**Definition**

1.
,

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 "same"
as x in

the expressions . However, i does have the fol lowing properties .

**Remark: **The powers of I repeat with EVERY 4-th
power.

**Example 1**

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

ope rations for real numbers carry over:

**• Equality: **a+bi = c+di ⇔ a=c and b=d.

**• Addition: **(a+bi)+(c+di) ⇔ (a+c)+(b+d)i.

**• Subtration:** (a+bi)-(c+di) ⇔ (a-c)+(b-d)i.

**• Multiplication : **(a+bi)(c+di)=ac+adi+bci-bd=(ac-bd)+(ad+bc)i.

The following operations are different from real numbers:

**• Conjugate:** if z=a+bi, then the conjugate of z is a-bi, denoted by
. (Conjugate

is to negate the imaginary part.)

,this trick is
important in doing complex division. Actually we can

understand this as

**Extra Credit**

Show that the above 6 identities are true. Can we inter pret the identity

by using the difference of squares formula ?

**• Division:** if z=a+bi and w=c+di, then

**Example 2**

3. Solving Quadratic Equations with NEGATIVE Discriminants

Recall:|

Δ=b^{2} − 4ac >0: TWO x -intercepts

Δ=b^{2} − 4ac =0: ONE x -intercept

Δ= b^{2} − 4ac <0: NO x-intercept (i.e. no real solutions). However, after we have

introduced complex numbers, we know that, by the quadratic formula,

, the quadratic function with real
coefficients , in this case, will

have a PAIR of CONJUGATE complex numbers as two solutions. (THEY ARE NOT

X- INTERCEPTS THOUGH .)

Remark: How to take square root of a negative number:
.

**Example 3**