Here is my powerpoint presentation on COMPLEX NUMBERS..

2. If a and b are real numbers and i is the imaginary unit, then a + bi is called a complex number . ▪ a is the real part ▪ bi is the imaginary part . Definition of Complex Numbers

3. Definition: The number i , called the imaginary unit , is the number such that i = ____ √-1 __ and i 2 = __ -1 ______

4. Powers of i

6. Let a + bi and c + di be complex numbers. 1. Multiply the binomials. 2. Convert i 2 to -1 and add the like terms. (3 + 2i)(4 + 5i) = (3 × 4) + (3 × (5i)) + ((2i) × 4) + ((2i) × (5i)) = 12 + 15i + 8i + 10i² = 12 + 23i -10 (Remenber that 10i² = 10(-1) = -10) = 2 + 23i Therefore, (3 + 2i)(4 + 5i) = 2+23i

8. Division Of Complex Numbers Let a + bi and c + di be complex numbers. Multiply the numerator and denominator of the fraction by the Complex Conjugate of the Denominator . Then to perform the operation 2+6i x 4-i = (2+6i) (4-i) = 14+22i = 14 + 22 i 4+i 4-i (4+i) (4-i) 17 17 17

9. Real Axis Imaginary Axis y x  The angle formed from the real axis and a line from the origin to ( x , y ) is called the argument of z , with requirement that 0   < 2  . modified for quadrant and so that it is between 0 and 2  Let a complex number be Z such that : z Modulus and Argument of Complex Numbers The magnitude or modulus of z denoted by z is the distance from the origin to the point ( x , y ).

10. The Principal Argument is between -  and  Real Axis Imaginary Axis y x z = r  1  The unique value of θ such that – π < θ < π is called principle value of the argument. but in Quad II

11. The magnitude or modulus of z is the same as r. We can take complex numbers given as and convert them to polar form : Real Axis Imaginary Axis y x z = r  Plot the complex number: Find the polar form of this number. 1  factor r out but in Quad II