Question

How many complex numbers have a modulus of 5?

Answer 1

Infinite
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Answer 2

• To find a modulus of a complex number

      z = a+bi

     |z| = $\sqrt{a^{2}+b^{2} }$

• so we have complex number which gives the modulus 5 are
• z = 3+4i      |z| = $\sqrt{3^{2}+4^{2} }$              =5

• z = 3-4i       |z| = $\sqrt{3^{2}+(-4)^{2} }$        =5

• z = -3+4i     |z| = $\sqrt{(-3)^{2}+4^{2} }$        =5
• z = -3-4i      |z| = $\sqrt{(-3)^{2}+(-4)^{2} }$   =5
• z = 4+3i      |z| = $\sqrt{4^{2}+3^{2} }$               =5
• z = 4-3i       |z| = $\sqrt{4^{2}+(-3)^{2} }$         =5
• z = -4+3i     |z| = $\sqrt{(-4)^{2}+3\\^{2} }$         =5
• z = -4-3i      |z| = $\sqrt{(-4)^{2}+(-3)^{2} }$   =5
• z = 5+0i      |z| = $\sqrt{5^{2}+0^{2} }$              =5
• z = -5+0i     |z| = $\sqrt{(-5)^{2}+0^{2} }$       =5
• z = 0+5i      |z| = $\sqrt{0^{2}+5^{2} }$             =5
• z = 0-5i       |z| = $\sqrt{(0)^{2}+(-5)^{2} }$   =5
• So 12 complex numbers have modulus 5.