Complex Numbers in Polar Form Imaginary and Complex

Complex Numbers in Polar Form

Imaginary and Complex Numbers • Imaginary Number – Basic definition: i = √-1 = Sqrt(-1) i 2 = -1 i 3 = i*i 2 = i*(-1) = -i i 4 = ? • Complex Number – Basic definition: A number that has both a real and imaginary part: z = a + bi ( a – bi is called the complex conjugate) For example: z = 5 + 3 i or z = 1. 4 + 2. 9 i

Basic operations on complex numbers 1) Addition/subtraction: combine all real parts together and all imaginary parts together 2) Multiplication: expand first and then combine real and imaginary parts together 3) Division: to get a real number in the denominator, we multiply the top and bottom of the fraction by the complex conjugate

Graphing Complex Numbers • Cartesian Form in the complex plane: – The real part goes on the x-axis – The imaginary part goes on the y-axis – z = a + bi • Polar Form: y-axis x-axis – r is the distance from the origin to the point – θ is the angle measured up from the x-axis • Examining the diagram, we can see that: – a = r cos θ b = r sin θ

Polar form of a complex number • Plug in the expression for a and b to get: – z = r cis θ • • r is the modulus, aka magnitude or length θ is the argument, aka angle the absolute value of any complex number is: |z| = r Examine the right triangle to find: r 2 = a 2 + b 2 & θ = tan-1(b/a) • Recap of definitions: z = a + bi = r cis θ a = r cos θ & b = r sin θ r 2 = a 2 + b 2 & θ = tan-1(b/a)

Operations in polar form: • • • 1) multiply 2) reciprocal 3) divide 4) exponents 5) roots

Operations in polar form: • 1) Multiply two complex numbers together: z 1 z 2 – But we see there is a shortcut: – Multiply the moduli, add the arguments – z 1 z 2 = r 1 r 2 cis (θ 1 + θ 2)

Operations in polar form: • 2) Find the reciprocal: 1/z – But we see there is a shortcut: – Take the reciprocal of the modulus, and negative θ – 1/z = 1/r cis (-θ)

Operations in polar form: • 3) Divide two complex numbers: z 1/z 2 – Apply the two tricks we just learned – But we see there is a shortcut: – Divide the moduli, subtract the arguments – z 1/z 2 = r 1/r 2 cis (θ 1 - θ 2)

Operations in polar form: • 4) Raising a complex number to the nth power: zn – First using the tricks we have learned – But we see there is a shortcut: – Raise the modulus to the nth power, multiply θ by n – zn = rn cis (n*θ) – This is known as De Moivre’s Theorem

Operations in polar form: • 5) Taking an nth root of complex numbers: n√z = z 1/n – Here we have to be careful to include all possible results – Result: An nth root will have n total solutions, evenly spaced around the pole in the complex plane