The Real Number System Notes and Worksheets - Lindsay Bowden - Free Printable
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Step-by-step solution for: The Real Number System Notes and Worksheets - Lindsay Bowden
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Step-by-step solution for: The Real Number System Notes and Worksheets - Lindsay Bowden
Let’s go through each problem one by one, step by step, and solve them carefully.
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Problem 1:
Find the product, then determine if the factors and product are rational or irrational.
√6 • 3
Step 1: Multiply √6 and 3 → 3√6
Step 2: Is √6 rational? No — it’s irrational (can’t be written as a fraction).
Step 3: Is 3 rational? Yes — it’s a whole number.
Step 4: Is 3√6 rational? No — multiplying an irrational number by a non-zero rational number gives an irrational number.
✔ Factors: √6 (irrational), 3 (rational)
✔ Product: 3√6 (irrational)
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Problem 2:
Fill in the blank.
The product of a non-zero rational number and an irrational number is always __________.
This is a rule:
If you multiply a non-zero rational number (like 2, -5, 1/3) by an irrational number (like √2, π), the result is always irrational.
Example: 2 × √3 = 2√3 → still irrational.
✔ Answer: irrational
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Problem 3:
Finish the sentence.
The sum of a rational number and an irrational number is always __________.
Rule: Adding a rational number (like 5 or -1/2) to an irrational number (like √7 or π) always gives an irrational number.
Example: 4 + √2 → can’t simplify to a fraction → irrational.
✔ Answer: irrational
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Problem 4:
Find the sum, then determine if the addends and sum are rational or irrational.
√24 + 7
Step 1: Simplify √24 → √(4×6) = 2√6 → still irrational
Step 2: Add 7 → 2√6 + 7 → still irrational (adding rational to irrational stays irrational)
Step 3: Check addends:
- √24 → irrational
- 7 → rational
Step 4: Sum → irrational
✔ Addends: √24 (irrational), 7 (rational)
✔ Sum: 2√6 + 7 (irrational)
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Problem 5:
Find the product, then determine if the factors and product are rational or irrational.
√6 • 0
Step 1: Multiply → √6 × 0 = 0
Step 2: Is √6 rational? No → irrational
Step 3: Is 0 rational? Yes → zero is rational
Step 4: Is 0 rational? Yes → product is 0, which is rational
⚠️ Note: This is a special case! Multiplying any number by 0 gives 0, which is rational — even if one factor is irrational.
✔ Factors: √6 (irrational), 0 (rational)
✔ Product: 0 (rational)
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Problem 6:
Find the sum, then determine if the addends and sum are rational or irrational.
-0.37 + 1/2
Step 1: Convert to same form. Let’s use decimals.
1/2 = 0.5
So: -0.37 + 0.5 = 0.13
Step 2: Are addends rational?
- -0.37 → yes (can write as -37/100)
- 1/2 → yes
Step 3: Is 0.13 rational? Yes → 13/100
✔ Addends: both rational
✔ Sum: 0.13 (rational)
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Problem 7:
Prove this statement with two examples:
“The product of two rational numbers is always rational.”
We need two examples where we multiply two rational numbers and get a rational result.
Example 1: 2 × 3 = 6 → 6 is rational
Example 2: 1/4 × 2/5 = 2/20 = 1/10 → 1/10 is rational
Both results are fractions or integers → rational.
✔ Proven with examples.
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Problem 8:
Find the difference. Then determine if each number (including the answer) is rational or irrational.
√24 – 54
Step 1: Simplify √24 → 2√6 → irrational
Step 2: Subtract 54 → 2√6 – 54 → still irrational (subtracting rational from irrational stays irrational)
Step 3: Check each part:
- √24 → irrational
- 54 → rational
- Result → irrational
✔ Numbers: √24 (irrational), 54 (rational), difference (irrational)
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Problem 9:
Determine if the statement is true or false. If it’s false, prove it with an example.
“The product of a rational number and an irrational number will always be irrational.”
Wait — is this *always* true?
No! There’s an exception: if the rational number is 0.
Because 0 × anything = 0 → and 0 is rational.
Example:
Rational = 0, Irrational = √5
Product = 0 × √5 = 0 → which is rational, not irrational.
So the statement is FALSE.
✔ False. Example: 0 × √5 = 0 (rational)
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Problem 10:
Find the product, then determine if the factors and product are rational or irrational.
-1/5 • 7
Step 1: Multiply → (-1/5) × 7 = -7/5
Step 2: Is -1/5 rational? Yes → fraction
Step 3: Is 7 rational? Yes → integer
Step 4: Is -7/5 rational? Yes → fraction
✔ Factors: -1/5 (rational), 7 (rational)
✔ Product: -7/5 (rational)
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Final Answer:
1. Product: 3√6; factors: √6 (irrational), 3 (rational); product: irrational
2. irrational
3. irrational
4. Sum: 2√6 + 7; addends: √24 (irrational), 7 (rational); sum: irrational
5. Product: 0; factors: √6 (irrational), 0 (rational); product: rational
6. Sum: 0.13; addends: -0.37 (rational), 1/2 (rational); sum: rational
7. Example 1: 2 × 3 = 6 (rational); Example 2: 1/4 × 2/5 = 1/10 (rational)
8. Difference: 2√6 – 54; numbers: √24 (irrational), 54 (rational), difference: irrational
9. False. Example: 0 × √5 = 0 (rational)
10. Product: -7/5; factors: -1/5 (rational), 7 (rational); product: rational
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Problem 1:
Find the product, then determine if the factors and product are rational or irrational.
√6 • 3
Step 1: Multiply √6 and 3 → 3√6
Step 2: Is √6 rational? No — it’s irrational (can’t be written as a fraction).
Step 3: Is 3 rational? Yes — it’s a whole number.
Step 4: Is 3√6 rational? No — multiplying an irrational number by a non-zero rational number gives an irrational number.
✔ Factors: √6 (irrational), 3 (rational)
✔ Product: 3√6 (irrational)
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Problem 2:
Fill in the blank.
The product of a non-zero rational number and an irrational number is always __________.
This is a rule:
If you multiply a non-zero rational number (like 2, -5, 1/3) by an irrational number (like √2, π), the result is always irrational.
Example: 2 × √3 = 2√3 → still irrational.
✔ Answer: irrational
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Problem 3:
Finish the sentence.
The sum of a rational number and an irrational number is always __________.
Rule: Adding a rational number (like 5 or -1/2) to an irrational number (like √7 or π) always gives an irrational number.
Example: 4 + √2 → can’t simplify to a fraction → irrational.
✔ Answer: irrational
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Problem 4:
Find the sum, then determine if the addends and sum are rational or irrational.
√24 + 7
Step 1: Simplify √24 → √(4×6) = 2√6 → still irrational
Step 2: Add 7 → 2√6 + 7 → still irrational (adding rational to irrational stays irrational)
Step 3: Check addends:
- √24 → irrational
- 7 → rational
Step 4: Sum → irrational
✔ Addends: √24 (irrational), 7 (rational)
✔ Sum: 2√6 + 7 (irrational)
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Problem 5:
Find the product, then determine if the factors and product are rational or irrational.
√6 • 0
Step 1: Multiply → √6 × 0 = 0
Step 2: Is √6 rational? No → irrational
Step 3: Is 0 rational? Yes → zero is rational
Step 4: Is 0 rational? Yes → product is 0, which is rational
⚠️ Note: This is a special case! Multiplying any number by 0 gives 0, which is rational — even if one factor is irrational.
✔ Factors: √6 (irrational), 0 (rational)
✔ Product: 0 (rational)
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Problem 6:
Find the sum, then determine if the addends and sum are rational or irrational.
-0.37 + 1/2
Step 1: Convert to same form. Let’s use decimals.
1/2 = 0.5
So: -0.37 + 0.5 = 0.13
Step 2: Are addends rational?
- -0.37 → yes (can write as -37/100)
- 1/2 → yes
Step 3: Is 0.13 rational? Yes → 13/100
✔ Addends: both rational
✔ Sum: 0.13 (rational)
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Problem 7:
Prove this statement with two examples:
“The product of two rational numbers is always rational.”
We need two examples where we multiply two rational numbers and get a rational result.
Example 1: 2 × 3 = 6 → 6 is rational
Example 2: 1/4 × 2/5 = 2/20 = 1/10 → 1/10 is rational
Both results are fractions or integers → rational.
✔ Proven with examples.
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Problem 8:
Find the difference. Then determine if each number (including the answer) is rational or irrational.
√24 – 54
Step 1: Simplify √24 → 2√6 → irrational
Step 2: Subtract 54 → 2√6 – 54 → still irrational (subtracting rational from irrational stays irrational)
Step 3: Check each part:
- √24 → irrational
- 54 → rational
- Result → irrational
✔ Numbers: √24 (irrational), 54 (rational), difference (irrational)
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Problem 9:
Determine if the statement is true or false. If it’s false, prove it with an example.
“The product of a rational number and an irrational number will always be irrational.”
Wait — is this *always* true?
No! There’s an exception: if the rational number is 0.
Because 0 × anything = 0 → and 0 is rational.
Example:
Rational = 0, Irrational = √5
Product = 0 × √5 = 0 → which is rational, not irrational.
So the statement is FALSE.
✔ False. Example: 0 × √5 = 0 (rational)
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Problem 10:
Find the product, then determine if the factors and product are rational or irrational.
-1/5 • 7
Step 1: Multiply → (-1/5) × 7 = -7/5
Step 2: Is -1/5 rational? Yes → fraction
Step 3: Is 7 rational? Yes → integer
Step 4: Is -7/5 rational? Yes → fraction
✔ Factors: -1/5 (rational), 7 (rational)
✔ Product: -7/5 (rational)
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Final Answer:
1. Product: 3√6; factors: √6 (irrational), 3 (rational); product: irrational
2. irrational
3. irrational
4. Sum: 2√6 + 7; addends: √24 (irrational), 7 (rational); sum: irrational
5. Product: 0; factors: √6 (irrational), 0 (rational); product: rational
6. Sum: 0.13; addends: -0.37 (rational), 1/2 (rational); sum: rational
7. Example 1: 2 × 3 = 6 (rational); Example 2: 1/4 × 2/5 = 1/10 (rational)
8. Difference: 2√6 – 54; numbers: √24 (irrational), 54 (rational), difference: irrational
9. False. Example: 0 × √5 = 0 (rational)
10. Product: -7/5; factors: -1/5 (rational), 7 (rational); product: rational
Parent Tip: Review the logic above to help your child master the concept of the real number system worksheet.
