Page 162 - Algebra

8.6. Complex Numbers
In a real number system, you will not find a solution to x2 = –2. But, in the imaginary number system, the
equation has a solution.
You can solve such an equation using the imaginary unit ‘i’
i = √−1 i2 = 1
6 + 5i
A complex number is of the form a + bi, where a and b are the real numbers and i = √−1
When a real number is combined with an imaginary number, you get a complex number.
For example,
Remember
• • • •
•
• • •
i3 = -i
All real numbers are complex numbers. You can write 8 as 8 + 0i.
When adding or subtracting complex numbers, group all the real parts and imaginary parts.
When multiplying, use the same rules as binomials.
Power of i
i0 = 1
i1 = i
i2 = – 1
When solving higher powers of i, always remember that i2 = –1, and
–1even number = 1
–1odd number= –1
(a + bi) (a – bi) are called conjugate complex numbers.
(a + bi) (a – bi) = a2 + b2
a + bi = c + di, only when a = c and b = d
Worked Example
If i = √−1, find the value of i
415
.
Solution:
= –i
i415 = i.i414
= i.(i2)207
= i.(–1)207
Page 161 of 177
Algebra I & II