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ACT Math

ACT Math Help: Real Numbers

Review real example questions for Real Numbers in ACT Math.

Question 1

What is the absolute value of −73-\frac{7}{3}−37​?

  1. −73-\frac{7}{3}−37​
  2. 73\frac{7}{3}37​
  3. 37\frac{3}{7}73​
  4. 777
Explanation: The absolute value of a number is its distance from zero on the number line, always positive or zero. The absolute value of −73-\frac{7}{3}−37​ is the distance from −73-\frac{7}{3}−37​ to 0, which is 73\frac{7}{3}37​ units. Therefore, ∣−73∣=73|-\frac{7}{3}| = \frac{7}{3}∣−37​∣=37​.

Question 2

Which number is greatest? −0.5-0.5−0.5, 0.10.10.1, −0.1-0.1−0.1, 0.050.050.05

  1. −0.5-0.5−0.5
  2. 0.10.10.1
  3. −0.1-0.1−0.1
  4. 0.050.050.05
Explanation: We need to identify the greatest among -0.5, 0.1, -0.1, and 0.05. On the number line, positive numbers are greater than negative numbers, and among positive numbers, larger values are to the right. Comparing the positive values: 0.1 > 0.05, and both are greater than the negative values -0.5 and -0.1. Therefore, 0.1 is the greatest.

Question 3

What is the absolute value of −94-\frac{9}{4}−49​?

  1. −94-\frac{9}{4}−49​
  2. 94\frac{9}{4}49​
  3. 49\frac{4}{9}94​
  4. 999
Explanation: 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 −94-\frac{9}{4}−49​ is the distance from −94-\frac{9}{4}−49​ to 0, which is 94\frac{9}{4}49​ units. Therefore, ∣−94∣=94\left| -\frac{9}{4} \right| = \frac{9}{4}​−49​​=49​.

Question 4

Which number is smallest? −2-2−2, −34-\frac{3}{4}−43​, 0.50.50.5, 111

  1. 0.50.50.5
  2. −34-\frac{3}{4}−43​
  3. −2-2−2
  4. 111
Explanation: We need to identify the smallest among -2, -3/4, 0.5, and 1. Convert to decimals: -2 = -2, -3/4 = -0.75, 0.5 = 0.5, 1 = 1. On the number line, negative numbers are to the left of positive numbers, and among negative numbers, the one with greater absolute value is smaller. Since |-2| = 2 > |-0.75| = 0.75, -2 is smaller than -0.75.

Question 5

Which expression represents a real number? −1\sqrt{-1}−1​, ln⁡(−1)\ln(-1)ln(−1), 40.54^{0.5}40.5, 10\frac{1}{0}01​

  1. −1\sqrt{-1}−1​
  2. ln⁡(−1)\ln(-1)ln(−1)
  3. 40.54^{0.5}40.5
  4. 10\frac{1}{0}01​
Explanation: We need to identify which expression represents a real number. √(-1) is undefined in real numbers (square root of negative), ln(-1) is undefined (logarithm of negative), 4^0.5 = √4 = 2 is a positive real number, and 1/0 is undefined (division by zero). Only 4^0.5 represents a real number.

Question 6

Which of the following lists the numbers 3.143.143.14, π\piπ, and 227\frac{22}{7}722​ in order from least to greatest? (Note: π≈3.14159...\pi \approx 3.14159...π≈3.14159...)

  1. 3.14<227<π3.14 < \dfrac{22}{7} < \pi3.14<722​<π
  2. 3.14<π<2273.14 < \pi < \dfrac{22}{7}3.14<π<722​
  3. π<3.14<227\pi < 3.14 < \dfrac{22}{7}π<3.14<722​
  4. 227<π<3.14\dfrac{22}{7} < \pi < 3.14722​<π<3.14
Explanation: This is an ordering of real numbers question testing number sense with irrational numbers. Choice B (3.14 < π < 22/7) is correct — converting to decimals: 3.14 = 3.1400..., π ≈ 3.14159..., 22/7 ≈ 3.14286. The correct order from least to greatest is: 3.14 < π < 22/7. Choice A (3.14 < 22/7 < π) places 3.14 correctly but swaps π and 22/7 — a very common error since 22/7 is often introduced as a shorthand for π, but it is actually slightly larger than π. Choice C (π < 3.14 < 22/7) incorrectly places π below 3.14, reversing their actual relationship — π ≈ 3.14159, which is greater than 3.14. Choice D (22/7 < π < 3.14) inverts the entire order, placing 22/7 as the smallest when it is actually the largest. Pro tip: Convert all three to decimals before comparing: 22 ÷ 7 ≈ 3.142857. This removes any ambiguity. Remember: 22/7 is a common approximation for π, but it overestimates π by about 0.001.

Question 7

What is the approximate value of 3\sqrt{3}3​?

  1. 1.71.71.7
  2. 1.731.731.73
  3. 1.81.81.8
  4. 1.831.831.83
Explanation: To approximate 3\sqrt{3}3​, we find perfect squares near 3. Since 12=11^2 = 112=1 and 22=42^2 = 422=4, 3\sqrt{3}3​ is between 1 and 2. More precisely, 1.72=2.891.7^2 = 2.891.72=2.89 and 1.82=3.241.8^2 = 3.241.82=3.24, so 3\sqrt{3}3​ is between 1.7 and 1.8. Calculating: 1.732=2.9929≈31.73^2 = 2.9929 \approx 31.732=2.9929≈3, so 3≈1.73\sqrt{3} \approx 1.733​≈1.73.

Question 8

Which number is greatest? −5-5−5, −2-2−2, 000, 333.

  1. −5-5−5
  2. −2-2−2
  3. 000
  4. 333
Explanation: We need to identify which number is greatest among -5, -2, 0, and 3. On the number line from left to right: -5 < -2 < 0 < 3. The rightmost position represents the greatest value. Therefore, 3 is the greatest number. Choice A incorrectly thought -5 was greatest, confusing the magnitude of negative numbers with their actual value.

Question 9

Which represents a rational number? 11\sqrt{11}11​, 0.750.750.75, π\piπ, eee

  1. 11\sqrt{11}11​
  2. 0.750.750.75
  3. π\piπ
  4. eee
Explanation: A rational number can be expressed as a fraction p/q where p and q are integers and q ≠ 0. √11 is irrational, 0.75 = 75/100 = 3/4 is rational (terminating decimal), π is irrational, and e is irrational. The decimal 0.75 terminates, making it rational.

Question 10

Which number is greatest? −3.5-3.5−3.5, −2.75-2.75−2.75, −4-4−4, −1.25-1.25−1.25.

  1. −3.5-3.5−3.5
  2. −2.75-2.75−2.75
  3. −4-4−4
  4. −1.25-1.25−1.25
Explanation: We need to identify which number is greatest among -3.5, -2.75, -4, and -1.25. All numbers are negative, so the one closest to zero is greatest. On the number line from left to right: -4 < -3.5 < -2.75 < -1.25. The rightmost position represents the greatest value. Therefore, -1.25 is the greatest number. Choice A incorrectly selected -3.5, not understanding that among negative numbers, the one with smallest magnitude is greatest.