Which number produces an irrational number when multiplied by [tex]0.5[/tex]?

A. [tex]\sqrt{16}[/tex]

B. [tex]\sqrt{3}[/tex]

C. [tex]\frac{1}{3}[/tex]

D. [tex]0.555 \ldots[/tex]



Answer :

To determine which number produces an irrational number when multiplied by [tex]\(0.5\)[/tex], let's analyze each of the given choices step-by-step:

1. Choice A: [tex]\(\sqrt{16}\)[/tex]

[tex]\[ \sqrt{16} = 4 \][/tex]

When multiplied by [tex]\(0.5\)[/tex]:

[tex]\[ 4 \times 0.5 = 2 \][/tex]

Since 2 is a rational number, Choice A does not produce an irrational number.

2. Choice B: [tex]\(\sqrt{3}\)[/tex]

[tex]\[ \sqrt{3} \][/tex]

[tex]\(\sqrt{3}\)[/tex] is an irrational number by itself. When multiplied by [tex]\(0.5\)[/tex]:

[tex]\[ \sqrt{3} \times 0.5 = \frac{\sqrt{3}}{2} \][/tex]

Since the product of an irrational number and a nonzero rational number is still irrational, [tex]\(\frac{\sqrt{3}}{2}\)[/tex] is irrational. Hence, Choice B produces an irrational number.

3. Choice C: [tex]\(\frac{1}{3}\)[/tex]

When multiplied by [tex]\(0.5\)[/tex]:

[tex]\[ \frac{1}{3} \times 0.5 = \frac{1}{6} \][/tex]

Since [tex]\(\frac{1}{6}\)[/tex] is a rational number, Choice C does not produce an irrational number.

4. Choice D: [tex]\(0.555\ldots\)[/tex] (repeating decimal)

This value is a rational number because repeating decimals can be expressed as fractions. For example:

[tex]\[ 0.555\ldots = \frac{5}{9} \][/tex]

When multiplied by [tex]\(0.5\)[/tex]:

[tex]\[ 0.555\ldots \times 0.5 = \frac{5}{9} \times 0.5 = \frac{5}{18} \][/tex]

Since [tex]\(\frac{5}{18}\)[/tex] is a rational number, Choice D does not produce an irrational number.

From the analysis, we see that the only choice that results in an irrational product when multiplied by [tex]\(0.5\)[/tex] is Choice B: [tex]\(\sqrt{3}\)[/tex].

Thus, the answer is:
[tex]\[ \boxed{B} \][/tex]