The Number System - Review

Which of the following numbers is an irrational number?

• 0.75

• 2.5

Answer explanation

The number \(\sqrt{5}\) is irrational because it cannot be expressed as a fraction of two integers. In contrast, \(\frac{3}{4}\), 0.75, and 2.5 are all rational numbers.

Tags

CCSS.8.NS.A.1

Answer explanation

To convert the repeating decimal 0.333... into a fraction, let x = 0.333.... Then, multiplying by 10 gives 10x = 3.333.... Subtracting the original equation from this results in 9x = 3, leading to x = 1/3. Thus, the correct answer is \frac{1}{3}.

Tags

CCSS.8.NS.A.1

• 1.4

• 1.41

• 1.42

• 1.5

Answer explanation

The value of \( \sqrt{2} \) is approximately 1.414. Among the options, 1.41 is the closest rational approximation, making it the best choice compared to 1.4, 1.42, and 1.5.

Tags

CCSS.8.NS.A.2

• 6.28

• 8.86

• 9.87

• 10.24

Answer explanation

The value of \( \pi \approx 3.14 \). Therefore, \( \pi^2 \approx 3.14^2 \approx 9.87 \). This makes 9.87 the closest estimate for \( \pi^2 \) among the given options.

Tags

CCSS.8.NS.A.2

Which of the following numbers has a decimal expansion that repeats?

Answer explanation

The number \( \frac{1}{3} \) has a decimal expansion of 0.333..., which repeats. In contrast, \( \sqrt{3} \), \( \pi \), and \( \sqrt{2} \) are irrational numbers with non-repeating decimal expansions.

Tags

CCSS.8.NS.A.1

• Between 2 and 3

• Between 3 and 4

• Between 4 and 5

• Between 5 and 6

Answer explanation

The value of \sqrt{10} \approx 3.16, which is between 3 and 4. Therefore, the correct location on the number line is between 3 and 4.

Tags

CCSS.8.NS.A.2

Answer explanation

To convert 0.666... to a fraction, let x = 0.666... . Then, 10x = 6.666... . Subtracting these gives 9x = 6, so x = 6/9, which simplifies to 2/3. Thus, the correct answer is \frac{2}{3}.

Tags

CCSS.8.NS.A.1

Which of the following is an irrational number?

• 0.25

• 1.75

Answer explanation

An irrational number cannot be expressed as a fraction of two integers. Among the options, \(\sqrt{7}\) is irrational, while 0.25, \(\frac{7}{8}\), and 1.75 are all rational numbers.

Tags

CCSS.8.NS.A.1

• 6.5

• 7.1

• 7.5

• 8.0

Answer explanation

To estimate \( \sqrt{50} \), note that it is between \( \sqrt{49} = 7 \) and \( \sqrt{64} = 8 \). A closer approximation is \( 7.1 \), as \( 50 \) is closer to \( 49 \) than to \( 64 \). Thus, the best estimate is 7.1.

Tags

CCSS.8.NS.A.2

Which of the following numbers is rational?

Answer explanation

A rational number can be expressed as a fraction of two integers. The number \(\frac{5}{6}\) is a fraction, making it rational. In contrast, \(\sqrt{11}\), \(\pi\), and \(\sqrt{3}\) are irrational.

Tags

CCSS.8.NS.A.1