<
There is a new “number” that we associate negative numbers with under the square root. These are called COMPLEX NUMBERS, and they are indicated with the letter i, where: \(i=\sqrt{-1}\)
We can do regular operations like addition, subtraction, multiplication, and division with complex numbers. Keep the following in mind, to make complex multiplication simpler.
\(i=\sqrt{-1}\)
\(i^{2}=i\times i=\sqrt{-1}\times \sqrt{-1}=-1\)
\(i^{3}=i\times i\times i=\sqrt{-1}\times \sqrt{-1}\times \sqrt{-1}= -1\times \sqrt{-1}=-i\)
\(i^{4}=i^{2}\times i^{2}=-1\times -1=1\)
\(i^{5}=i^{4}\times i=i\)
\(i^{6}=i^{4}\times i^{2}=-1\)
\(i^{7}=i^{4}\times i^{3}=-i\)
\(i^{8}=i^{4}\times i^{4}=1\)
Solved Example 1
Find the value of \(i^{203}\)
A. i
B. -i
C. 1
D. -1
Solution: Now, we need to first break up \(i^{203}\: to\: i^{202+1}\)
\(i^{202}\times i^{1}\)
\((i^{2})^{101}\times i\)
\((-1)^{101}\times i\)
-1 x i
= -i
The correct answer is B.
Adding and subtracting complex numbers is similar to adding and subtracting polynomials. Just add or subtract the real and imaginary parts.
Example: Solve: (-6 - 5i) - (3 - 4i)
(-6 - 5i) - (3 - 4i) = (-6 - 5i) + (-3 + 4i)
= [-6 + (-3)] + (-5 + 4)i
= -9 - i
Multiplying complex numbers is similar to multiplying polynomials.
Solve: (2 + 5i) (6 + 4i)
(2 + 5i) (6 + 4i) = 12 + 8i + 30i + 20i2
= 12 + 38i + 20(-1)
= 12 + 38i - 20
= -8 + 38i
For complex division, we need to know what a conjugate is. For starters, all complex numbers take the form of a + bi, where a is the x-coordinate and b is the y-coordinate. A complex conjugate is changing the + in a + bi to a − bi or vice versa. When we have complex division, we multiply by a clever form of “one,” which will be the conjugate of the denominator.
Example: Write each quotient in the form \(a+bi:\frac{6}{7-4i}\)
\(\frac{6}{7-4i}=\frac{6}{7-4i}\times \frac{7+4i}{7+4i}\)
\(\frac{42+24i}{49-16i^{2}}\)
\(\frac{42+24i}{49-16(-1)}\)
\(\frac{42+24i}{65}\)
\(\frac{42}{65}+\frac{25}{65}i\)
