5.4 Complex Numbers

1. 5.4 Complex Numbers Algebra 2

2. Learning Target • I can simplify radicals containing negative radicands • I can multiply pure imaginary numbers, and • I can solve quadratic equations that have pure imaginary solutions • I can add, subtract, and multiply complex numbers.

3. Definition of a complex number • A complex number is any number that can be written in the form a + bi where a and b are real numbers and i is the imaginary unit. a is called the real part , and bi is called the imaginary part.

4. Definition of a complex number • A real number is also a complex number. For example, 3 can be expressed as 3 +0i. The imaginary part is 0. A complex number is real only if the imaginary part is zero. • NOTE: the set of complex numbers has two independent subsets—the real and the imaginary numbers.

5. The Complex Numbers Non-reals Reals, R Q Z W N Pure Imaginary Pure Imaginary The diagram below shows the relationship between the various sets of numbers we have studied and the complex numbers. R = reals I = irrationals Q = rationals Z = Integers W = wholes N = Naturals Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, a + bi = c + di if and only if a = c and b = d

6. Ex. 1: Find values for x and y such that 3x + 4yi = 12 + 8i 3x + 4yi = 12 + 8i 3x = 12 and 4y = 8 x = 4 y = 2 Check: 3x + 4yi = 12 + 8 3(4) + 4(2)i = 12 + 8i 12 + 8i = 12 + 8i ?

7. Ex 2: Simplify (2 + 5i) + (4 – i) • To add or subtract complex numbers, we must combine like terms, that is, combine the real parts and combine the imaginary parts. (2 + 5i) + (4 – i) = (2 + 4) + (5i - i) = 6 + 4i

8. Ex 2: Simplify (8 - 2i) – (6 – 4i) • To add or subtract complex numbers, we must combine like terms, that is, combine the real parts and combine the imaginary parts. (8 - 2i) – (6 – 4i) = (8 - 2i) + (-6 + 4i) = (8 – 6) + (-2i + 4i) = 2 + 2i

9. The complex plane is also known as the Gaussian plan or an Argand diagram • The complex numbers can also be graphed on a complex plane, where the horizontal axis represents the real part of the complex number and the vertical axis represents the imaginary part. The complex numbers are represented by segments whose endpoint are the origin and a point whose coordinates are the real part and the imaginary part of the complex number.

10. The complex plane is also known as the Gaussian plan or an Argand diagram • Addition of complex numbers can also be represented by graphing. First, graph the two numbers to be added. Then complete the parallelogram that has two sides represented by the segments. The segments from the origin to the fourth vertex of the parallelogram represents the sum of the two original numbers.

11. Graph each complex number using the real part as the x- coordinate and the imaginary part as the y – coordinate. Connect each point to the origin. Next complete the parallelogram. The fourth vertex has coordinates (1, 5) or 1 + 5i Check algebraically. Ex. 4: Graph -4 + 3i and 5 + 2i on the complex plane. Find their sum geometrically.

12. Ex. 5: Simplify (9 – 3i)(2 + 2i). • You can multiply complex numbers using the FOIL method. • (9 – 3i)(2 + 2i) = 18 + 18i – 6i – 6i2 = 18 + 12i –(6)(-1) = 18 + 12i + 6 = 24 + 12i

13. Ex. 6: Simplify (-2 + 3i)(3 – i). • You can multiply complex numbers using the FOIL method. • (-2 + 3i)(3 – i) = -6 + 2i + 9i – 3i2 = -6 + 11i – (3)(-1) = -6 + 11i + 3 = -3 + 11i

14. Summary Chart

15. The Assignment • pp. #29 - 63 (even)