### Complex Numbers

A complex number is a number of the form a + bi, where i = and a and b are real numbers. For example, 5 + 3i, - + 4i, 4.2 - 12i, and - - i are all complex numbers. a is called the real part of the complex number and bi is called the imaginary part of the complex number. In the complex number 6 - 4i, for example, the real part is 6 and the imaginary part is -4i.

### Adding and Subtracting Complex Numbers

To add two complex numbers, add their real parts and add their imaginary parts: (a1 + b1i) + (a2 + b2i) = (a1 + a2) + (b1 + b2)i.

Examples:

(12 + 6i) + (11 + 5i) = (12 + 11) + (6 + 5)i = 23 + 11i
(5 - 7i) + (4 + i) = (5 + 4) + (- 7 + 1)i = 9 - 6i.
(2 - 4i) + (- 6 - 5i) = (2 - 6) + (- 4 - 5)i = - 4 - 9i.

To subtract two complex numbers, subtract their real parts and subtract their imaginary parts: (a1 + b1i) - (a2 + b2i) = (a1 - a2) + (b1 - b2)i.

Examples:

(4 + 5i) - (2 + 3i) = (4 - 2) + (5 - 3)i = 2 + 2i.
(3 - 7i) - (4 + 6i) = (3 - 4) + (- 7 - 6)i = - 1 - 13i
(- 4 + 2i) - (3 - 11i) = (- 4 - 3) + (2 - (- 11))i = - 7 + 13i
(6 - 9i) - (- 3 - 4i) = (6 - (- 3)) + (- 9 - (- 4))i = 9 - 5i

### Multiplying a Complex Number by a Scalar

To multiply a complex number by a scalar, multiply the real part by the scalar and multiply the imaginary part by the scalar: c(a + bi) = ca + cbi.