2. A complex number is any number of the form a + bi where a and b are real numbers.
To add or subtract two complex numbers, you add or subtract the real parts and the imaginary parts.
(a + bi) + (c + id) = (a + c) + (b + d)i.
(a + bi) - (c + id) = (a - c) + (b - d)i.
(3 - 5i) + (6 + 7i) = (3 + 6) + (-5 + 7)i = 9 + 2i.
(3 - 5i) - (6 + 7i) = (3 - 6) + (-5 - 7)i = -3 - 12i.
Let's take specific complex numbers to multiply, say 2 + 3i and 2 - 5i.
(2 + 3i)(2 - 5i) = 4 - 10i + 6i - 15i2 = 4 - 4i - 15i2
The definition of i tells us that i2 = -1 . Therefore,
(2 + 3i)(2 - 5i) = 4 - 4i -15(-1) = 19 - 4i.
If you generalize this example, you'll get the general rule for multiplication
(x + yi)(u + vi) = (xu - yv) + (xv + yu)i
We define the conjugate of a + bi as
Conjugates are important because of the fact that a complex number times its conjugate is real.
We define modulus or absolute value of complex number a + bi as . We write modulus of a + bi as |a + bi|.
|3 + 4i| =
The process of division of complex numbers:
step 1: Find the conjugate of a denominator.
step 2: Multiply the complex fraction, both top and bottom complex number.
Here is the complete division problem:
Now, we can write down a general formula for division of complex numbers
Every nth - order polynomial possess exactly n complex roots.