Complex Numbers

Complex number is a number in the form z = a+bi, where a and b are real numbers and i is imaginary. Here a is the real part and b is the imaginary part of z. Where Examples: If r is a positive real number, then

Geometry Plot the points 3 + 4i and –2 – 2i in the complex plane. Imaginary axis Real axis 2 4 – 2 2 (3, 4) or 3 + 4i (– 2, – 2) or – 2 – 2i

Operations on Complex Numbers Which one is true? or

Absolute Value The absolute value of the complex number z = a + bi is the distance between the origin (0, 0) and the point (a, b). Example: Plot z = 3 + 6i and find its absolute value. Imaginary axis Real axis 4 4 – 2 – z = 3 + 6i

Trigonometric Representation of Complex Number To write a complex number z = x + yi in trigonometric form, let  be the angle from the positive real axis (measured counter clockwise) to the line segment connecting the origin to the point (x, y). x = r cos  y = r sin  Imaginary axis Real axis y r x z = (x, y)  z = x + yi = r (cos  + i sin  ) The number r is the modulus of z, and  is the argument of z. Example: modulus argument Let and

Example Write the complex number z = –7 + 4i in trigonometric form. Imaginary axis Real axis z = –7 + 4i °

Standard form Write the complex number in standard form a + bi. Example:

De Moivre’s Theorem Expanding a power of a complex number in rectangular form is tedious. The best way to expand one of these is using Pascal’s triangle and binomial expansion. It’s much nicer in trig form. We just raise the r to the power and multiply theta by the exponent.

Nth Root of a Complex Number ; k =0,1,…,n-1 Put K=0,1,2,3 into the above equation we get 4 roots as follows: Hint: 1=1+i.0=1(cos0+isin0)