Lecture Notes: Introduction to Complex Numbers
1. Motivation: Why do we need Complex Numbers?
In mathematics, we solve various types of quadratic equations.
Case 1: x 2−5 x+ 6=0. By middle-term factoring, we get ( x−3 )( x−2 ) =0,
so x=2,3 . These are Real Numbers.
Case 2: x 2−2 x+1=0. Using the formula ( a−b )2 =a2−2 ab +b2, we get
( x−1 )2=0, so x=1.
Case 3: x 2+ 1=0. This leads to x 2=−1. To solve for x , we need to find
the square root of a negative number: x=± √ −1.
Because no real number squared equals a negative value, we define a
new type of number called an Imaginary Number.
The Imaginary Unit (i ): i stands for "imaginary" and is defined as
i=√ −1.
2. Powers of i (Cyclic Property)
Understanding the powers of i is fundamental to complex algebra. They
follow a cycle of four:
1
i =i
2
i =( √ −1 ) =−1
2
i 3=i 2 ⋅ i= (−1 ) ⋅ i=−i
4 2 2
i =i ⋅ i =(−1 ) ⋅ (−1 )=1
Key Rule: Since i 4 =1, any power of i that is a multiple of 4 (i 4 n ) will equal
1.
i 4 n=1
i 4 n +1=i
i 4 n +2=−1
4 n +3
i =−i
i 4 n +P =i P
3. Calculating Large Powers of i
To find the value of i n where n is a very large number, divide the power by
4 and look at the remainder ( R ).
1. Motivation: Why do we need Complex Numbers?
In mathematics, we solve various types of quadratic equations.
Case 1: x 2−5 x+ 6=0. By middle-term factoring, we get ( x−3 )( x−2 ) =0,
so x=2,3 . These are Real Numbers.
Case 2: x 2−2 x+1=0. Using the formula ( a−b )2 =a2−2 ab +b2, we get
( x−1 )2=0, so x=1.
Case 3: x 2+ 1=0. This leads to x 2=−1. To solve for x , we need to find
the square root of a negative number: x=± √ −1.
Because no real number squared equals a negative value, we define a
new type of number called an Imaginary Number.
The Imaginary Unit (i ): i stands for "imaginary" and is defined as
i=√ −1.
2. Powers of i (Cyclic Property)
Understanding the powers of i is fundamental to complex algebra. They
follow a cycle of four:
1
i =i
2
i =( √ −1 ) =−1
2
i 3=i 2 ⋅ i= (−1 ) ⋅ i=−i
4 2 2
i =i ⋅ i =(−1 ) ⋅ (−1 )=1
Key Rule: Since i 4 =1, any power of i that is a multiple of 4 (i 4 n ) will equal
1.
i 4 n=1
i 4 n +1=i
i 4 n +2=−1
4 n +3
i =−i
i 4 n +P =i P
3. Calculating Large Powers of i
To find the value of i n where n is a very large number, divide the power by
4 and look at the remainder ( R ).
