Rational Numbers

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The natural numbers and the integers are only able to describe quantities that are whole or complete. For example you can have 4 apples, but what happens when you divide one apple into 4 equal pieces and share it among your friends? Then it is not a whole apple anymore and a different type of number is needed to describe the apples. This type of number is known as a rational number. A rational number is any number which can be written as: [math]\frac{a}{b}[/math] where a and b are integers and b [math] \neq [/math] 0. The following are examples of rational numbers:

  • [math]\frac{20}{9}[/math], [math]\frac{-1}{2}[/math], [math]\frac{20}{10}[/math], [math]\frac{3}{15}[/math]

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Extension exercise

Notation Tip

Rational numbers are any number that can be expressed in the form:
[math]\frac{a}{b}[/math];[math]a[/math], [math]b \in Z[/math], [math]b \neg 0[/math] which means "the set of numbers [math]\frac {a}{b}[/math] when a and b are integers".

Mathematicians use the symbol Q to mean the set of all rational numbers. The set of rational numbers contains all numbers which can be written as terminating or repeating decimals.

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Extension exercise

Rational Numbers: all integers are rational numbers with denominator 1.

You can add and multiply rational numbers and still get a rational number at the end, which is very useful. If we have 4 integers, a, b, c and d, then the rules for adding and multiplying rational numbers are
[math]\frac {a}{b}+\frac{c}{d}= \frac {ad+bc}{bd}[/math]
[math]\frac {a}{b}x\frac{c}{d}= \frac{ac}{bd}[/math]

Extension: Notation Tip: the statement "4 integers a, b, c and d" can be written formally as [math]{a, b, c, d} \in Z[/math] because the [math]\in[/math] symbol means in and we say that [math]a, b, c[/math] and [math]d[/math] are in the set of integers. Two rational numbers ( [math]\frac{a}{b}[/math] and [math]\frac{c}{d}[/math]) represent the same number if [math]ad = bc[/math]. It is always best to simplify any rational number so that the denominator is as small as possible. This can be achieved by dividing both the numerator and the denominator by the same integer. For example, the rational number 1000/10000 can be divided by 1000 on the top and the bottom, which gives [math]\frac{1}{10}[/math]. [math]\frac{2}{3}[/math] of a pizza is the same as [math]\frac{8}{12}[/math]

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8/12 of the pizza is the same as 2/3 of the pizza.

You can also add rational numbers together by finding the lowest common denominator and then adding the numerators. Finding a lowest common denominator means finding the lowest number that both denominators are a factor 5 of. A factor of a number is an integer which evenly divides that number without leaving a remainder. The following numbers all have a factor of 3:

  • 3, 6, 9, 12, 15, 18, 21, 24, . . .

and the following all have factors of 4:

  • 4, 8, 12, 16, 20, 24, 28, . . .

The common denominators between 3 and 4 are all the numbers that appear in both of these lists, like 12 and 24. The lowest common denominator of 3 and 4 is the smallest number that has both 3 and 4 as factors, which is 12. For example, if we wish to add 3 4 + 2 3 , we first need to write both fractions so that their denominators are the same by finding the lowest common denominator, which we know is 12. 5Some people say divisor instead of factor.

We can do this by multiplying 3 4 by 3 3 and 2 3 by 4 4 . 3 3 and 4 4 are really just complicated ways of writing 1. Multiplying a number by 1 doesn’t change the number. 3 4 + 2 3 = 3 4 × 3 3 + 2 3 × 4 4 (2.35) = 3 × 3 4 × 3 + 2 × 4 3 × 4 = 9 12 + 8 12 = 9 + 8 12 = 17 12 Dividing by a rational number is the same as multiplying by its reciprocal, as long as neither the numerator nor the denominator is zero: a b ÷ c d = a b . d c = ad bc (2.36) A rational number may be a proper or improper fraction. Proper fractions have a numerator that is smaller than the denominator. For example, −1 2 , 3 15 , −5 −20 are proper fractions. Improper fractions have a numerator that is larger than the denominator. For example, −10 2 , 15 13 , −53 −20 are improper fractions. Improper fractions can always be written as the sum of an integer and a proper fraction.

Converting Rationals into Decimal Numbers

Converting rationals into decimal numbers is very easy. If you use a calculator, you can simply divide the numerator by the denominator. If you do not have a calculator, then you have to use long division. Since long division was first taught in primary school, it will not be discussed here. If you have trouble with long division, then please ask your friends or your teacher to explain it to you.