#### Complex Numbers

- i² = -1
- The general form for all complex numbers is a + bi
- From this, we say:
- Re(a + bi) = a, and
- Im(a + bi) = -a
**NOTE:**Re – Real, and Im – Imaginary

- The complex number z and its conjugate z
^{+}are- z = a + bi, and
- z
^{+}= a – bi

- Arithmetic Operations of Complex Numbers:
- Addition and Subtraction: Add and subtract real, and imaginary parts with each other
- Multiplication: Carry out algebraic expansion, if i² is present, convert it to -1
- Division: Rationalize the denominator by multiplying conjugate pairs
- Equivalence: Equate coefficients

- For quadratics:
- Use the quadratic formula:
- b² – 4ac is a negative value
- pull out a negative and replace it with i²
- simplify to its general form

- Use sum of two squares
__Examples__- Solve z
^{2}+ 4z + 13 = 0- Convert to completed square form: (z + 2)
^{2}+ 9 = 0 - Utilize i
^{2}as -1 to make it a difference of two squares: (z + 2)^{2 }– 9i^{2}= 0 - Proceed with the general difference of two squares method:
- (z + 2 + 3i) (z + 2 – 3i) = 0
- ∴ z = -2 + 3i, and -2 – 3i

- Convert to completed square form: (z + 2)

- Solve z

- Use the quadratic formula:

- For Square Roots:
__Examples__- Find the square roots of 4 + 3i
- We can say that √(4 + 3i) = a + bi
- Square both sides: a
^{2}+ b^{2}+ 2abi = 4 + 3i - Equate the real and imaginary parts: a – b = 4, 2ab = 3
- Solve the equations simultaneously:
- a =
^{3√2}/_{4}, b =^{√2}/_{2}

- a =
- ∴ √(4 + 3i) =
^{3√2}/_{4}+ i^{√2}/_{2 }or –^{3√2}/_{4}– i^{√2}/_{2}

- Find the square roots of 4 + 3i

__Argand Diagram__

__Argand Diagram__

- For the complex number z = a + bi:
- Its magnitude is defined as |z| = √(a² + b²)
- Its argument is defined as arg z = tan
^{-1}^{b}/_{a} - Simply plot the imaginary (y-axis) against the real (x-axis)

- Arguments: Always -π < θ < π

- The position of z
^{+}is a reflection in the x-axis of z

__Locus__

__Locus__

- |z – w| = r
- The locus of a point z such that |z – w| = r, is a circle with its center at w and with radius r.

- arg(z – w) = θ
- The locus of a point z such that arg(z – w) = θ is a ray from w, making an angle θ with positive real axis

- The locus of a point z such that |z – w| = |z – v| is the perpendicular bisector of the line joining w and v

__Examples__

__Examples__

- On a sketch of an Argand diagram, shade the region whose points represent the complex numbers z which satisfy the inequality |z – 3i| ≤ 2. Find the greatest value of arg z for points in this region.
- The part shaded blue is the answer
- To find the greatest value of arg z within this region, we must use the tangent at a point on the circle which has the greatest value of θ from the horizontal (red line)

- The triangle magnified:

- sin α =
^{2}/_{3} - α = 0.730
- θ = α +
^{π}/_{2} - θ = 0.730 + π/2 = 2.30

- On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities |z – 2 + 2i|, arg z ≤ -π
^{1}/_{4}, and Re z ≥ 1- Calculate the greatest possible value of Re z for points lying in the shaded region
- The greatest value for the real part of z would be the one which is furthest right on the Re axis but within the limits of the shaded area. Graphically:

- Now using circle and Pythagoras theorems, we can find the value of x:
- x = 2 × cos
^{π}/_{4} - x = √2
- ∴ Greatest value of Re z = 2 + √2

- x = 2 × cos

- The greatest value for the real part of z would be the one which is furthest right on the Re axis but within the limits of the shaded area. Graphically:

- Calculate the greatest possible value of Re z for points lying in the shaded region

__Polar Form__

__Polar Form__

- For a complex number z withe magnitude R and argument θ:
- z = R(cos θ + i sin θ) = Re
^{iθ} - ∴ cos θ + i sin θ = e
^{iθ}

- z = R(cos θ + i sin θ) = Re
- Polar Form to General Form:
__Examples__- Convert z = 4e
^{π/4i}to the general form- R = 4, arg z =
^{π}/_{4} - ∴ z = 4(cos
^{π}/_{4 }+ i sin^{π}/_{4}) - z = 4 (
^{√2}/_{2}+^{√2}/_{2}i) - z = 2√2 + (2√2)i

- R = 4, arg z =

- Convert z = 4e

- General Form to Polar Form:
__Examples__- Convert z = 2√2 + (2√2)i to polar form
- z = 2√2 + (2√2)i
- R = √((2√2)² + (2√2)²) = 4
- θ = tan
^{-1}^{2√2}/_{2√2}=^{π}/_{4} - ∴4(cos
^{π}/_{4}+ i sin^{π}/_{4}) = 4e^{π/4i}

- Convert z = 2√2 + (2√2)i to polar form

__Multiplication and Division in Polar Form__

__Multiplication and Division in Polar Form__

- To find the product of two complex numbers in polar form:
- Multiply their magnitudes
- Add their arguments
- z
_{1}z_{2}= |z_{1}||z_{2}|(arg z_{1}+ arg z_{2})__Examples__- Find z
_{1}z_{2}in polar form given, z_{1}= 2(cos^{π}/_{4}+ i sin^{π}/_{4}), z_{2}= 4(cos^{π}/_{8}+ i sin^{π}/_{8})- z
_{1}z_{2 }= (2 × 4)(cos^{π}/_{4 }+^{π}/_{8}) + i sin (^{π}/_{4 }+^{π}/_{8}) - ∴ z
_{1}z_{2 }= 8(cos^{3π}/_{8}+ i sin^{3π}/_{8})

- z

- Find z

- To find the quotient of two complex numbers in polar form:
- Divide their arguments
- Subtract their magnitudes
^{z1}/_{z2}=^{|z1|}/_{|z2|}(arg z_{1}– arg z_{2})__Examples__- Find
^{z1}/_{z2}in polar form given, z_{1}= 2(cos^{π}/_{4}+ i sin^{π}/_{4}), z_{2}= 4(cos^{π}/_{8}+ i sin^{π}/_{8})^{z1}/_{z2}= (^{2}/_{4})(cos^{π}/_{4 }–^{π}/_{8}) + i sin (^{π}/_{4 }–^{π}/_{8})- ∴
^{z1}/_{z2}=^{1}/_{2}(cos^{π}/_{8}+ i sin^{π}/_{8})

- Find

__De Moivre’s Theorem__

__De Moivre’s Theorem__

- z
^{n}= R^{n}(cos nθ + i sin nθ) = R^{n}e^{inθ}