Rational Irrational Numbers Worksheet - Answer Key

Name:

Date:

📚 Part 1: Classification - Multiple Choice

1. Which of the following is a rational number?

√2

✓

0.75

√5

Answer: 0.75 is rational because it can be written as 3/4

2. Which number is irrational?

4/5

✓

√3

0.333...

-7

Answer: √3 is irrational because it cannot be expressed as a fraction and has a non-terminating, non-repeating decimal

3. The decimal 0.142857142857... is:

✓

Rational

Irrational

Answer: Rational because it has a repeating pattern (142857 repeats)

4. Which of these are irrational numbers? (Select all that apply)

✓

√9

✓

√7

2.5

Answer: π and √7 are irrational. √9 = 3 (rational), 2.5 = 5/2 (rational)

✏️ Part 2: Explanations and Analysis

5. Explain why √16 is rational but √15 is irrational.

Sample Answer: √16 = 4, which is a whole number and can be written as 4/1, making it rational. √15 cannot be simplified to a whole number or fraction and has a non-terminating, non-repeating decimal expansion, making it irrational.

6. A student claims that 0.999... is irrational because "it goes on forever." Explain why this reasoning is incorrect.

Sample Answer: 0.999... is rational because it has a repeating pattern (the digit 9 repeats). It can be expressed as the fraction 1/1 = 1. A decimal "going on forever" doesn't make it irrational - it must be non-repeating AND non-terminating to be irrational.

7. Complete the table by writing R for rational or I for irrational:

√25 = R (because √25 = 5)

π/2 = I (because π is irrational, so π/2 is also irrational)

0.125 = R (because 0.125 = 1/8)

√2 + 3 = I (because √2 is irrational, adding 3 doesn't change this)

-4/7 = R (because it's already expressed as a fraction)

8. Using a calculator, approximate √2 to 5 decimal places and explain why this doesn't make √2 rational.

Sample Answer: √2 ≈ 1.41421 (calculator approximation). This doesn't make √2 rational because the calculator can only display a limited number of digits. The actual decimal expansion of √2 continues infinitely without repeating, so it cannot be expressed as an exact fraction.

🎯 Part 3: Problem Solving

9. Create your own example of a rational number and explain why it's rational.

Sample Answer: 3/8 = 0.375. This is rational because it can be expressed as a fraction where both numerator (3) and denominator (8) are integers, and the denominator is not zero. The decimal form terminates after 3 digits.

10. Explain the difference between rational and irrational numbers using mathematical vocabulary.

Sample Answer: Rational numbers can be expressed as a fraction a/b where a and b are integers and b ≠ 0. Their decimal expansions either terminate or repeat. Irrational numbers cannot be expressed as fractions and have decimal expansions that are non-terminating and non-repeating.