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Complex Numbers: four rules

Consider complex numbers in the form $a+bi$

Examples

$(2+i)+(4+i)=(2+4)+(1+1)i=6+2i$

$(3+5i)+(-2-3i)=(3-2)+(5-3)i=1+2i$

$(4-i)+(-4+7i)=(4-4)+(-1+7)i=6i$

Subtracting complex numbers

When subtracting complex numbers we subtract the real part and then we subtract the imaginary part

Examples

$(2+i)-(4+i)=(2-4)+(1-1)i=-2$

$(3+5i)-(-2-3i)=(3+2)+(5+3)i=5+8i$

$(4-i)-(-4+7i)=(4+4)+(-1-7)i=8-8i$

Multiplying complex numbers

When multiplying complex numbers we need to remember that $i\times i = -1$

Examples

$(2+i)(4+i)=8+2i+4i+i^2=7+6i$

$(3+5i)(-2-3i)=-6-9i-10i-15i^2=9-19i$

$(4-i)(-4+7i)=-16+28i+4i-7i^2=-9+32i$

Dividing complex numbers

Dividing complex numbers requires us to get rid of the imaginary part from the denominator.

We can do this by multiplying the denominator and numerator by the complex conjugate of the denominator.

Examples

$\displaystyle z=\frac{2+8i}{1+i}$

$\displaystyle \implies z=\frac{2+8i}{1+i}\times \frac{1-i}{1-i}$

$\displaystyle \implies z=\frac{10+6i}{2}=5+3i$

$\displaystyle w=\frac{3-i}{4+i}$

$\displaystyle \implies w=\frac{3-i}{4+i}\times \frac{4-i}{4-i}$

$\displaystyle \implies w=\frac{11-7i}{17}$

$\displaystyle \implies w=\frac{11}{17}- \frac{7}{17}i$