**Thei-th Continuum**

Acomplex numberis any number than have the right to be to express in the form:
Where i iscalled the imaginary unit: The The |

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

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

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

= **0 + 26i**

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

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

= **-7 + 24i**

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

= **-3 + 4i**

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