Operations with Complex Numbers

Let z=7-4i and w=10+3i. Find (z+w) and (z)(w). Simplify

 

z+w=(7-4i)+(10+3i)

  • real: 7, 10
  • imaginary: -4i, 3i

z+w=(7+10)+(-4i+3i)=17-i

(z)(w)=(7-4i)(10+3i)=(7)(10)+(7)(3i)+(-4i)(10)+(-4i)(3i)

  • First                 =70+21i-40i-12i^2
  • Outside          =70-19i-12(-1)
  • Inside              =70-19i+12
  • Last                  =82-19i

 

Let z=12+6i and w=1+i. Find (z-w) and (\frac{z}{w}). Simplify.

 

z-w=(12+6i)-(1+i)=(12-1)+(6i-i)=11+5i

\frac{z}{w}=\frac{12+6i}{1+i}\times\frac{1-i}{1-i}=\frac{12-12i+6i-(6i)(i)}{1+i-i-12}

   =\frac{12-6i-6i^2}{1-i^2}=\frac{12-6i-6(-1)}{1--1}=\frac{12-6i+6}{2}=\frac{18-6i}{2}

   =9-3i

 

Find the value of (a+bi)(a-bi). Use that to find the value of (2+\sqrt{3}i)(2-\sqrt{3}i).

(a+bi)(a-bi)=(a)(a)+(a)(-bi)+(bi)(a)+(bi)(-bi)

                      =a^{2}-b^{2}i^{2}

                      =a^{2}-b^{2}(-1)=a^{2}+b^{2}

 

(2+\sqrt{3}i)(2-\sqrt{3}i)=a^{2}+b^{2}=2^{2}+\sqrt{3}^{2}=4+3=7

a=2, b=\sqrt{3}