Introduction Complex Numbers The Argand Diagram Modulus

2. Introduction Complex Numbers The Argand Diagram Modulus

3. Introduction • Mathematicians have a concept called completeness. It is the need to be able to answer every single question. • Historically it is this need for completeness which led the Hindus to discover negative numbers. • Later the Greeks developed the idea of irrational numbers.

4. –1 = i, an imaginary number. Rafaello Bombelli • During the sixteenth century, an Italian called Rafaello Bombelli came up with the question: “If the square root of +1 is both +1 and –1, then what is the square root of –1?” • He answered the question himself and declared that • In your own life you develop through all these stages of number.

5. Rafaello Bombelli • N: Natural numbers – these are whole positive numbers. These are the first numbers people understand, e.g. you are three years old and fighting with your little sister because she has three sweets and you have only two! • Z: Integers – positive and negative whole numbers. Later in life, perhaps when you are five years old you understand the idea of minus numbers. You may have five sweets but you owe your friend two sweets, so you realise that in fact, you really only have three sweets, 5 – 2 = 3.

6. He gets of the bar and you get of it. 3 8 5 8 __ __ – 1 4 4 , 2·114, –3·49, 3 , 9  1015 Rafaello Bombelli • R: Real Numbers – all numbers on the number line. • Later again in life you realise that there are fractions and decimals. You may divide a bar of chocolate with eight squares and give your brother three squares and keep five for yourself. • Other real numbers include: THIS IS PROBABLY WHERE YOU ARE NOW!

7. –1 = i i2 = –1 Complex Numbers • These are numbers with a real and an imaginary part. • They are written as a + ib where a and b are real numbers.

8. Addition Simply add the real parts, then add the imaginary parts. 3 + 5i 6 +2i + –––––– 9 + 7i

9. Subtraction ‘Change the sign on the lower line and add’ 8 – 3i 3 2i – + – – + + –––––– 11 – 5i

10. i2 = –1 Multiplication Each part of the first complex number must be multiplied by each part of the second complex number.

11. Multiplication 4(3 + 2i) =12 +8i 4i(3 + 2i) = 12i + 8i2 i2=–1 = 12i+ 8 (–1) =12i– 8 =– 8 + 12i

12. Multiplication =12 +15i + 8i + 10i2 i2=–1 (3 + 2i)(4 + 5i) =12 + 15i+ 8i+10 (–1) =12 + 15i+ 8i – 10 =2 + 23i Collect real and imaginary parts

13. Division To divide complex numbers we need the concept of the complex conjugate. The conjugate of a complex number is the same number with the sign of the imaginary part changed. The conjugate of 5 + 3i is 5 – 3i.

14. Division To divide we multiply the top and bottom by the conjugate of the bottom.

15. 5 + 2i 3 + 4i 3 – 4i 3 – 4i × 15 – 20i+ 6i + 8 9 + 16 –––––––––––––– = 23 – 14i 25 ––––––– = Division 15 –20i +6i – 8i2 (–1) i2=–1 –––––––––––––––––– = 9 +12i – 12i – 16i2 (–1)

16. Application A farmer has 100 m of fence to surround a small vegetable plot. The farmer wants to enclose a rectangular area of 400 m2. How long and wide should it be? x 2 lengths + 2 widths = 100 y 50 – x 2x + 2y = 100 x + y = 50

17. The area = Length  width 400 =x(50 –x) 400 = 50x–x2 x2–50x+ 400 = 0 x 50 – x

18. 2 b- 4ac -b± x = 2a ––––––––––––––– –(–50)± (–50)2 –4(1)(400) –––––––––––––––––––––––– x = 2(1) ––––––––––– 50 ± 2500–1600 ––––––––––––––––– = 2 50±30 ––––––– = 2 x2–50x+ 400 = 0 a= 1 b= –50 c= 400 =40 or 10

19. Application A farmer has 100 m of fence to surround a small vegetable plot. The farmer wants to enclose a rectangular area of 650 m2. How long and wide should it be? x 2 lengths + 2 widths = 100 y 50 – x 2x + 2y = 100 x + y = 50

20. The area = Length  width 650 =x(50 –x) 650 = 50x–x2 x2–50x+ 650 = 0 x 50 – x

21. 2 b- 4ac -b± x = 2a ––––––––––––––– –(–50)± (–50)2 –4(1)(650) –––––––––––––––––––––––– x = 2(1) ––––––––––– –––––– 50 ± 2500–2600 ––––––––––––––––– 50 ± –100 = –––––––––––– 2 = i2 = –1 2 ––– ––  –1 50 ± 100 50 ± 10i –––––––––––––– –––––––– = = 2 2 x2–50x+ 650 = 0 a= 1 b= –50 c= 650 = 25 ± 5i

22. The Argand Diagram • The German mathematician Carl Fredrich Gauss (1777 – 1855) proposed the Argand diagram. • This has the real numbers on the x-axis and the imaginary ones on the y-axis. • All complex numbers can be plotted and are usually called z1, z2 etc.

23. Im 4 3 2 1 Re -3 -2 -1 1 2 3 -1 -2 -3 -4 The Argand Diagram z1 z1 = (2 + 4i) z2= (–2 – 4i) z3= (4 + 0i) z4= (–3 + 0i) z5= (–3 + 2i) z5 z4 z3 z2

24. z= x2+y2 |2 + 4i|=22+ 42 |4 + 0i|=42+ 02 = 4 + 16 =20 =16 Modulus z1= 2 + 4i z3= 4 + 0i =4