Complex Numbers Operations

Operations with Complex Numbers

• The relationship between voltage (E), current (I), and resistance (Z) is expressed as E = IZ.
• Example: For I = 3 + 2i and Z = 2 - i, the voltage E is calculated to be 8 + i.

Additive Inverse

• The additive inverse of a complex number reverses its sign.
• For -8 + 3i, the additive inverse is 8 - 3i.

Associative Property of Addition

• This property showcases that the way numbers are grouped in addition does not affect the sum.
• Example: (-4 + i) + 4i equals -4 + (i + 4i).

Commutative Property of Multiplication

• Indicates that the order of factors does not change the product.
• For any complex numbers x = a + bi and y = c + di, x × y = y × x illustrates this property.

Commutative Property of Addition

• Similar to multiplication, the order of addends does not affect the sum.
• Example: a + bi + c + di can be rearranged to a + c + bi + di.

Distributive Property Mistake

• An example of incorrect application of the distributive property.
• Donte incorrectly simplified 4(1 + 3i), failing to distribute correctly.

Total Score Calculation

• When combining complex scores, add real and imaginary parts separately.
• Example: Melissa (5 - 4i) and Tomas (3 + 2i) have a total score of 8 - 2i.

Inverse Property of Addition

• This property confirms that adding a number and its additive inverse results in zero.
• Example: If x = a + bi and y = -a - bi, then x + y = 0.

Product of Complex Numbers

• The product of two conjugate complex numbers can be calculated to yield a real number.
• Example: (3 - 2i)(3 + 2i) results in 13.

Additive Identity Property

• This property indicates that adding zero to a number does not change its value.
• Example: (7 + 4i) + 0 equals 7 + 4i.