1. Complex Numbers MATH 109 - Precalculus S. Rook

2. Overview Section 2.4 in the textbook: – Imaginary numbers & complex numbers – Adding & subtracting complex numbers – Multiplying complex numbers – Dividing complex numbers 2

3. Imaginary Numbers & Complex Numbers

4. 4 Imaginary Numbers In the real number system, recall that a value under a radical must be greater than or equal to 0 – Otherwise the value is non-real Consider if we decomposed and rewrote it as – This step is called “poking out the i” – We know how to evaluate Imaginary unit: – Thus, – Any number with an i is called an imaginary number – Also by definition:

5. 5 Multiplying and Dividing Imaginary Numbers First step is to ALWAYS “poke out the i” WRONG CORRECT After “poking out the i” use the product or quotient rule for radicals on the REAL roots: – After checking whether the REAL roots can be simplified of course – Only acceptable to have i in the final answer – i 2 can be simplified to -1

6. 6 Complex Numbers Complex Number: a number written in the standard format a + bi where: – a and b are real numbers – a is the real part – bi is the imaginary part The set of real numbers along with the set of imaginary numbers comprises the set of complex numbers – i.e. a complex number is exclusively real when b = 0 and exclusively imaginary when a = 0

7. Imaginary Numbers (Example) Ex 1: Perform the following operations and write the answer in standard form: a) b) 7

8. Adding & Subtracting Complex Numbers

9. 9 To add complex numbers: – Add the real parts – Add the imaginary parts – The real and imaginary parts cannot be combined any further To subtract two complex numbers: – Distribute the negative to the second complex number – Treat as adding complex numbers

10. Adding & Subtracting Complex Numbers (Example) Ex 2: Add or subtract and write the result in standard form: a)(5 + i) + (6 – 2i) b)13i – (14 – 7i) 10

11. Multiplication of Complex Numbers

12. 12 Multiplication of Complex Numbers To multiply 3i · 2i: – Multiply the real numbers first: 6 – Multiply the i s: i · i = i 2 3i · 2i = 6i 2 = -6 Remember that it is only acceptable to leave i in the final answer To multiply complex numbers in general – Use the distributive property or FOIL

13. Multiplication of Complex Numbers (Example) Ex 3: Multiply and write in standard form: a)(4 + 5i) 2 b)(1 + i)(3 – 2i) 13

14. Division of Complex Numbers

15. 15 Complex Conjugate Consider (3 + i)(3 – i) – What is noticeable? Complex conjugate: the same complex number with real parts a and imaginary part bi except with the opposite sign – Very similar to conjugates when we discussed rationalizing – What would be the complex conjugate of (2 – i)?

16. 16 Division of Complex Numbers Goal is to write the quotient of complex numbers in the standard format a + bi To divide complex numbers: – Multiply the numerator and denominator by the complex conjugate of the denominator (dealing with an expression) – The numerator simplifies to a complex number – The denominator simplifies to a single real number – Divide the denominator into each part of the numerator and write the result in a + bi format

17. Division of Complex Numbers (Example) Ex 4: Divide and write in standard form: a) b) 17

18. Summary After studying these slides, you should be able to: – Understand the principles of imaginary and complex numbers – Understand the standard form for a complex number – Add, subtract, multiply, and divide complex numbers Additional Practice – See the list of suggested problems for 2.4 Next lesson – Zeros of Polynomial Functions (Section 2.5) 18