Algebra
Textbooks
Boundless Algebra
Complex Numbers and Polar Coordinates
Complex Numbers
Algebra Textbooks Boundless Algebra Complex Numbers and Polar Coordinates Complex Numbers
Algebra Textbooks Boundless Algebra Complex Numbers and Polar Coordinates
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 10
Created by Boundless

Introduction to Complex Numbers

A complex number has the form a+bia+bia+bi, where aaa and bbb are real numbers and iii is the imaginary unit.

Learning Objective

  • Describe the properties of complex numbers and the complex plane


Key Points

    • A complex number is a number that can be expressed in the form a+bia+bia+bi, where aaa and bbb are real numbers and iii is the imaginary unit.
    • The real number aaa is called the real part of the complex number z=a+biz=a+biz=a+bi and is denoted Re{a+bi}=a\text{Re}\{a+bi\}=aRe{a+bi}=a. The real number bbb is called the imaginary part of z=a+biz=a+biz=a+bi and is denoted Im{a+bi}=b\text{Im}\{a+bi\}=bIm{a+bi}=b.

Terms

  • real number

    An element of the set of real numbers. The set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers.

  • imaginary number

    a number of the form aiaiai, where aaa is a real number and iii the imaginary unit

  • complex

    a number, of the form a+bia+bia+bi, where aaa and bbb are real numbers and iii is the square root of −1-1−1.


Full Text

The Complex Number System

A complex number is a number that can be put in the form a+bia+bia+bi where aaa and bbb are real numbers and iii is called the imaginary unit, where i2=−1i^2=-1i​2​​=−1. In this expression, aaa is called the real part and bbb the imaginary part of the complex number. We will write Re{a+bi}=a\text{Re}\{a+bi\}=aRe{a+bi}=a to indicate the real part of the complex number, and Im{a+bi}=b\text{Im}\{a+bi\}=bIm{a+bi}=b to indicate the imaginary part. 

For example, to indicate that the real part of the number 2+3i2+3i2+3i is 222, we would write Re{2+3i}=2\text{Re}\{2+3i\}=2Re{2+3i}=2. To indicate that the imaginary part of 4−5i4-5i4−5i is −5-5−5, we would write Im{4−5i}=−5\text{Im}\{4-5i\} = -5Im{4−5i}=−5.

Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a+bia+bia+bi can be identified with the point (a,b)(a,b)(a,b). Thus, for example, complex number −2+3i-2+3i−2+3i would be associated with the point (−2,3)(-2,3)(−2,3) and would be plotted in the complex plane as shown below. 

The complex point −2+3i-2+3i−2+3i

The complex number −2+3i-2+3i−2+3i is plotted in the complex plane, 222 to the left on the real axis, and 333 up on the imaginary axis.

 

A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the set of ordinary real numbers can be thought of as a subset of the set of complex numbers. It is beneficial to think of the set of complex numbers as an extension of the set of real numbers. This extension makes it possible to solve certain problems that can't be solved within the realm of the set of real numbers. 

Complex numbers are used in many scientific fields, including engineering, electromagnetism, quantum physics, and applied mathematics, such as chaos theory. 

Complex numbers allow for solutions to certain equations that have no real number solutions. For example, the equation:

(x+1)2=−9(x + 1)^2 = -9(x+1)​2​​=−9

has no solution if we restrict ourselves to the real numbers, since the square of a real number is never negative. However, we can see that the complex numbers 1+3i1+3i1+3i and 1−3i1-3i1−3i are solutions, since

(1+3i−1)2=(3i)2=9i2=−9\begin{aligned}\left(1+3i-1\right)^2&=(3i)^2\\&=9i^2\\&=-9\end{aligned}​(1+3i−1)​2​​​​​​​=(3i)​2​​​=9i​2​​​=−9​​ 

and

 (1−3i−1)2=(−3i)2=9i2=−9\begin{aligned} \left(1-3i-1\right)^2 &=(-3i)^2 \\& =9i^2\\&=-9 \end{aligned}​(1−3i−1)​2​​​​​​​=(−3i)​2​​​=9i​2​​​=−9​​

It turns out that if we allow xxx to be a complex number, then any polynomial equation in xxx of degree nnn will have nnn (not necessarily unique) solutions. 

[ edit ]
Edit this content
Prev Concept
Other Curves in Polar Coordinates
Addition and Subtraction of Complex Numbers
Next Concept
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.