratio of L to M is 5:8.
This is the sense in which rational numbers represent ratios of integers.
The integers can be thought of as beads or points in a row going to infinity in both directions. The rationals go to infinity in both directions, too, but:
That is because between any two rational numbers there is another one.
In general, if r and s are any distinct rational numbers, then is a rational number between them. This number is the average (or mean) of r and s, so it makes sense that it is between them. The preceding sentence is an example of using the rich view to see why something is true. Here is a rigorous proof:
Theorem. Let r and s be distinct rational numbers. Assume WLOG that r < s. Then .
Proof. Let and , where a, b, c and d are integers. Then by AMD,
which is rational because and are integers.
I recommend that you check using AMD that if and , then .
Warning: is not the only rational number between r and s. In fact, between any two distinct rational numbers there are infinitely many other rational numbers. This means that if you are given a rational number r, there is no “next largest” rational number (or next smallest, either).
These properties are discussed for all real numbers here.