comparing rational and irrational numbers worksheet

Lemon

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04-02-2025

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This worksheet will help you solidify your understanding of rational and irrational numbers and how to compare them. Remember, a rational number can be expressed as a fraction p/q, where p and q are integers, and q is not zero. An irrational number cannot be expressed as a fraction of two integers; its decimal representation is non-terminating and non-repeating.

Section 1: Identifying Rational and Irrational Numbers

Instructions: Identify each number below as either rational (R) or irrational (I).

1. √9
2. π
3. 0.75
4. √2
5. -3/5
6. 0.121212... (repeating)
7. √16/9
8. 0.1010010001... (non-repeating, non-terminating)
9. 2.345
10. 1/√4

Section 2: Converting Rational Numbers to Decimals

Instructions: Convert each rational number below into its decimal form. Identify whether the decimal is terminating or repeating.

1. 2/5
2. 1/3
3. 7/8
4. 5/6
5. -4/11

Section 3: Comparing Rational and Irrational Numbers

Instructions: Compare the following pairs of numbers using the symbols <, >, or =. You may need to approximate the value of irrational numbers using a calculator. Remember to show your work or reasoning.

1. √4 and 2.1
2. π and 3.14
3. √10 and 3.2
4. -1/2 and -0.6
5. √25 and 5
6. 0.7 and √2/2
7. 3.14159 and π

Section 4: Ordering Numbers

Instructions: Arrange the following numbers in ascending order (from least to greatest):

√2, 1.5, 2, π, 0, -1, -0.5, √9

Section 5: Challenging Questions

1. Explain why the number 0.999... (repeating) is considered a rational number.
2. Can you provide an example of an irrational number between 2 and 3? Explain how you found it.
3. Is the sum of two rational numbers always rational? Explain your answer.
4. Is the sum of a rational number and an irrational number always irrational? Explain your answer.
5. Is the product of two irrational numbers always irrational? Explain your answer with examples.

Answer Key (Provided separately to the student to allow for self-checking and learning)

This worksheet provides a comprehensive exercise in understanding and comparing rational and irrational numbers. The variety of questions, from simple identification to more challenging problem-solving, helps students build a strong foundation in this mathematical concept. Remember to consult your textbook or teacher for further assistance if needed. Good luck!