Question

part a: identifying rational numbers
circle the numbers that are rational.

1. −3
2. ⅔
3. √5 - (2.2360679)
4. 0.25

Explanation:

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 $q
eq0$. Integers, fractions, terminating decimals, and repeating decimals are rational.

Step2: Analyze -3

-3 is an integer, so it can be written as $\frac{-3}{1}$, so it's rational.

Step3: Analyze $\frac{2}{3}$

$\frac{2}{3}$ is a fraction with integer numerator and non - zero integer denominator, so it's rational.

Step4: Analyze $\sqrt{5}-(2.2360679)$

$\sqrt{5}\approx2.2360679775\cdots$, so $\sqrt{5}-2.2360679\approx2.2360679775 - 2.2360679=0.0000000775\cdots$, but actually $\sqrt{5}$ is irrational, and subtracting a rational number from an irrational number gives an irrational number. Wait, no, the expression is $\sqrt{5}-(2.2360679)$, but $2.2360679$ is an approximation of $\sqrt{5}$. In reality, $\sqrt{5}- \sqrt{5}=0$? Wait, no, the problem has $\sqrt{5}-(2.2360679)$, but $2.2360679$ is a decimal approximation of $\sqrt{5}$. If we consider the exact value, $\sqrt{5}$ is irrational, and $2.2360679$ is rational. But if we compute $\sqrt{5}-2.2360679$, since $\sqrt{5}$ is irrational, the result is irrational. But maybe there is a typo, and it's supposed to be $\sqrt{5}- \sqrt{5}$? No, as per the problem, let's re - evaluate. Wait, maybe the problem is that $2.2360679$ is the decimal expansion of $\sqrt{5}$, so $\sqrt{5}-2.2360679$ is approximately $0$, but actually, $\sqrt{5}$ is irrational, so $\sqrt{5}-2.2360679$ is irrational.

Step5: Analyze 0.25

0.25 is a terminating decimal, and it can be written as $\frac{25}{100}=\frac{1}{4}$, so it's rational.

Answer:

The rational numbers are - 3, $\frac{2}{3}$, 0.25. So we circle - 3, $\frac{2}{3}$, and 0.25.