Section 2-5

Complex Numbers

Section 2-5

• complex numbers and i

• operations with complex numbers (+,-, x)

• complex conjugates and division

• solving quadratic equations with complex solutions

• plotting complex numbers

• absolute value of complex numbers

Complex Numbers

• we learned back in Algebra 1 that the square root of a negative number is not a real number

• there is a way to work with these numbers using the imaginary unit, i

• we use this simple definition:

• for example:

1i

9 3 and 8 2 2i i

Complex Numbers

• all numbers we work with are part of the set of complex numbers

• this set consists of all real numbers and all imaginary numbers (contain i)

• all complex numbers can be written in the form a + bi

• a is the real part, b is the imaginary part

Operations With Complex Numbers

• to add complex numbers, add their like parts (same for subtraction)

• to multiply complex numbers, use FOIL

• use the fact that 2 1i

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

a bi c di a c b d i

a bi c di a c b d i

Division of Complex Numbers

• if a + bi is a complex number, then its complex conjugate is a – bi

• in order to simplify the division of two complex numbers, multiply the top and bottom of the fraction by the conjugate of the denominator

• use FOIL on both top and bottom; the bottom will no longer contain i

Solving Quad.’s

• now, when you solve a quadratic equation for which the discriminant is negative, you can find its complex solutions

• the solutions will be complex conjugates

Plotting Complex Numbers

• complex numbers cannot be plotted on a single number line because they have both a real and imaginary part

• instead, we plot them on a complex plane which looks a lot like a coordinate plane we use for ordered pairs

• the axes of this plane are the real axis and the imaginary axis

Plotting Complex Numbers

a

b

-5 + 3i

3 – 6i

Absolute Value of Complex Numbers

• remember that absolute value means distance from 0 on a number line

• for complex numbers, it’s the distance from the origin

• we use the distance formula to compute it

2 2a bi a b