Thei-th Continuum
Acomplex numberis any number than have the right to be to express in the form:

a + bi

Where a and b are actual numbers (

*
and also i iscalled the imaginary unit:
*

*

The standard form of facility numbers is a + bi where a is dubbed the real part and also bis referred to as the imaginary part.

The conjugateof a + bi is a - bi and (a+ bi)(a - bi) = a2+ b2 (a real number)

I. Convert the following to the conventional form the thecomplex number a + bi
Problem Examples Standard form: a+ bi
(1) 5 + 3i 5 + 3i(Already in conventional form)
(2) 4 - 5i 4 +(-5)i
(3)
*
*
(note
*
is best written as
*
)

(4) -6i 0 +(-6)i (when a=0, bi is referred to as a pureimaginary number)
(5) 12 12 + 0i

II. Include the following complex number:

Principles: (a + bi) + (c + di) = (a + c) + (b+ d)i

(a + bi)- (c + di) =(a - c) + (b - d)i

(a + bi)+ (-a - di) =0 (where -a - diis dubbed the additive inverse)

Examples - Add and substract as indicated

(1) (-7 - 3i) +(-4 + 4i) = (-7 -4) + (-3 +4)i = -11 + i

(2) (-2 - 3i) -(-1 - i) = (-2 -(-1)) + (-3- (-1))i = -1 - 2i

(3)

*

III. Mutipication and division of complicated Numbers

Principles:

(1)

*

(2)

*

*

Use multiplication rules:

(1) (a + bi)(c + di) = a(c +di) + bi(c + di)

(2) (a + bi)2 = a2+ 2abi + (bi)2

(3) (a - bi)2 = a2- 2abi + (bi)2

(4) (a + bi)(a - bi) = a2 - (bi)2 = a2 + b2 (conjugate provided tosimplify the quotient that 2 facility numbers)

Examples: Express every products in typical form:

(1) (4i)(3-2i) = 4i(3)+ 4i(-2i) =12i- 8i2 = 12i - 8(-1) = 12i +8 = 8 + 12i

(2) (3 + 2i)(4 + 6i) = 3(4 + 6i)+ 2i(4 + 6i) =12 + 18i+ 8i + 12i2 = 12 + 26i + 12(-1) = 0 + 26i

= 0 + 26i

(3) (4 + 5i)(2 - 9i) = 4(2 - 9i)+ 5i(2 - 9i) = 8 - 36i+ 10i - 45i2 = 8 - 26i - 45(-1) = 53 - 26i

(4) (3 + 4i)2= (3 + 4i)(3 + 4i) = 32 + 2(3)(4)i + (4i)2= 9 + 24i + 16i2 = 9 + 24i + 16(-1)

= -7 + 24i

(5) (-1 - 2i)2= (-1 - 2i)( -1 - 2i) = (-1)2 - 2(-1)(2)i + (2i)2= 1 + 4i + 4i2 = 1 + 4i + 4(-1)

= -3 + 4i

(6) Conjugate:(3 + 4i)(3 - 4i) = (-3)2 - (4i)2 = 9 - 16i2 = 9 - 16(-1) = 9 + 16 =25