# Write a complex number in trigonometric form

Such plots are named after Jean-Robert Argand —although they were first described by Norwegian—Danish land surveyor and mathematician Caspar Wessel — Cartesian form and definition via ordered pairs[ edit ] A complex number can thus be identified with an ordered pair Re zIm z in the Cartesian plane, an identification sometimes known as the Cartesian form of z. In fact, a complex number can be defined as an ordered pair a, bbut then rules for addition and multiplication must also be included as part of the definition see below.

Complex plane Figure 1: A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram see Pedoe and Solomentsevnamed after Jean-Robert Argand.

The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical see Figure 1. These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form. A position vector may also be defined in terms of its magnitude and direction relative to the origin. These are emphasized in a complex number's polar form. Using the polar form of the complex number in calculations may lead to a more intuitive interpretation of mathematical results.

Notably, the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: History in brief[ edit ] Main section: History The solution in radicals without trigonometric functions of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible the so-called casus irreducibilis.

This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around though his understanding was rudimentary.

Work on the problem of general polynomials ultimately led to the fundamental theorem of algebrawhich shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed fieldwhere any polynomial equation has a root.

Many mathematicians contributed to the full development of complex numbers.

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The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli. Equality and order relations[ edit ] Two complex numbers are equal if and only if both their real and imaginary parts are equal. That is, complex numbers z.There are a few points to make.

The modulus operator finds the remainder after an integer divide. The print command output a floating point number on the divide, but an integer for the rest. The string concatenate operator is confusing, since it isn't even visible.

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This looks really complicated, so let’s go through some examples (and get answers in either degrees or radians, per indicated). Let’s first work with roots with either the real or imaginary part of the complex number (but not both).

The trigonometric form of a complex number z= a+ biis z= r(cos + isin); where r= ja+ bijis the modulus of z, and tan = b a.

## Trigonometric Form of Complex Numbers Calculator

is called the argument of z. Normally, we will require 0. Section , Trigonometric Form of a Complex Number Homework: #1, 3, 5, 11{17 odds, 21, 31{37 odds, 45{57 odds, 71, 77, 87, 89, 91, , 1 Review of Complex Numbers Complex numbers can be written as z= a+bi, where aand bare real numbers, and i= p 1.

This form, a+ bi, is called the standard form of a complex number.

## How to Write Complex Numbers in Trigonometric Form - Tutorial, Example

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