Complex Number and Geometry Formula Sheet

Complex Numbers

Definition

A complex number z can be expressed as: z = a + bi

where a and b are real numbers, and i = (-1)

Basic Operations

• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
• Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
• Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
• Division: (a + bi)/(c + di) = ((ac + bd)/(c + d)) + ((bc - ad)/(c + d))i

Polar Form

A complex number can also be expressed in polar form:

z = r(cos + i sin ) = re^(i)

where r is the modulus and is the argument

Modulus and Argument

• Modulus: |z| = (a + b)
• Argument: arg(z) = = tan^(-1)(b/a)

De Moivre's Theorem

(cos + i sin )^n = cos(n) + i sin(n)

Useful for finding powers and roots of complex numbers

Roots of Complex Numbers

The n-th roots of z = re^(i) are:

z_k = r^(1/n) [cos(( + 2k)/n) + i sin(( + 2k)/n)], k = 0, 1, ..., n-1

Euler's Formula

e^(i) = cos + i sin

Connects exponential, trigonometric, and complex functions

Geometry Formulas

Plane Geometry

Triangle

• Area: A = (1/2)bh (base height)
• Heron's formula: A = (s(s-a)(s-b)(s-c)) where s = (a+b+c)/2
• Perimeter: P = a + b + c
• Pythagorean theorem: a + b = c (right triangle)

Circle

• Circumference: C = 2r
• Area: A = r
• Arc length: L = r ( in radians)
• Sector area: A = (1/2)r

Quadrilateral

• Rectangle: Area = lw, Perimeter = 2(l+w)
• Parallelogram: Area = bh, Perimeter = 2(a+b)
• Trapezoid: Area = (1/2)(a+b)h
• Square: Area = s, Perimeter = 4s

Solid Geometry

Cube

• Volume: V = s
• Surface area: A = 6s

Rectangular Prism

• Volume: V = lwh
• Surface area: A = 2(lw + lh + wh)

Sphere

• Volume: V = (4/3)r
• Surface area: A = 4r

Cylinder

• Volume: V = rh
• Lateral surface area: A = 2rh
• Total surface area: A = 2r(r+h)

Cone

• Volume: V = (1/3)rh
• Lateral surface area: A = r(r+h)
• Total surface area: A = r(r+(r+h))

Pyramid

• Volume: V = (1/3)Bh (B = base area, h = height)
• Lateral surface area: A = (1/2)Pl (P = perimeter of base, l = slant height)

Coordinate and Analytical Geometry

Distance Formulas

Distance between two points (x, y) and (x, y):

d = [(x-x) + (y-y)]

Distance from point (x, y) to line ax + by + c = 0:

d = |ax + by + c|/(a + b)

Midpoint Formula

Midpoint of segment with endpoints (x, y) and (x, y):

M = ((x+x)/2, (y+y)/2)

Slope Formula

Slope of line through (x, y) and (x, y):

m = (y-y)/(x-x)

Line Equations

• Slope-intercept form: y = mx + b
• Point-slope form: y - y = m(x - x)
• Standard form: ax + by + c = 0
• Two-point form: (y - y)/(y - y) = (x - x)/(x - x)

Conic Sections

Circle

Standard form with center (h, k) and radius r:

(x - h) + (y - k) = r

Ellipse

Standard form with center (h, k), semi-major axis a, and semi-minor axis b:

(x - h)/a + (y - k)/b = 1

Parabola

Vertical form with vertex (h, k):

(x - h) = 4p(y - k)

Horizontal form with vertex (h, k):

(y - k) = 4p(x - h)

Hyperbola

Horizontal form with center (h, k):

(x - h)/a - (y - k)/b = 1

Vertical form with center (h, k):

(y - k)/a - (x - h)/b = 1

Three-Dimensional Geometry

Distance between points (x, y, z) and (x, y, z):

d = [(x-x) + (y-y) + (z-z)]

Sphere equation with center (h, k, l) and radius r:

(x - h) + (y - k) + (z - l) = r

Applications of Complex Numbers in Geometry

Complex Plane Representation

Every complex number z = a + bi corresponds to a point (a, b) in the complex plane:

• The real axis corresponds to the x-axis
• The imaginary axis corresponds to the y-axis

Geometric Interpretations

• |z - z|
• arg(z) = angle between the positive real axis and the line from origin to z
• arg(z/z) = angle between lines from origin to z and z

Complex Transformations

• Translation: z z + c (shifts points by c)
• Rotation: z ze^(i) (rotates points by angle )
• Scaling: z kz (scales distances from origin by factor k)
• Reflection: z z (reflects across the real axis)

Representing Geometric Figures

• Line through points a and b: {z : (z-a)/(b-a) is real}
• Circle with center c and radius r: {z : |z-c| = r}
• Perpendicular bisector of segment AB: {z : |z-a| = |z-b|}

Complex Numbers and Trigonometry

Using De Moivre's theorem to find trigonometric identities:

• cos(n) = Re[(cos + i sin )^n]
• sin(n) = Im[(cos + i sin )^n]

Mbius Transformations

A Mbius transformation is of the form:

f(z) = (az + b)/(cz + d) where ad - bc 0

These transformations map circles and lines to circles and lines in the complex plane.