To work with complex numbers, it's important to know their standard form. This is written as ( z = a + bi ). Here, ( a ) is the real part, and ( b ) is the imaginary part. The letter ( i ) represents the imaginary unit, which means ( i^2 = -1 ).

Adding and Subtracting Complex Numbers

• If we have two complex numbers, ( z_1 = a + bi ) and ( z_2 = c + di ):
  • To add them, you would do: [ z_1 + z_2 = (a + c) + (b + d)i ]
  • To subtract them, you would do: [ z_1 - z_2 = (a - c) + (b - d)i ]

Multiplying Complex Numbers

• For the same ( z_1 ) and ( z_2 ):
  • To multiply them, you would calculate: [ z_1 \times z_2 = (ac - bd) + (ad + bc)i ]

Dividing Complex Numbers

If you want to divide ( z_1 ) by ( z_2 ):

• You need to multiply both the top (numerator) and the bottom (denominator) by the conjugate of the bottom number. The formula looks like this: [ \frac{z_1}{z_2} = \frac{(a + bi)(c - di)}{c^2 + d^2} ]

Special Features of Complex Numbers

1. Magnitude: The size or length of ( z ), written as ( |z| ), is found using: [ |z| = \sqrt{a^2 + b^2} ]

2. Conjugate: The conjugate of ( z ) is written as ( \overline{z} = a - bi ). This shows ( z ) mirrored over the real number line.

3. Polar Form: You can also express complex numbers in polar form. This looks like: [ z = r(\cos \theta + i \sin \theta) ] Here, ( r = |z| ) (the magnitude) and ( \theta = \tan^{-1}(b/a) ) (the angle).

Understanding these basics helps you work confidently with complex numbers!