To determine which of the following complex numbers has an absolute value (or modulus) of 5, we must first understand how to calculate the absolute value of a complex number. For a complex number [tex]\(a + bi\)[/tex], its absolute value is calculated as follows:
[tex]\[ |a + bi| = \sqrt{a^2 + b^2} \][/tex]
Let's evaluate each given complex number step-by-step.
1. For the complex number [tex]\(-3 + 4i\)[/tex]:
[tex]\[
| -3 + 4i | = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
So, the absolute value of [tex]\(-3 + 4i\)[/tex] is 5.
2. For the complex number [tex]\(2 + 3i\)[/tex]:
[tex]\[
| 2 + 3i | = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6
\][/tex]
The absolute value of [tex]\(2 + 3i\)[/tex] is approximately 3.6, which is not 5.
3. For the complex number [tex]\(7 - 2i\)[/tex]:
[tex]\[
| 7 - 2i | = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \approx 7.28
\][/tex]
The absolute value of [tex]\(7 - 2i\)[/tex] is approximately 7.28, which is not 5.
4. For the complex number [tex]\(9 + 4i\)[/tex]:
[tex]\[
| 9 + 4i | = \sqrt{9^2 + 4^2} = \sqrt{81 + 16} = \sqrt{97} \approx 9.85
\][/tex]
The absolute value of [tex]\(9 + 4i\)[/tex] is approximately 9.85, which is not 5.
Given these calculations, the complex number [tex]\(-3 + 4i\)[/tex] has an absolute value of 5.
