1.3: Rational and Irrational Numbers | Algebra 1 Honors

How it works

A rational number is a ratio of integers with a nonzero denominator. Its decimal expansion terminates or repeats. An irrational number cannot be written as such a ratio. A square root is rational when the radicand is a perfect square rational number; simplify before classifying.

\sqrt{ab}=\sqrt{a}\sqrt{b}\quad(a,b\ge0)

Worked examples

Example 1

Simplify $\displaystyle \sqrt{72}$ .

Factor out a perfect square: $\displaystyle \sqrt{36\cdot2}=6\sqrt{2}$ . This is irrational.

Example 2

Write $\displaystyle 0.375$ as a fraction.

$\displaystyle 0.375=375/1000=3/8$ , so it is rational.

Problems

Problem 1.3.1

Classify $\displaystyle \sqrt{49}$ and $\displaystyle \sqrt{50}$ as rational or irrational.

Show worked solution

$\displaystyle \sqrt{49}=7$ is rational. $\displaystyle \sqrt{50}=5\sqrt{2}$ is irrational.

Problem 1.3.2

Simplify $\displaystyle \sqrt{48}+\sqrt{12}$ .

Show worked solution

$\displaystyle \sqrt{48}=4\sqrt{3}$ and $\displaystyle \sqrt{12}=2\sqrt{3}$ , giving $\displaystyle 6\sqrt{3}$ .

Problem 1.3.3

Between which consecutive integers does $\displaystyle \sqrt{30}$ lie?

Show worked solution

Since $\displaystyle 25<30<36$ , taking square roots gives $\displaystyle 5<\sqrt{30}<6$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 1.3.1

Problem 1.3.2

Problem 1.3.3

Need help with this topic?