Question

classify each number below as a rational number or an irrational number.
-10π
22.\overline{53}
-\sqrt{16}
92.38
-\sqrt{3}

Explanation:

Step1: Recall definitions

A rational number can be expressed as a fraction $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$. An irrational number cannot be expressed as such and has a non - repeating, non - terminating decimal expansion.

Step2: Classify $- 10\pi$

$\pi$ is an irrational number (its decimal expansion is non - repeating and non - terminating: $3.1415926\cdots$). When we multiply an irrational number by a non - zero integer ($- 10$ here), the result is still irrational. So $-10\pi$ is irrational.

Step3: Classify $22.\overline{53}$

A repeating decimal can be expressed as a fraction. Let $x = 22.\overline{53}=22.535353\cdots$. Then $100x=2253.535353\cdots$. Subtract $x$ from $100x$: $100x - x=2253.5353\cdots-22.5353\cdots$, which gives $99x = 2231$, so $x=\frac{2231}{99}$. Since it can be written as a fraction of two integers, $22.\overline{53}$ is rational.

Step4: Classify $-\sqrt{16}$

We know that $\sqrt{16} = 4$, so $-\sqrt{16}=- 4=-\frac{4}{1}$. Since it can be written as a fraction of two integers, $-\sqrt{16}$ is rational.

Step5: Classify $92.38$

$92.38=\frac{9238}{100}=\frac{4619}{50}$. It can be written as a fraction of two integers, so $92.38$ is rational.

Step6: Classify $-\sqrt{3}$

$\sqrt{3}\approx1.7320508\cdots$ has a non - repeating, non - terminating decimal expansion, so it is irrational. Multiplying it by $- 1$ (a non - zero integer) still gives an irrational number, so $-\sqrt{3}$ is irrational.

Answer:

• For $-10\pi$: irrational (select the circle under "irrational")
• For $22.\overline{53}$: rational (select the circle under "rational")
• For $-\sqrt{16}$: rational (select the circle under "rational")
• For $92.38$: rational (select the circle under "rational")
• For $-\sqrt{3}$: irrational (select the circle under "irrational")