Real Numbers - ACT Math

1. Which represents a rational number? $\sqrt{8}$, $0.4$, $\pi$, $\ln(3)$
   Answer: $0.4$ A rational number can be expressed as a fraction p/q where p and q are integers and q ≠ 0. √8 is irrational, 0.4 = 4/10 = 2/5 is rational (terminating decimal), π is irrational, and ln(3) is irrational. The decimal 0.4 terminates, making it rational.
2. Order the following from least to greatest: $-2$, $-\dfrac{7}{3}$, $-\sqrt{5}$, $-2.2$.
   Answer: $-\dfrac{7}{3},\ -\sqrt{5},\ -2.2,\ -2$ To order negative numbers, convert each to a decimal and remember that the more negative a value is, the smaller it is. Here $-\dfrac{7}{3} \approx -2.333$, $-\sqrt{5} \approx -2.236$, $-2.2$ is already decimal, and $-2$ is the largest, so from least to greatest the order is $-\dfrac{7}{3},\ -\sqrt{5},\ -2.2,\ -2$. The ordering $-\sqrt{5},\ -\dfrac{7}{3},\ -2.2,\ -2$ swaps the first two by treating $-2.236$ as more negative than $-2.333$. The ordering $-\dfrac{7}{3},\ -2.2,\ -\sqrt{5},\ -2$ places $-2.2$ before $-\sqrt{5}$, missing that $-2.236$ lies further left on the number line than $-2.2$. The ordering $-2,\ -2.2,\ -\sqrt{5},\ -\dfrac{7}{3}$ runs from greatest to least, the reverse of what was asked. Convert every value to a decimal first and read the list off the number line from left to right, checking the direction the question requests.
3. Which expression represents a real number? $\sqrt{-9}$, $\frac{7}{0}$, $5.5$, $\ln(0)$
   Answer: $5.5$ We need to identify which expression represents a real number. √(-9) is undefined in real numbers (square root of negative), 7/0 is undefined (division by zero), 5.5 is a real rational number, and ln(0) is undefined (logarithm of zero). Only 5.5 represents a real number.
4. A temperature change is represented by $x=-8.5$. What is the absolute value of $x$?
   Answer: $8.5$ The absolute value represents the distance from zero on the number line, which is always positive or zero. For x = -8.5, the absolute value |x| = |-8.5| = 8.5. The absolute value removes the negative sign and gives the positive distance from zero. Choice A incorrectly keeps the negative sign, but absolute value is always non-negative.
5. Order the following numbers from least to greatest: $-3$, $\frac{-1}{2}$, $0.75$, $-\frac{3}{4}$.
   Answer: $-3, -\frac{3}{4}, \frac{-1}{2}, 0.75$ We need to order the numbers -3, -1/2, 0.75, and -3/4 from least to greatest. Converting to decimal form: -3 = -3.0, -1/2 = -0.5, 0.75 = 0.75, -3/4 = -0.75. On the number line from left to right: -3 < -0.75 < -0.5 < 0.75. Therefore, the order from least to greatest is -3, -3/4, -1/2, 0.75. Choice B incorrectly placed -1/2 first, not recognizing that -3 is the smallest negative number.
6. Order the following numbers from greatest to least: $-\frac{3}{5}$, $0.8$, $-0.9$, $\frac{1}{4}$.
   Answer: $0.8, \frac{1}{4}, -\frac{3}{5}, -0.9$ We need to order the numbers from greatest to least. Converting to decimal form: -3/5 = -0.6, 0.8 = 0.8, -0.9 = -0.9, 1/4 = 0.25. Ordering from greatest to least: 0.8 > 0.25 > -0.6 > -0.9. Therefore, the order is 0.8, 1/4, -3/5, -0.9. Choice B incorrectly placed -3/5 before 1/4, not recognizing that positive numbers are greater than negative numbers.
7. A student estimates a square root using nearby perfect squares. What is the approximate value of $\sqrt{3}$ to the nearest hundredth?
   Answer: $1.73$ To approximate √3, we use nearby perfect squares: 1² = 1 and 2² = 4, so √3 is between 1 and 2. Since 3 is closer to 4 than to 1, √3 is closer to 2. More precisely, √3 ≈ 1.732, which rounds to 1.73 to the nearest hundredth. Choice A (1.41) is actually the approximation for √2, not √3.
8. What is the absolute value of $-12$?
   Answer: $12$ The absolute value of a number is its distance from zero on the number line, which is always positive or zero. The absolute value of -12 is the distance from -12 to 0, which is 12 units. Therefore, |-12| = 12.
9. Order the following from least to greatest: $-\frac{1}{3}$, $0$, $0.25$, $\frac{1}{2}$.
   Answer: $-\frac{1}{3}, 0, 0.25, \frac{1}{2}$ We need to order $-\frac{1}{3}$, $0$, $0.25$, and $\frac{1}{2}$ from least to greatest. Convert to decimals: $-\frac{1}{3}$ ≈ $-0.333$, $0$ = $0$, $0.25$ = $0.25$, $\frac{1}{2}$ = $0.5$. On the number line: $-0.333 < 0 < 0.25 < 0.5$. The correct order from least to greatest is $-\frac{1}{3}$, $0$, $0.25$, $\frac{1}{2}$.
10. A student evaluates a distance formula and gets $x=-\sqrt{49}$. What is the absolute value of $x$?
    Answer: $7$ The absolute value represents the distance from zero, which is always positive. First, we evaluate $x = -\sqrt{49} = -7$ since $\sqrt{49} = 7$. Then the absolute value $|x| = |-7| = 7$. The absolute value removes the negative sign and gives the positive distance from zero. Choice A incorrectly keeps the negative sign, but absolute value is always non-negative.
11. Which expression represents a rational number? $\sqrt{5}$, $\pi$, $\frac{2}{3}$, $\sqrt{2}$.
    Answer: $\frac{2}{3}$ We need to identify which expression represents a rational number. A rational number can be expressed as a fraction of two integers and has a decimal that either terminates or repeats. √5, π, and √2 are all irrational numbers with non-terminating, non-repeating decimals. The fraction 2/3 is rational because it can be expressed as the ratio of two integers. Choice A incorrectly selected √5, which is irrational since 5 is not a perfect square.
12. What is the absolute value of $-\frac{7}{3}$?
    Answer: $\frac{7}{3}$ The absolute value of a number is its distance from zero on the number line, always positive or zero. The absolute value of $-\frac{7}{3}$ is the distance from $-\frac{7}{3}$ to 0, which is $\frac{7}{3}$ units. Therefore, $|-\frac{7}{3}| = \frac{7}{3}$.
13. A calculator is not available. What is the approximate value of $\sqrt{50}$?
    Answer: $7.1$ To approximate √50 without a calculator, find perfect squares near 50 and use estimation. Since 7² = 49 and 8² = 64, √50 lies between 7 and 8. Because 50 is very close to 49, √50 is slightly more than 7, approximately 7.07. Among the choices, 7.1 is the closest approximation. Choice A gives 5, which would be √25, and choice C gives 10, which would be √100.
14. A set of values includes integers, fractions, and radicals. Which number is greatest?
    Answer: $\dfrac{19}{4}$ To find the greatest number, we convert all to decimal form for comparison. Converting: √20 ≈ 4.472, 9/2 = 4.5, 4.4 = 4.4, and 19/4 = 4.75. Comparing these values: 4.4 < 4.472 < 4.5 < 4.75, so 19/4 is the greatest at 4.75. Choice B incorrectly suggests 9/2 is greatest, but 4.5 < 4.75.
15. Which expression represents a real number? $\sqrt{-6}$, $\ln(-2)$, $\frac{5}{5}$, $\frac{8}{0}$
    Answer: $\frac{5}{5}$ We need to identify which expression represents a real number. $\sqrt{-6}$ is undefined in real numbers (square root of negative), $\ln(-2)$ is undefined (logarithm of negative), $\frac{5}{5} = 1$ is a real number, and $\frac{8}{0}$ is undefined (division by zero). Only $\frac{5}{5}$ represents a real number.
16. What is the approximate value of $\sqrt{5}$?
    Answer: 2.24 We need to approximate $\sqrt{5}$. Since $2^2 = 4$ and $3^2 = 9$, $\sqrt{5}$ lies between 2 and 3. More precisely, since $2.2^2 = 4.84$ and $2.3^2 = 5.29$, $\sqrt{5}$ is approximately 2.24. Using a calculator confirms $\sqrt{5} \approx 2.236$. Therefore, 2.24 is the best approximation. Choice A gave 2.2, which is slightly too small since $2.2^2 = 4.84 < 5$.
17. Which number is greatest?
    Answer: $0$ To find the greatest number, we convert all to decimal form for comparison. Converting: -0.04 = -0.04, -1/20 = -0.05, 0 = 0, and -√0.0016 = -0.04. On the number line: -0.05 < -0.04 = -0.04 < 0, so 0 is the greatest value. Choice A incorrectly suggests a negative number could be greater than zero, but all negative numbers are less than zero.
18. A calculator display shows the value of $\sqrt{50}$ rounded to the nearest tenth. What is the approximate value of $\sqrt{50}$?
    Answer: $7.1$ To approximate √50, we identify nearby perfect squares: 7² = 49 and 8² = 64, so √50 is between 7 and 8. Since 50 is very close to 49, √50 is slightly greater than 7. Calculating more precisely: √50 ≈ 7.071, which rounds to 7.1 when rounded to the nearest tenth. Choice C (7.5) would be closer to √56.25, which is too large.
19. A measurement is written as $\sqrt{0.09}$. What is the approximate value of $\sqrt{0.09}$?
    Answer: $0.3$ To find the value of √0.09, we recognize this as a perfect square. Since 0.09 = 9/100 = (3/10)², we have √0.09 = 3/10 = 0.3. Alternatively, since (0.3)² = 0.09, the square root of 0.09 is 0.3. Choice A incorrectly gives 0.03, which would be the square root of 0.0009.
20. Order the following real numbers from least to greatest: $-2.1$, $-\dfrac{9}{4}$, $-\sqrt{5}$, $-2.05$.
    Answer: $-\dfrac{9}{4},\ -\sqrt{5},\ -2.1,\ -2.05$ To order negative numbers from least to greatest, convert each to decimal form and remember that more negative means smaller. Converting: -√5 ≈ -2.236, -9/4 = -2.25, -2.1 = -2.1, and -2.05 = -2.05. On the number line from left to right (least to greatest): -2.25 < -2.236 < -2.1 < -2.05. This gives the order: -9/4, -√5, -2.1, -2.05. Choice A incorrectly places -√5 first, forgetting that -2.25 is more negative than -2.236.

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