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Fields Medal Prize Winners (1998)




TUTORIALS:


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Fractions
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Roots
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Powers
Rewriting Algebraic Fractions
Exponents
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Polar Representation of Complex Numbers

The Argand diagram

In two dimensional Cartesian coordinates (x,y), we are used to plotting the function y(x) with y on the vertical axis and x on the horizontal axis.

In an Argand diagram, the complex number z = x + iy is plotted as a single point with coordinates (x,y). The horizontal axis is called the real axis (x-axis) and the vertical axis is called the imaginary axis (y-axis).

As in usual Cartesian coordinates, the distance from the origin to a point (x, y) is equal to . This is equal to the modulus | z | of the complex number z = x + iy.

The Argand diagram may also be called the complex plane. It stresses that complex numbers are a generalisation of real numbers, that lie on the horizontal axis only.

The expression z = x + iy is known as the Cartesian form or the rectangular form of the complex number z. Using the Argand diagram, we can see that the addition of complex numbers behaves like the addition of vectors.

If we express z = x + iy as an ordered pair (x, y), then the addition of two complex numbers may be defined by

in the same way as the addition of two vectors.

 

Polar coordinates

A position vector of a point in two dimensions may be expressed in terms of Cartesian coordinates (x,y) and plotted with y on the vertical axis and x on the horizontal axis.

It is also possible to express the two dimensional position vector in terms of polar coordinates (r,θ) where r is the magnitude of the vector (distance from origin to the point) and θ is the angle between the position vector and the positive x-axis.

The Cartesian and polar coordinates are related by:

In the same way, the complex number z = x + iy may be expressed in polar coordinates (r, θ), in its polar form:

where

 

The modulus of a complex number

In polar coordinates (r, θ) the magnitude r of the distance from the origin to the point represented by z is equal to the modulus of the complex number | z |:

 

The argument of a complex number

In polar coordinates (r, θ) the angle θ is known as the argument of the complex number z, denoted θ = arg(z).

There is a complication because a single point on the Argand diagram does not correspond to a single complex number. The reason is that we can add 2π to the value of the argument θ in order to produce a different complex number, but when plotted on the Argand diagram, the two numbers are plotted in the same place.

Principal value: If we want to uniquely define the value of the argument θ we can impose the condition −π<θ≤π so that θ is known as the principal value of the argument.

For the complex number z = x + iy, the argument θ is given by the solution of the equations:

or

If the second expression tan θ = y/x is used to determine θ, it is wise to plot z = x + iy on an Argand diagram to check that the answer is correct.