Rational Numbers
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(Back to Algebra I)
Rational numbers are those numbers that include integers and that lie between the integers. As the name suggests, rational numbers are numbers that can be expressed as ratios or fractions.
Contents |
Vocabulary
- Numerator - the number or expression on the top of the ratio or fraction.
- Denominator - the number or expression on the bottom of the ratio or fraction.
- rational numbers - numbers that can be written as a ratio (or fraction) of two integers.
- fraction - a portion, or part, of a whole number or integer.
Lesson 1
A rational number is a fraction, written p/q where p and q are integers. p is called the numerator and q the denominator. Applied to a cake, it means p parts of a cake divided equally into q parts. For example 1/2 means a half. But note that p and q can be negative. +1/2 means gaining a half and -1/2 means losing a half.
Example Problems
I have been given 1 piece of cake, my father who is very hungry has received 2. My mother has taken 1. And my sister has taken 1 too. There were 10 pieces. What fraction of the cake has been eaten?
1/10 + 2/10 + 1/10 + 1/10 = 5/10 of the cake, which is the half
Note that in this case, the addition is very simple because the denominator is always 10. We just have to add the numerators.
Fractions of negative numbers
If p and q are positive, then the fraction or rational number is positive. This is the way we commonly think of fractions (1/3 of a cake...).
There is no difference whether p is negative or q is negative. The reason for this is simple : if you talk about losing parts of a cake (-p/q), or about parts of a lost cake (p/-q), in both cases, you talk about lost parts. In these cases, the fraction is said to be negative.
Finally, if p and q are negative, then their effect is canceled by each other and the fraction is positive. As rational numbers are on one axis, the second time you take the opposite you obtain the original fraction. Thus, the fraction -p/-q is the fraction p/q.
Lesson 2
It is easy to add fractions when the denominators are equal. For example. adding 3/10 and 2/10 is very simple, just add the numerators and you have the numerator of the resulting fraction:
Notice the simplification: five parts out of ten is the half of the parts. Unfortunately, it is not always so simple. Sometimes we need to add fractions that have different denominators. Before we can add them, we must alter the fractions so that their denominators are the same. We can do this by multiplying each fraction by the number one which doesn't change the value of the fraction). However, the form of the number one will itself be represented as a fraction whose denominator and numerator are equal, and under our control. For example, all of these fractions are equal to one:
Knowing this, we can change the denominators of the fractions so that the denominators of both are the same. For example:
In this case we changed both fractions so that they each had a denominator of 6.
Practice with simple fractions
Calculate the following additions:
results:
Lesson 3
Subtracting fractions when the denominators are equal is also easy
Notice that when the denominators are the same for either adding or subtracting fractions we only add or subtract the numerators. You can treat the subtraction of fractions with uncommon denominators the same as we did in the addition lesson above, keeping in mind to subtract the numerators (only).
