Middle School Math Quiz: Understand Irrational Numbers

What this quiz covers

This quiz focuses on Understand Irrational Numbers, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The student is correct; the first notation shows termination while the second shows repetition
2. The student is incorrect; the first is rational while the second is irrational due to its length
3. The student is partially correct; they have the same value but different levels of precision
4. The student is incorrect; both notations represent the same rational number with identical decimal expansions (correct answer)

Question 2

1. A number is irrational if its decimal terminates.
2. A number is irrational if it can be written as $\frac{a}{b}$ with integers $a,b$ and $b\ne 0$.
3. A number is rational if its decimal terminates or repeats. (correct answer)
4. A number is rational if its decimal is non-terminating and non-repeating.

Question 3

1. $\frac{27}{100}$
2. $\frac{2}{7}$
3. $\frac{27}{90}=\frac{3}{10}$
4. $\frac{27}{99}=\frac{3}{11}$ (correct answer)

Question 4

1. It does not terminate and does not repeat, so it is irrational. (correct answer)
2. It terminates, so it is rational.
3. It repeats a pattern, so it is irrational.
4. It does not terminate, so it must be rational.

Question 5

1. $0.333\ldots$ is rational only if it terminates.
2. $0.333\ldots$ is irrational because it is a decimal.
3. $0.333\ldots$ is irrational because it never ends.
4. $0.333\ldots$ is rational because it repeats, and all repeating decimals can be written as a fraction. (correct answer)

Question 6

1. Rational, because all square roots are rational
2. Irrational, because $17$ is not a perfect square (correct answer)
3. Irrational, because it can be written exactly as $\frac{17}{2}$
4. Rational, because it is close to $4$

Question 7

1. $\pi$ is rational because its decimal repeats.
2. $\pi$ is irrational, and $\frac{22}{7}$ is a rational approximation (not exactly equal). (correct answer)
3. $\pi$ is irrational because it terminates.
4. $\pi$ is rational because $\frac{22}{7}$ equals $\pi$ exactly.

Question 8

1. Convert to decimal form to check: $\frac{45}{99} = 0.454545...$ which confirms the answer
2. Simplify the fraction: $\frac{45}{99} = \frac{5}{11}$ by dividing both numerator and denominator by 9 (correct answer)
3. Leave the answer as $\frac{45}{99}$ since this is the most accurate form of the fraction
4. Check by multiplying: $99 \times \frac{45}{99} = 45$ which proves the fraction is correct

Question 9

1. $\frac{15}{37}$, because 37 is a prime number
2. $0.123456789101112...$ where digits follow the pattern of consecutive integers (correct answer)
3. $\frac{8}{25}$, because the denominator contains only factors of 2 and 5
4. $0.285714285714...$ because it has a long repeating block

Question 10

1. The calculator is rounding $\pi$ to fit the display, but $\pi$ actually has infinitely many non-repeating digits (correct answer)
2. The calculator display proves $\pi$ is rational because any number that can be displayed must be expressible as a fraction
3. The calculator is truncating $\pi$, so the student's conclusion is correct but only applies to the truncated value, not $\pi$ itself
4. The decimal terminates on screen because $\pi$'s digits eventually repeat, but the pattern is too long to display

Question 11

1. Both are rational because they both follow predictable patterns that can be described mathematically
2. Both are irrational because they both continue indefinitely without terminating
3. $0.\overline{3}$ is rational because it repeats, while $0.3010010001...$ is irrational because it never repeats (correct answer)
4. $0.\overline{3}$ is irrational because it never terminates, while $0.3010010001...$ is rational because it has a pattern

Question 12

1. The student thinks any decimal that follows a describable pattern must be rational (correct answer)
2. The student believes that decimals containing only digits 0-9 are automatically rational numbers
3. The student assumes that decimals starting with small digits are more likely to be rational
4. The student confuses the decimal with $0.123456789$ which terminates and is therefore rational

Question 13

1. $\sqrt{2} = 1.41421356\ldots$
2. $0.101001000100001\ldots$
3. $0.75$ (correct answer)
4. $\pi = 3.14159265\ldots$

Question 14

1. $\sqrt{4}$
2. $0.777\ldots$
3. $0.125$
4. $\sqrt{3}$ (correct answer)

Question 15

1. $\sqrt{9}$ and $\pi$
2. $\sqrt{2}$, $\sqrt{9}$, and $\pi$
3. $0.75$ and $0.333\ldots$
4. $\sqrt{2}$ and $\pi$ (correct answer)

Question 16

1. $\pi$ is irrational, and $\frac{22}{7}$ is only a rational approximation. (correct answer)
2. $\pi$ is rational because $\frac{22}{7}$ equals $\pi$ exactly.
3. $\pi$ is rational because its decimal repeats.
4. $\pi$ is irrational only if you round it.

Question 17

1. $\frac{3}{2}$
2. $\frac{6}{10}$
3. $\frac{2}{3}$ (correct answer)
4. $\frac{6}{9}$

Question 18

1. Both $A$ and $B$ are rational because they are decimals.
2. Both $A$ and $B$ are irrational because they do not terminate.
3. $A$ is irrational and $B$ is rational.
4. $A$ is rational because it repeats, and $B$ is irrational because it does not repeat. (correct answer)

Question 19

1. Irrational, because all decimals are irrational.
2. Rational, because any decimal with a pattern is repeating.
3. Irrational, because it does not terminate and does not repeat a fixed block. (correct answer)
4. Rational, because it does not terminate.

Question 20

1. $\sqrt{3}$ (correct answer)
2. $\sqrt{25}$
3. $0.4$
4. $\frac{7}{8}$