Imaginary Unit and Standard Complex Form Name______Block______
The Imaginary Unit is defined as i = .
It is said that the term "imaginary" was coined by René Descartes in the seventeenth century and was meant to be a derogatory reference since, obviously, such numbers did not exist. Today, we find the imaginary unit being used in mathematics and science. Electrical engineers use the imaginary unit (which they represent as j ) in the study of electricity.
Imaginary numbers occur when a quadratic equation has
no roots in the set of real numbers.
An imaginary number is a number whose square is negative.
Examples:pure imaginary
numbers /
A pure imaginary number can be written in bi form where bis a real number and i is .
A complex number is any number that can be written in the
standard form a + bi, where a and b are real numbers and i is
the imaginary unit.
standard a + biform / a / bi
7 + 2i / 7 / 2i
8i / 0 / 8i
The set of real numbersand the set of imaginary numbers are subsets of the set of complex numbers.
Adding and Subtracting Complex Numbers *Add like terms*
REMEMBER: Final answer must be in simplest form.
Multiplying and Dividing Complex Numbers
Distributive Multiplication:(2 + 3i) • (4 + 5i) = 2(4 + 5i) + 3i(4 + 5i)
= 8 + 10i + 12i + 15i2
= 8 + 22i + 15(-1)
= 8 + 22i -15
= -7 + 22i Answer
Be sure to replace i2 with (-1) and proceed with
the simplification. Answer should be in a + bi form.
The product of two complex numbers is a complex number.
proof: (a+bi)(c+di) = a(c+di) + bi(c+di)
= ac + adi + bci + bdi2
= ac + adi + bci + bd(-1)
= ac + adi + bci - bd
= (ac - bd) + (adi + bci)
= (ac -bd) + (ad + bc)i answer in
a+bi form
is a real number, and is always positive.
(a + bi)(a - bi) = a2 + abi - abi - b2i2
= a2 - b2 (-1) (the middle terms drop out)
= a2 + b2Answer
This is a real number ( no i's ) and since both values are squared, the answer is positive. /
When dividing two complex numbers,
1. / write the problem in fractional form,
2. / rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.
(Remember that a complex number times its conjugate will give a real number.This process will remove the i from the denominator.)
Example:
Answer
Cyclic Nature of the Powers of i
To be cyclic means to be repetitive in nature. When the imaginary unit, i, is raised to increasingly larger powers, it creates a cyclic pattern.
The powers of i repeat in a definite pattern:( i, -1, -i, 1 )
Powers of i / / / / / / / / / ...Simplified form / i / -1 / -i / 1 / i / -1 / -i / 1 / ...
You need to remember that:
Think about what happens wheni is raised to a given power:
LOOK OUT!!!
False /TRUE: /
Whenever the exponent is greater than or equal to 5, you can
use the fact that to simplify a power of i.
When raising i to any integral power, the answer is always i, -1, -i or 1.
Whenever the exponent is greater than or equal to 5, you can
use the fact that to simplify a power of i.
Let's examine two ways to simplify :
Using the patterns shown in the robot table above:
Looking at remainders when dividing by 4:
with a remainder of 3,
which means the answer is
i 3 = -i.