Question

classify each number as rational or irrational and explain.\\(\frac{2}{3}\\) \\(\sqrt{25}\\) \\(-0.7\overline{5}\\) \\(\sqrt{2}\\) \\(7,548,123\\)

Explanation:

for $\boldsymbol{\frac{2}{3}}$:

Step1: Recall rational number definition

A rational number is any number that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q
eq0$.

Step2: Check $\frac{2}{3}$

Here, $p = 2$, $q = 3$ (both integers, $q
eq0$). So $\frac{2}{3}$ is rational.

for $\boldsymbol{\sqrt{25}}$:

Step1: Simplify $\sqrt{25}$

$\sqrt{25}=5$ (since $5\times5 = 25$).

Step2: Check rationality

$5$ can be written as $\frac{5}{1}$ ( $p = 5$, $q = 1$, integers, $q
eq0$). So $\sqrt{25}$ is rational.

for $\boldsymbol{- 0.7\overline{5}}$:

Step1: Recall repeating decimal property

A repeating decimal is a rational number because it can be expressed as a fraction of two integers.

Step2: Confirm $-0.7\overline{5}$

$-0.7\overline{5}$ is a repeating decimal (the $5$ repeats), so it is rational.

for $\boldsymbol{\sqrt{2}}$:

Step1: Recall irrational number definition

An irrational number is a number that cannot be expressed as a fraction of two integers and has a non - repeating, non - terminating decimal expansion.

Step2: Analyze $\sqrt{2}$

The decimal expansion of $\sqrt{2}\approx1.41421356\cdots$ is non - repeating and non - terminating, and it cannot be written as $\frac{p}{q}$ ( $p,q$ integers, $q
eq0$). So $\sqrt{2}$ is irrational.

Answer:

• $\boldsymbol{\frac{2}{3}}$: Rational (can be expressed as $\frac{2}{3}$, $p = 2,q = 3$ integers, $q

eq0$).

• $\boldsymbol{\sqrt{25}}$: Rational ( $\sqrt{25}=5=\frac{5}{1}$, $p = 5,q = 1$ integers, $q

eq0$).

• $\boldsymbol{-0.7\overline{5}}$: Rational (repeating decimal, can be expressed as a fraction of two integers).
• $\boldsymbol{\sqrt{2}}$: Irrational (non - repeating, non - terminating decimal, cannot be expressed as $\frac{p}{q}$ with $p,q$ integers, $q

eq0$).

• $\boldsymbol{7,548,123}$: Rational (integer, can be written as $\frac{7548123}{1}$, $p = 7548123,q = 1$ integers, $q

eq0$).