This worksheet provides a thorough exploration of rational and irrational numbers, focusing on comparison and understanding their fundamental differences. We'll move beyond simple definitions to delve into practical application and problem-solving. Understanding this distinction is crucial for success in algebra and beyond.
What are Rational and Irrational Numbers?
Before we begin comparing them, let's review the definitions:
Rational Numbers: These numbers can be expressed as a fraction p/q, where 'p' and 'q' are integers, and 'q' is not zero. This includes:
- Integers: Whole numbers (positive, negative, and zero). Examples: -3, 0, 5
- Terminating Decimals: Decimals that end. Examples: 0.75, 2.5, -3.125
- Repeating Decimals: Decimals with a pattern that repeats infinitely. Examples: 0.333..., 0.142857142857...
Irrational Numbers: These numbers cannot be expressed as a fraction p/q. Their decimal representation is non-terminating and non-repeating. Examples:
- √2: The square root of 2 (approximately 1.41421356...)
- π (pi): The ratio of a circle's circumference to its diameter (approximately 3.14159...)
- e (Euler's number): The base of the natural logarithm (approximately 2.71828...)
Comparing Rational and Irrational Numbers
Comparing rational and irrational numbers often involves approximating irrational numbers to their decimal form and then comparing them to rational numbers in decimal form as well. Let's work through some examples:
Example 1:
Compare √9 and 3.1
Solution: √9 = 3. Since 3 < 3.1, √9 < 3.1
Example 2:
Compare π and 22/7
Solution: π ≈ 3.14159... and 22/7 ≈ 3.142857... In this case, 22/7 is a slightly better approximation of π than 3.14159, therefore, π < 22/7 (though they are very close!)
Example 3:
Which is greater: √5 or 2.2?
Solution: √5 ≈ 2.236. Therefore, √5 > 2.2
Example 4:
Order the following numbers from least to greatest: 2.5, √7, 17/7, 2.7
Solution: First, approximate: √7 ≈ 2.646 and 17/7 ≈ 2.429. Ordering from least to greatest: 17/7, 2.5, √7, 2.7
Practice Problems
Now it's your turn! Compare the following numbers using <, >, or =. Show your work where necessary by approximating irrational numbers to at least three decimal places.
- √16 and 4
- π and 3.14
- √2 and 1.5
- 2.75 and 11/4
- √10 and 3.2
- 0.666... and 2/3
- √3 and 1.73
- Order from least to greatest: 2.8, √8, 22/9, 3
Advanced Problems (Challenge)
- Explain why the sum of a rational number and an irrational number is always irrational.
- Can the product of two irrational numbers be rational? Provide an example.
This worksheet aims to strengthen your understanding of rational and irrational numbers and their comparison. Remember to practice regularly to improve your skills. Good luck!