Question

try it!
classify each number as rational or irrational and explain.
\\(\frac{2}{3}\\) \\(sqrt{25}\\) \\(-0.75\\) \\(sqrt{2}\\) \\(7,548,123\\)

Explanation:

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

Step1: Recall rational number definition

A rational number is a number that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q
eq0$.
$\frac{2}{3}$ is in the form $\frac{p}{q}$ with $p = 2$, $q = 3$ (integers, $q
eq0$).

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

Step1: Simplify the square root

Calculate $\sqrt{25}$. Since $5\times5 = 25$, $\sqrt{25}=5$.

Step2: Classify the result

$5$ is an integer, and integers are rational (as they can be written as $\frac{5}{1}$).

for $\boldsymbol{-0.75}$:

Step1: Express as a fraction

$-0.75=-\frac{75}{100}=-\frac{3}{4}$.

Step2: Classify using rational definition

$-\frac{3}{4}$ is in $\frac{p}{q}$ form ($p=-3$, $q = 4$, integers, $q
eq0$), so it is rational.

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

Step1: Recall perfect square and irrationality

A perfect square is a square of an integer. $2$ is not a perfect square (no integer $n$ with $n\times n=2$). By the rule, if $b$ is not a perfect square, $\sqrt{b}$ is irrational. So $\sqrt{2}$ is irrational.

Answer:

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

eq0$).

• $\boldsymbol{\sqrt{25}}$: Rational ($\sqrt{25}=5$, an integer, hence rational).
• $\boldsymbol{-0.75}$: Rational (expressed as $-\frac{3}{4}$, a fraction of integers).
• $\boldsymbol{\sqrt{2}}$: Irrational (2 is not a perfect square, so $\sqrt{2}$ is irrational).
• $\boldsymbol{7,548,123}$: Rational (an integer, can be written as $\frac{7548123}{1}$).