Search

15.4: Complex Numbers
https://math.libretexts.org/Bookshelves/Applied_Mathematics/Math_in_Society_(Lippman)/15%3A_Fractals/15.04%3A_Complex_Numbers
When we add $-1+3 i$ , we add -1 to the real part, moving the point 1 units to the left, and we add 5 to the imaginary part, moving the point 5 units vertically. \(\begin{array}{ll} (2+5 i)(4+i) & \t...When we add $-1+3 i$ , we add -1 to the real part, moving the point 1 units to the left, and we add 5 to the imaginary part, moving the point 5 units vertically. $\begin{array}{ll} (2+5 i)(4+i) & \text{Expand} \\ =8+20 i+2 i+5 i^{2} & \text{Since }i=\sqrt{-1}, i^{2}=-1 \\ =8+20 i+2 i+5(-1) & \text{Simplify} \\ =3+22 i \end{array}$ $i \cdot 1+i \cdot 2 i = i+2 i^{2}=i+2(-1)=-2+i$ $(1+2 i)(1+i)=1+i+2 i+2 i^{2}=1+3 i+2(-1)=-1+3 i$ $(-1+3 i)(1+i) =-1-i+3 i+3 i^{2} =-1+2 i+3(-1) =-4+2 i$
6.1: Complex Numbers, Vectors and Matrices
https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/06%3A_Complex_Analysis_I/6.01%3A_Complex_Numbers_Vectors_and_Matrices
$\frac{z_{1}}{z_{2}} \equiv \frac{z_{1}}{z_{2}} \frac{\overline{z_{2}}}{\overline{z_{2}}} = \frac{x_{1}x_{2}+y_{1}y_{2}+i(x_{2}y_{1}-x_{1}y_{2})}{x_{2}^{2}+y_{2}^{2}} \nonumber$ \[\begin{align*} z_{... $\frac{z_{1}}{z_{2}} \equiv \frac{z_{1}}{z_{2}} \frac{\overline{z_{2}}}{\overline{z_{2}}} = \frac{x_{1}x_{2}+y_{1}y_{2}+i(x_{2}y_{1}-x_{1}y_{2})}{x_{2}^{2}+y_{2}^{2}} \nonumber$ $\begin{align*} z_{1}z_{2} &= |z_{1}||z_{2}|(\cos(\theta_{1})\cos(\theta_{2})-\sin(\theta_{1})\sin(\theta_{2})+i(\cos(\theta_{1}) \sin(\theta_{2})+\sin(\theta_{1}) \cos(\theta_{2}))) \\[4pt] &=|z_{1}||z_{2}|(\cos(\theta_{1}+\theta_{2})+i \sin(\theta_{1}+\theta_{2})) \end{align*}$
16.4.1: Complex Numbers
https://math.libretexts.org/Bookshelves/Applied_Mathematics/Developmental_Math_(NROC)/16%3A_Radical_Expressions_and_Quadratic_Equations/16.04%3A_Complex_Numbers/16.4.01%3A_Complex_Numbers
Up to now, you’ve known it was impossible to take a square root of a negative number. This is true, using only the real numbers. But here you will learn about a new kind of number that lets you work w...Up to now, you’ve known it was impossible to take a square root of a negative number. This is true, using only the real numbers. But here you will learn about a new kind of number that lets you work with square roots of negative numbers! Like fractions and negative numbers, this new kind of number will let you do what was previously impossible.
1.1: Complex Numbers
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/01%3A_Preliminaries/1.01%3A_Complex_Numbers
The complex numbers were originally invented to solve problems in algebra. It was later recognized that the algebra of complex numbers provides an elegant set of tools for geometry in the plane. This ...The complex numbers were originally invented to solve problems in algebra. It was later recognized that the algebra of complex numbers provides an elegant set of tools for geometry in the plane. This section presents the basics of the algebra and geometry of the complex numbers.
7.7: Complex Numbers and Their Operations
https://math.libretexts.org/Workbench/Hawaii_CC_Intermediate_Algebra/07%3A_Radical_Functions_and_Equations/7.07%3A_Complex_Numbers_and_Their_Operations
There is no real number that when squared results in a negative number. We begin to resolve this issue by defining the imaginary unit, i , as the square root of −1 . In this way any square root of a...There is no real number that when squared results in a negative number. We begin to resolve this issue by defining the imaginary unit, i , as the square root of −1 . In this way any square root of a negative real number can be written in terms of the imaginary unit. Such a number is often called an imaginary number.
5.1: The Complex Number System
https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Trigonometry_(Sundstrom_and_Schlicker)/05%3A_Complex_Numbers_and_Polar_Coordinates/5.01%3A_The_Complex_Number_System
To make sense of solutions of quadratic equations that are not real, we introduce complex numbers. Although complex numbers arise naturally when solving quadratic equations, their introduction into ma...To make sense of solutions of quadratic equations that are not real, we introduce complex numbers. Although complex numbers arise naturally when solving quadratic equations, their introduction into mathematics came about from the problem of solving cubic equations.
5.3: DeMoivre’s Theorem and Powers of Complex Numbers
https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Trigonometry_(Sundstrom_and_Schlicker)/05%3A_Complex_Numbers_and_Polar_Coordinates/5.03%3A_DeMoivres_Theorem_and_Powers_of_Complex_Numbers
The trigonometric form of a complex number provides a relatively quick and easy way to compute products of complex numbers. As a consequence, we will be able to quickly calculate powers of complex num...The trigonometric form of a complex number provides a relatively quick and easy way to compute products of complex numbers. As a consequence, we will be able to quickly calculate powers of complex numbers, and even roots of complex numbers.
6.1: Complex Numbers
https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/06%3A_Complex_Numbers/6.01%3A_Complex_Numbers
Although very powerful, the real numbers are inadequate to solve equations such as $x^2+1=0$ , and this is where complex numbers come in.
2: Introduction to Complex Numbers
https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/02%3A_Introduction_to_Complex_Numbers
Let $\mathbb{R}$ denote the set of $\textbf{real numbers}$ , which should be a familiar collection of numbers to anyone who has studied Calculus. In this chapter, we use $\mathbb{R}$ to build the...Let $\mathbb{R}$ denote the set of $\textbf{real numbers}$ , which should be a familiar collection of numbers to anyone who has studied Calculus. In this chapter, we use $\mathbb{R}$ to build the equally important set of so-called complex numbers. Isaiah Lankham, Mathematics Department at UC Davis Bruno Nachtergaele, Mathematics Department at UC Davis Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.
1.3: Terminology and Basic Arithmetic
https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/01%3A_Complex_Algebra_and_the_Complex_Plane/1.03%3A_Terminology_and_Basic_Arithmetic
If $z = x + iy$ then its conjugate is $\bar{z} = x - iy$ and we read this as "z-bar = $x - iy$ ". The following is a very useful property of conjugation: If $z = x + iy$ then In the next sectio...If $z = x + iy$ then its conjugate is $\bar{z} = x - iy$ and we read this as "z-bar = $x - iy$ ". The following is a very useful property of conjugation: If $z = x + iy$ then In the next section we will discuss the geometry of complex numbers, which gives some insight into the meaning of the magnitude of a complex number. The magnitude of the complex number $x + iy$ is defined as The norm is the sum of $x^2$ and $y^2$ .
6.1: Complex Numbers
https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/06%3A_Complex_Numbers/6.01%3A_Complex_Numbers
Although very powerful, the real numbers are inadequate to solve equations such as $x^2+1=0$ , and this is where complex numbers come in.

Search

Text Color

Text Size

Margin Size

Font Type

Support Center

How can we help?