Which of the following numbers will produce an irrational number when multiplied by 0.4?
A. $\sqrt{13}$
B. $3 \pi$
C. $0.444 \ldots$
D. $\frac{2}{7}$
Answer (2)
Convert 0.4 to a fraction: 0.4 = 5 2 .
Multiply each option by 5 2 and determine if the result is irrational.
5 2 ⋅ 13 = 5 2 13 which is irrational.
5 2 ⋅ 3 π = 5 6 π which is irrational.
5 2 ⋅ 0.444 … = 5 2 ⋅ 9 4 = 45 8 which is rational.
5 2 ⋅ 7 2 = 35 4 which is rational.
The numbers that produce an irrational number when multiplied by 0.4 are 13 and 3 π .
Explanation
Understanding the Problem We are given four numbers: 13 , 3 π , 0.444 … , and 7 2 . We need to determine which of these, when multiplied by 0.4, results in an irrational number.
Key Concepts
4 can be expressed as the fraction 5 2 . A number is rational if it can be expressed as a fraction q p , where p and q are integers and q is not zero. Otherwise, the number is irrational. The product of a rational and irrational number is irrational, unless the rational number is zero. The product of two rational numbers is rational.
Analyzing Option A A. 0.4 ⋅ 13 = 5 2 13 . Since 13 is irrational, and 5 2 is rational, the product is irrational.
Analyzing Option B B. 0.4 ⋅ 3 π = 5 2 ⋅ 3 π = 5 6 π . Since π is irrational, and 5 6 is rational, the product is irrational.
Analyzing Option C C. 0.4 ⋅ 0.444 … = 5 2 ⋅ 9 4 = 45 8 . This is rational.
Analyzing Option D D. 0.4 ⋅ 7 2 = 5 2 ⋅ 7 2 = 35 4 . This is rational.
Conclusion Therefore, the numbers 13 and 3 π will produce an irrational number when multiplied by 0.4.
Examples
Understanding irrational numbers is crucial in fields like engineering and physics, where precise calculations are essential. For instance, when designing circular structures or calculating wave patterns, irrational numbers like π and square roots frequently appear. Knowing how these numbers behave in calculations ensures accuracy and prevents errors in real-world applications.
The numbers that will produce an irrational number when multiplied by 0.4 are 13 and 3 π . Option C, 0.444 … , and Option D, 7 2 , produce rational numbers when multiplied by 0.4.
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