Rational Numbers and the Number Line
Define rational numbers as ratios of integers and plot them precisely on the number line. Compare and order positive and negative fractions and decimals, and understand density of rationals between any two points.
A lesson of Pre-Algebra, in Number Sense & Operations, about 10 minutes.
Rational Numbers
By the end of this lesson, you will be able to classify numbers as rational or irrational, place rational numbers on a number line, and convert between forms.
What Is a Rational Number?
Rational number: any number expressible as $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.
As a decimal, rational numbers either terminate (end) or repeat.
Rational examples: $\frac{3}{4} = 0.75$ (terminates), $\frac{1}{3} = 0.\overline{3}$ (repeats), $-2$, $0$, $5$
Irrational examples: $\pi = 3.14159\ldots$ (infinite, non-repeating), $\sqrt{2} = 1.41421\ldots$
Converting Forms
Fraction → Decimal: divide numerator by denominator.
$\frac{5}{8} = 5 \div 8 = 0.625$
Terminating decimal → Fraction: use place value.
$0.36 = \frac{36}{100} = \frac{9}{25}$
Repeating decimal → Fraction (intro): $0.\overline{3} = \frac{1}{3}$; $0.\overline{6} = \frac{2}{3}$; $0.\overline{9} = 1$
Placing on a Number Line
Convert to decimals, then place between the appropriate integers.
$-\frac{7}{4} = -1.75$ → between $-2$ and $-1$, closer to $-2$
Comparing Rational Numbers
Convert to the same form (both decimals or both fractions with LCD), then compare.
$\frac{3}{5}$ vs. $\frac{5}{8}$: $\frac{24}{40}$ vs. $\frac{25}{40}$ → $\frac{3}{5} < \frac{5}{8}$
Summary
Rational: $\frac{p}{q}$; terminates or repeats as a decimal.
Irrational: non-terminating, non-repeating ($\pi$, $\sqrt{2}$).
Convert: fraction ÷ = decimal; decimal → fraction using place value.
Compare: convert to same form, then compare.