Complex Numbers
Complex Numbers Who uses them in real life? The navigation system in the space shuttle depends on complex numbers!
What is a complex number? It is a tool to solve an equation.
What is a complex number? It is a tool to solve an equation. It has been used to solve equations for the last 200 years or so.
What is a complex number? It is a tool to solve an equation. It has been used to solve equations for the last 200 years or so. It is defined to be i such that ;
What is a complex number? It is a tool to solve an equation. It has been used to solve equations for the last 200 years or so. It is defined to be i such that ; Or in other words;
Complex i is an imaginary number
Complex i is an imaginary number Or a complex number
Complex i is an imaginary number Or a complex number Or an unreal number
Complex? i is an imaginary number Or a complex number Or an unreal number The terms are inter-changeable
Some observations In the beginning there were counting numbers 1 2
Some observations In the beginning there were counting numbers And then we needed integers 1 2
Some observations In the beginning there were counting numbers And then we needed integers 1 2 -1 -3
Some observations In the beginning there were counting numbers And then we needed integers And rationals 1 0.41 2 -1 -3
Some observations In the beginning there were counting numbers And then we needed integers And rationals And irrationals 1 0.41 2 -1 -3
Some observations In the beginning there were counting numbers And then we needed integers And rationals And irrationals And reals 1 0.41 2 -1 -3
So where do unreals fit in ? We have always used them. 6 is not just 6 it is 6 + 0i. Complex numbers incorporate all numbers. 3 + 4i 2i 1 0.41 2 -1 -3
A number such as 3i is a purely imaginary number
A number such as 3i is a purely imaginary number A number such as 6 is a purely real number
A number such as 3i is a purely imaginary number A number such as 6 is a purely real number 6 + 3i is a complex number
A number such as 3i is a purely imaginary number A number such as 6 is a purely real number 6 + 3i is a complex number x + iy is the general form of a complex number
A number such as 3i is a purely imaginary number A number such as 6 is a purely real number 6 + 3i is a complex number x + iy is the general form of a complex number If x + iy = 6 – 4i then x = 6 and y = -4
A number such as 3i is a purely imaginary number A number such as 6 is a purely real number 6 + 3i is a complex number x + iy is the general form of a complex number If x + iy = 6 – 4i then x = 6 and y = – 4 The ‘real part’ of 6 – 4i is 6
Worked Examples Simplify
Worked Examples Simplify
Worked Examples Simplify Evaluate
Worked Examples Simplify Evaluate
Worked Examples 3. Simplify
Worked Examples 3. Simplify
Worked Examples 3. Simplify 4. Simplify
Worked Examples 3. Simplify 4. Simplify
Worked Examples 3. Simplify 4. Simplify 5. Simplify
Addition Subtraction Multiplication 3. Simplify 4. Simplify 5. Simplify
Division 6. Simplify
Division 6. Simplify The trick is to make the denominator real:
Division 6. Simplify The trick is to make the denominator real:
Solving Quadratic Functions
Powers of i
Powers of i
Powers of i
Powers of i
Powers of i
Useful rules
Useful rules
Useful rules
Useful rules
Argand Diagrams x y 1 2 3 2 + 3i
Argand Diagrams x y 1 2 3 2 + 3i We can represent complex numbers as a point.
Argand Diagrams x y 1 2 3
Argand Diagrams y x We can represent complex numbers as a vector. 1 2 3 A O We can represent complex numbers as a vector.
Argand Diagrams x y 1 2 3 B A O
Argand Diagrams C x y 1 2 3 B A O
Argand Diagrams C x y 1 2 3 B A O
Argand Diagrams C x y 1 2 3 B A O
Argand Diagrams C x y 1 2 3 B A O
Argand Diagrams C x y 1 2 3 B A O
Argand Diagrams C x y 1 2 3 B A O
Argand Diagrams C x y 1 2 3 B A O
This formula works for all values of n. De Moivre’s Theorem This formula works for all values of n.
Now we prove that , e iq = cos q + i sin q
The End