Basic Algebra/Working with Numbers/Rational Numbers

Vocabulary

• rational numbers
• fraction

Lesson

A rational number is a fraction, written ${\displaystyle {\frac {p}{q}}}$ where ${\displaystyle p}$ and ${\displaystyle q}$ are integers. ${\displaystyle p}$ is called the numerator and ${\displaystyle q}$ the denominator. Applied to a cake, it means ${\displaystyle p}$ parts of a cake divided equally into ${\displaystyle q}$ parts. For example ${\displaystyle {\frac {1}{2}}}$ means a half. But note that ${\displaystyle p}$ and ${\displaystyle q}$ can be negative. ${\displaystyle +{\frac {1}{2}}}$ means gaining a half and ${\displaystyle -{\frac {1}{2}}}$ means losing a half.

Example Problems

I have been given 1 piece of cake, my father who is very hungry has taken 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?

${\displaystyle {\frac {1}{10}}+{\frac {2}{10}}+{\frac {1}{10}}+{\frac {1}{10}}={\frac {5}{10}}={\frac {1}{2}}}$ of the cake, which is half of the cake.

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 ${\displaystyle p}$ and ${\displaystyle q}$ are positive, then the fraction or rational number is positive. This is the way we commonly think of fractions ( ${\displaystyle {\frac {1}{3}}}$ of a cake...).

There is no difference whether ${\displaystyle p}$ is negative or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): q is negative. The reason for this is simple : if you talk about losing parts of a cake ( ${\displaystyle -{\frac {p}{q}}}$ ), or about parts of a lost cake ( ${\displaystyle {\frac {p}{-q}}}$ ), in both cases, you talk about lost parts. In these cases, the fraction is said to be negative.

Finally, if ${\displaystyle p}$ and ${\displaystyle q}$ 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 ${\displaystyle {\frac {-p}{-q}}}$ is the fraction ${\displaystyle {\frac {p}{q}}}$.

Practice Games

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Practice Problems

(Note: put answer in parentheses after each problem you write)