Let w = 3 + 4i and z = 5 - 2i . Compute zz, |z|, and w/z.

Question

[latex] ext { Let } w=3+4 i ext { and } z=5-2 i ext {. Compute } z \bar{z},|z| ext {, and } w / z [/latex].

Accepted Answer

From equation (3),

[latex]z \bar{z}=(a+b i)(a-b i)=a^{2}-a b i+b a i-b^{2} i^{2}=a^{2}+b^{2}[/latex] (3).

[latex]z \bar{z}=5^{2}+(-2)^{2}=25+4=29[/latex].

For the absolute value, [latex]|z|=\sqrt{z \bar{z}}=\sqrt{29} . ext { To compute } w / z[/latex], first multiply both the numerator and the denominator by [latex]\bar{z}[/latex], the conjugate of the denominator. Because of (3), this eliminates the i in the denominator:

[latex]z \bar{z}=(a+b i)(a-b i)=a^{2}-a b i+b a i-b^{2} i^{2}=a^{2}+b^{2}[/latex] (3).

[latex]\frac{w}{z}=\frac{3+4 i}{5-2 i}[/latex].

[latex]=\frac{3+4 i}{5-2 i} \cdot \frac{5+2 i}{5+2 i}[/latex].

[latex]=\frac{15+6 i+20 i-8}{5^{2}+(-2)^{2}}[/latex].

[latex]=\frac{7+26 i}{29}[/latex].

[latex]=\frac{7}{29}+\frac{26}{29} i[/latex].

Question B.3: Let w = 3 + 4i and z = 5 - 2i . Compute zz, |z|, and w/z.