Section 3.5 Complex Numbers
(EXPONENTS AND RADICALS REQUIRED)
Definition
1. 1โ=i,
i is the building block of complex numbers. It handles the difficulty we usually have to
take a square root of a negative real number. i could be considered the โsameโ as x in
the expressions. However, i does have the following properties.
z 1
2โ=i
z iiii โ== 23
z 1)1(
234 =โโ=โ== iiii ; or 1)1()( 2224 =โ== ii
z iiii == 45
Remark: The powers of I repeat with EVERY 4-th power.
Example 1
โ
โ
โ
โ
โ
2. Complex number (standard form): a+bi, where a and b are real numbers. a is
called the real part. B is called the imaginary part. Treating i as x, the algebraic
operations for real numbers carry over:
z Equality: a+bi = c+di โ a=c and b=d.
z Addition: (a+bi)+(c+di)
โ
(a+c)+(b+d)i.
z Subtration: (a+bi)-(c+di)
โ
(a-c)+(b-d)i.
z Multiplication: (a+bi)(c+di)=ac+adi+bci-bd=(ac-bd)+(ad+bc)i.
The following operations are different from real numbers:
z Conjugate: if z=a+bi, then the conjugate of z is a-bi, denoted by z. (Conjugate
is to negate the imaginary part.)
azz 2=+
bizz 2=โ
22 bazz += , this trick is important in doing complex division. Actually we can
