Integrated Math 2 / Unit 1: Exponents and Radicals
1.6 Rational and Irrational Numbers
Notes
A rational number can be written as a fraction of integers. An irrational number cannot.
- Rational decimals end or repeat: $0.25$, $0.\overline{3}$. Irrational decimals never end or repeat: $\pi$, $\displaystyle \sqrt{2}$.
- The square root of a non-perfect-square integer is irrational: $\displaystyle \sqrt{20}$ is irrational, but $\displaystyle \sqrt{49} = 7$ is rational.
- Rational + rational = rational. Rational (nonzero) $\times$ irrational = irrational. Rational + irrational = irrational.
- Two irrationals can give a rational result: $\displaystyle \sqrt{2} \cdot \sqrt{8} = \sqrt{16} = 4$.
Practice
Which number is irrational?
$\displaystyle \sqrt{49} = 7$ and $\displaystyle 0.25 = \frac{1}{4}$ are rational, and $\displaystyle \frac{22}{7}$ is a fraction of integers. 20 is not a perfect square.
What is true about $\displaystyle 3 + \sqrt{2}$?
A rational plus an irrational is always irrational. Also $\displaystyle \sqrt{2} + 3 \neq \sqrt{5}$.
Which statement is true?
$\displaystyle \sqrt{2} \cdot \sqrt{8} = 4$ is rational. Repeating decimals are rational, and $\displaystyle \sqrt{9} = 3$ is rational.
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