let’s do some further maths what sorts of numbers do we already know about? natural numbers:...
TRANSCRIPT
Let’s do some Further Maths
What sorts of numbers do we already know about?
Natural numbers: 1,2,3,4...
Integers: 0, -1, -2, -3
Rational numbers: ½, ¼,... Irrational numbers: π, √2
Let’s do some Further Maths
Natural numbers: 1,2,3,4...
Integers: 0, -1, -2, -3
Rational numbers: ½, ¼,... Irrational numbers: π, √2
Complex numbers
Why would we ever need complex numbers? Solving quadratic equations: ax2+bx+c=0
Solving cubic equations square roots of negative numbers appear in
intermediate steps even when the roots are real Fundamental Theorem of Algebra- every
polynomial of degree n has n roots
We only need one new number
We define the number i = √-1 so that i2=-1 All square roots of negative numbers can be
written using i √-4 = √4 x √-1 = 2i
Exercise
Write the following using i:1. √-36 2. √-121 3. √-10 4. √-18 5. -√-75
We treat i a little like xExerciseSimplify:1. 3i + 2i 2. 16i – 5i 3. (2i)(3i) 4. i(4i)(6i)
5. Copy and complete: i0 =i1 =i2 =i3 =i4 =i5 =i6 =
6. i12 = 7. i25 = 8. i1026 =
Complex Numbers Multiples of i are imaginary numbers Real numbers can be added to imaginary numbers
to form complex numbers like 3+2i or -1/2 -√2i We can add, subtract and multiply complex
numbers (dividing is a little more complicated!) Exercise1. (3+2i) + (-1-4i) = 2. (2-i) – (1+5i) =
3. (2-2i)(1+3i) =
Is i positive or negative? Suppose i > 0. Then i2 > 0. In other words, -1 > 0.
So i < 0. Then i + (-i) < 0 +(-i). So –i > 0. And (-i)2 > 0. In other words, -1 > 0.
We need a 2-D number line!
A
B
CD
EF
G
H
I
Exercise
Name the complex numbers represented by each point on the Argand diagram.