Complex numbers are numbers of the form a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be added, subtracted, multiplied, and divided. When multiplying complex numbers, the real parts and imaginary parts are multiplied separately and combined. The conjugate of a + bi is a - bi. When a complex number is multiplied by its conjugate, the result is a real number equal to the modulus (magnitude) of the complex number squared. Complex numbers can also be expressed in polar form as r(cosθ + i sinθ), where r is the modulus and θ is the argument.

1. 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

5. Let a + bi and c + di be complex numbers. 1. Add/Subtract the Real parts. 2. Add/Subtract the Imaginary parts . (3 + 4i) + (2 - i) = (3 + 2) + (4i - i) = (5 + 3i) (7 + i) - (3 - i) = (7 - 3) + i(1 - (-1)) = 4 + 2i

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

7. A complex number z is a number of the form z = x + yi . Its conjugate is a number of the form = x - yi . The complex number and its conjugate have the same real part. Re( z ) = Re( ). The sign of the imaginary part of the conjugate complex number is reversed. Im( z ) = - Im( ). The conjugate numbers have the same modulus and opposite arguments. | z | = | |, arg( z ) = - arg( ). Any complex number multiplied by its complex conjugate is a real number, equal to the square of the modulus of the complex numbers z . z = ( x + yi )( x - yi ) = x 2+ y 2 = | z |2

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