Introduction to Complex Numbers
A complex number has the form
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
$a+bi$ , where$a$ and$b$ are real numbers and$i$ is the imaginary unit.  The real number
$a$ is called the real part of the complex number$z=a+bi$ and is denoted$\text{Re}\{a+bi\}=a$ . The real number$b$ is called the imaginary part of$z=a+bi$ and is denoted$\text{Im}\{a+bi\}=b$ .
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
$ai$ , where$a$ is a real number and$i$ the imaginary unit 
complex
a number, of the form
$a+bi$ , where$a$ and$b$ are real numbers and$i$ is the square root of$1$ .
Full Text
The Complex Number System
A complex number is a number that can be put in the form
For example, to indicate that the real part of the number
Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number
The complex point $2+3i$
The complex number
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:
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
and
It turns out that if we allow
Key Term Reference
 complex numbers
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Phasors, and Imaginary Numbers
 degree
 Appears in these related concepts: What Are Polynomials?, Solving Quadratic Equations by Factoring, and Adding and Subtracting Polynomials
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Graphs of Equations as Graphs of Solutions
 expression
 Appears in these related concepts: Compound Inequalities, Sets of Numbers, and Simplifying, Multiplying, and Dividing
 imaginary
 Appears in these related concepts: Standard Equations of Hyperbolas, Radical Functions, and Complex Conjugates and Division
 imaginary unit
 Appears in these related concepts: Complex Numbers and the Binomial Theorem and Multiplication of Complex Numbers
 number line
 Appears in these related concepts: Inequalities with Absolute Value, Introduction to Inequalities, and Absolute Value
 point
 Appears in these related concepts: The Intermediate Value Theorem, Graphing Equations, and Polynomial and Rational Functions as Models
 polynomial
 Appears in these related concepts: Domains of Rational and Radical Functions, Finding Polynomials with Given Zeroes, and Partial Fractions
 real numbers
 Appears in these related concepts: Piecewise Functions, Introduction to Domain and Range, and Linear Inequalities
 set
 Appears in these related concepts: Sequences, Introduction to Sequences, and Sequences of Mathematical Statements
 solution
 Appears in these related concepts: Electrolyte and Nonelectrolyte Solutions, Turning Your Claim Into a Thesis Statement, and What is an Equation?
 square
 Appears in these related concepts: Matrix Multiplication, Factoring a Difference of Squares, and Radical Equations
 subset
 Appears in these related concepts: Total Number of Subsets, Introduction to Influence and Negotiation, and Solving Systems of Linear Inequalities
 zero
 Appears in these related concepts: Rational Inequalities, Other Equations in Quadratic Form, and Historical Traditions of Numerical Systems
 zeros
 Appears in these related concepts: Parts of a Parabola, A Graphical Interpretation of Quadratic Solutions, and Applications of the Parabola
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