Home > Complex numbers > Complex Numbers: four rules

Complex Numbers: four rules

Consider complex numbers in the form a+bi

Adding complex numbers

When adding complex numbers we add the real part and then we add the imaginary part

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

Advertisements
Categories: Complex numbers
  1. No comments yet.
  1. No trackbacks yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: