Question

classify each number below as a rational number or an irrational number.

	rational	irrational
-15π	○	○
-√10	○	○
75.\overline{18}
√64

explanation
to do this problem, we need to know the definitions of rational and irrational numbers.

• a rational number is the ratio of two integers.

so ordinary fractions, like \\(\frac{1}{2}\\) and \\(\frac{3}{7}\\), are rational. some other types of numbers are ra

Explanation:

Step1: Recall Definitions

A rational number can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$. Irrational numbers cannot be expressed this way (non - repeating, non - terminating decimals, or roots of non - perfect squares, multiples of $\pi$ etc.).

Step2: Classify 73.8

73.8 can be written as $\frac{738}{10}=\frac{369}{5}$, so it is rational.

Step3: Classify $- 15\pi$

$\pi$ is irrational, and multiplying an irrational number by a non - zero integer ($-15$) gives an irrational number. So $-15\pi$ is irrational.

Step4: Classify $-\sqrt{10}$

10 is not a perfect square, so $\sqrt{10}$ is irrational. Multiplying by $- 1$ (a non - zero integer) keeps it irrational. So $-\sqrt{10}$ is irrational.

Step5: Classify $75.\overline{18}$

A repeating decimal can be expressed as a fraction. Let $x = 75.\overline{18}$, then $100x=7518.\overline{18}$, and $100x - x=7518.\overline{18}-75.\overline{18}$, $99x = 7443$, $x=\frac{7443}{99}$, so it is rational.

Step6: Classify $\sqrt{64}$

$\sqrt{64} = 8=\frac{8}{1}$, so it is rational.

Answer:

• 73.8: rational
• $-15\pi$: irrational
• $-\sqrt{10}$: irrational
• $75.\overline{18}$: rational
• $\sqrt{64}$: rational