Question

rational vs. irrational numbers
a rational number can be made by dividing two integers, as long as youre not dividing by 0. you can write any rational number as a fraction.
rational numbers written as decimals either terminate or repeat.
example written as a fraction
√49 7/1
1 5/6 11/6
-8.13 -813/100
4.3 13/3
an irrational number cannot be made by dividing two integers. it is impossible to write an irrational number as a fraction.
irrational numbers written as decimals go on forever without repeating in a pattern.
example written as a decimal
√21 4.58257569...
π 3.14159265...
-√8 -2.82842712...
10 + √3 11.73205080...
practice it! draw circles around the rational numbers, and draw squares around the irrational numbers.
3/4 √13 -9.5 -π √36 1,000 1/12
2.72 4.6 √61 2/5 -7 3/10 √9 -16/5
14/4 √25 1/50 π + 5 -4/8 1 - √32 -7
√90 3/11 √5 0 10.4 13 √100
3.6 -21.2 3π √4 + √5 -3/10 √14 -√1
√2 0.17 -2/36 8.3 √64 7/25 1.36

Explanation:

Step1: Recall rational - number definition

A rational number can be written as a fraction $\frac{a}{b}$ where $a,b$ are integers and $b
eq0$, and its decimal form either terminates or repeats.

Step2: Analyze each number

• $\frac{3}{4}$: It is a fraction, so rational.
• $\sqrt{13}$: Since $13$ is not a perfect - square, it is irrational.
• $-9.5=-\frac{19}{2}$, rational.
• $-\pi$: $\pi$ is irrational, so $-\pi$ is irrational.
• $\sqrt{36} = 6=\frac{6}{1}$, rational.
• $1000=\frac{1000}{1}$, rational.
• $\frac{1}{12}$, rational.
• $2.\overline{72}$: Repeating decimal, rational.
• $4.6=\frac{23}{5}$, rational.
• $\sqrt{61}$: $61$ is not a perfect - square, irrational.
• $\frac{2}{5}$, rational.
• $-7\frac{3}{10}=-\frac{73}{10}$, rational.
• $\sqrt{9} = 3=\frac{3}{1}$, rational.
• $-\frac{16}{5}$, rational.
• $\frac{14}{4}=\frac{7}{2}$, rational.
• $\sqrt{25} = 5=\frac{5}{1}$, rational.
• $\frac{1}{50}$, rational.
• $\pi + 5$: Since $\pi$ is irrational, $\pi + 5$ is irrational.
• $-\frac{4}{8}=-\frac{1}{2}$, rational.
• $1-\sqrt{32}$: $\sqrt{32}=4\sqrt{2}$ and $\sqrt{2}$ is irrational, so $1-\sqrt{32}$ is irrational.
• $-7=\frac{-7}{1}$, rational.
• $\sqrt{90}$: $90$ is not a perfect - square, irrational.
• $\frac{3}{11}$, rational.
• $\sqrt{5}$: Not a perfect - square, irrational.
• $0=\frac{0}{1}$, rational.
• $10.\overline{4}$: Repeating decimal, rational.
• $13=\frac{13}{1}$, rational.
• $\sqrt{100} = 10=\frac{10}{1}$, rational.
• $3.\overline{6}$: Repeating decimal, rational.
• $-21.2=-\frac{106}{5}$, rational.
• $3\pi$: Since $\pi$ is irrational, $3\pi$ is irrational.
• $\sqrt{4}+\sqrt{5}=2 + \sqrt{5}$, irrational.
• $-\frac{3}{10}$, rational.
• $\sqrt{14}$: Not a perfect - square, irrational.
• $-\sqrt{1}=-1=\frac{-1}{1}$, rational.
• $\sqrt{2}$: Not a perfect - square, irrational.
• $0.1\overline{7}$: Repeating decimal, rational.
• $-\frac{2}{36}=-\frac{1}{18}$, rational.
• $8.\overline{3}$: Repeating decimal, rational.
• $\sqrt{64} = 8=\frac{8}{1}$, rational.
• $\frac{7}{25}$, rational.
• $1.\overline{36}$: Repeating decimal, rational.

Answer:

Rational numbers: $\frac{3}{4},-9.5,\sqrt{36},1000,\frac{1}{12},2.\overline{72},4.6,\frac{2}{5},-7\frac{3}{10},\sqrt{9},-\frac{16}{5},\frac{14}{4},\sqrt{25},\frac{1}{50},-\frac{4}{8},-7,\frac{3}{11},0,10.\overline{4},13,\sqrt{100},3.\overline{6},-21.2,-\frac{3}{10},-\sqrt{1},0.1\overline{7},-\frac{2}{36},8.\overline{3},\sqrt{64},\frac{7}{25},1.\overline{36}$
Irrational numbers: $\sqrt{13},-\pi,\sqrt{61},\pi + 5,1-\sqrt{32},\sqrt{90},\sqrt{5},\sqrt{4}+\sqrt{5},3\pi,\sqrt{14},\sqrt{2}$