Glossary

5.2 Complex Numbers and Operations

5 min read•july 31, 2024

Complex numbers expand our number system, letting us solve equations that were once impossible. They're like a secret weapon in math, opening up new possibilities in algebra and beyond.

In this part of the chapter, we'll learn how to work with complex numbers. We'll cover their different forms, basic operations, and even see how they connect to quadratic equations. It's gonna be eye-opening!

Complex Number Representation

Definition and Components

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  • A is a number expressed in the form a+bia + bi, where aa and bb are real numbers, and ii is the defined as the square root of -1
  • The of a complex number is the aa value, and the is the bb value
  • Examples of complex numbers: 3+2i3 + 2i, −4−7i-4 - 7i, 6i6i

Rectangular and Polar Forms

  • In (a+bi)(a + bi), aa represents the point's horizontal position and bb represents the vertical position on the
  • The () of a complex number is the distance from the origin to the point representing the complex number on the complex plane, calculated using the Pythagorean theorem: ∣a+bi∣=a2+b2|a + bi| = \sqrt{a^2 + b^2}
  • In , a complex number is defined as r(cos⁡(θ)+isin⁡(θ))r(\cos(\theta) + i \sin(\theta)), where rr is the absolute value (modulus) and θ\theta is the angle () formed with the positive real axis on the complex plane
  • The argument (θ)(\theta) of a complex number can be calculated using the arctangent function: θ=arctan⁡(b/a)\theta = \arctan(b/a), with special consideration given to the signs of aa and bb to determine the quadrant
  • , eiθ=cos⁡(θ)+isin⁡(θ)e^{i\theta} = \cos(\theta) + i \sin(\theta), provides a connection between the polar and exponential forms of complex numbers

Complex Number Arithmetic

Addition, Subtraction, and Multiplication

  • Addition and are performed by adding or subtracting the real and imaginary parts separately: (a+bi)±(c+di)=(a±c)+(b±d)i(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i
    • Example: (3+2i)+(4−5i)=(3+4)+(2−5)i=7−3i(3 + 2i) + (4 - 5i) = (3 + 4) + (2 - 5)i = 7 - 3i
  • follows the distributive property and the rule i2=−1i^2 = -1: (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a + bi)(c + di) = (ac - bd) + (ad + bc)i
    • Example: (2+3i)(4−i)=(2⋅4−3⋅(−1))+(2⋅(−1)+3⋅4)i=11+10i(2 + 3i)(4 - i) = (2 \cdot 4 - 3 \cdot (-1)) + (2 \cdot (-1) + 3 \cdot 4)i = 11 + 10i
  • The of a complex number a+bia + bi is a−bia - bi. The product of a complex number and its conjugate is always a real number: (a+bi)(a−bi)=a2+b2(a + bi)(a - bi) = a^2 + b^2

Division and De Moivre's Theorem

  • is performed by multiplying the numerator and denominator by the conjugate of the denominator, then simplifying: (a+bi)÷(c+di)=(a+bi)(c−di)(c+di)(c−di)=ac+bdc2+d2+bc−adc2+d2i(a + bi) \div (c + di) = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i
    • Example: (3+4i)÷(1−2i)=(3+4i)(1+2i)(1−2i)(1+2i)=11+2i5=115+25i(3 + 4i) \div (1 - 2i) = \frac{(3 + 4i)(1 + 2i)}{(1 - 2i)(1 + 2i)} = \frac{11 + 2i}{5} = \frac{11}{5} + \frac{2}{5}i
  • states that for any complex number (cos⁡(θ)+isin⁡(θ))(\cos(\theta) + i \sin(\theta)) and any integer nn, (cos⁡(θ)+isin⁡(θ))n=cos⁡(nθ)+isin⁡(nθ)(\cos(\theta) + i \sin(\theta))^n = \cos(n\theta) + i \sin(n\theta). This theorem is useful for finding roots and powers of complex numbers

Complex Solutions to Quadratics

Quadratic Formula and Discriminant

  • The quadratic formula, x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, can be used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are real numbers and a≠0a \neq 0
  • When the (b2−4ac)(b^2 - 4ac) is negative, the quadratic equation has two complex solutions, which are complex conjugates of each other
  • To find the complex solutions, substitute the negative discriminant into the quadratic formula and simplify the expression, expressing the result in the form a+bia + bi
    • Example: For the quadratic equation x2+2x+5=0x^2 + 2x + 5 = 0, the discriminant is 22−4⋅1⋅5=−162^2 - 4 \cdot 1 \cdot 5 = -16. The complex solutions are x=−2±−162=−1±2ix = \frac{-2 \pm \sqrt{-16}}{2} = -1 \pm 2i

Sum and Product of Complex Solutions

  • The sum of the complex solutions of a quadratic equation is equal to −b/a-b/a, and the product of the solutions is equal to c/ac/a, which are the negatives of the coefficients of the quadratic equation in standard form
    • Example: For the quadratic equation x2+2x+5=0x^2 + 2x + 5 = 0, the sum of the solutions is −2/1=−2-2/1 = -2, and the product of the solutions is 5/1=55/1 = 5

Geometric Interpretation of Complex Numbers

Complex Plane (Argand Plane)

  • The complex plane () is a two-dimensional representation of complex numbers, where the horizontal axis represents the real part and the vertical axis represents the imaginary part
  • Each complex number a+bia + bi corresponds to a unique point (a,b)(a, b) on the complex plane, with aa being the x-coordinate and bb being the y-coordinate
  • The absolute value (modulus) of a complex number is the distance from the origin (0,0)(0, 0) to the point representing the complex number
  • The argument (angle) of a complex number is the angle formed between the positive real axis and the line segment connecting the origin to the point representing the complex number, measured in radians or degrees

Geometric Transformations

  • such as rotation, reflection, and dilation can be performed on complex numbers in the complex plane
  • Multiplication by ii corresponds to a 90-degree counterclockwise rotation in the complex plane, while multiplication by −i-i corresponds to a 90-degree clockwise rotation
    • Example: Multiplying 2+3i2 + 3i by ii results in −3+2i-3 + 2i, which is a 90-degree counterclockwise rotation of the point (2,3)(2, 3) in the complex plane
  • Reflection across the real axis is achieved by taking the complex conjugate of a complex number
    • Example: The complex conjugate of 2+3i2 + 3i is 2−3i2 - 3i, which is the reflection of the point (2,3)(2, 3) across the real axis in the complex plane

Key Terms to Review (24)

Absolute Value: Absolute value is the distance of a number from zero on the number line, regardless of its sign. It’s represented by two vertical bars surrounding a number, like |x|, and is always a non-negative value. This concept is crucial when dealing with inequalities and complex numbers, as it helps in defining the magnitude and provides clarity in comparisons and calculations involving these numbers.
Addition of Complex Numbers: Addition of complex numbers involves combining two complex numbers by adding their real parts and their imaginary parts separately. This process allows for the manipulation of complex numbers in mathematical operations, which is essential when working with polynomials, solving equations, and understanding functions in the complex plane. The addition follows the format $(a + bi) + (c + di) = (a + c) + (b + d)i$, where $a$ and $c$ are real parts, and $b$ and $d$ are imaginary parts.
Argand Plane: The Argand plane is a two-dimensional plane used to represent complex numbers graphically. In this system, the horizontal axis represents the real part of a complex number, while the vertical axis represents the imaginary part. This visual representation allows for easier understanding and manipulation of complex numbers and their operations.
Argument: In mathematics, particularly in the context of complex numbers, an argument refers to the angle formed between the positive x-axis and a line drawn from the origin to a point in the complex plane. It is an essential aspect that helps to represent complex numbers in polar form, connecting the geometric interpretation of complex numbers with their algebraic representation.
Complex Number: A complex number is a number that can be expressed in the form of 'a + bi', where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit defined by the property that $i^2 = -1$. Complex numbers allow for the solution of equations that have no real solutions and are essential in various fields such as engineering, physics, and applied mathematics. They can be represented graphically on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
Complex plane: The complex plane is a two-dimensional geometric representation of complex numbers, where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number. This visual representation helps in understanding complex numbers, their operations, and their relationships with one another. By plotting complex numbers as points or vectors on this plane, one can easily perform arithmetic operations and visualize concepts such as magnitude and angle.
Complex roots: Complex roots are solutions to polynomial equations that involve complex numbers, typically expressed in the form of 'a + bi', where 'a' and 'b' are real numbers and 'i' is the imaginary unit defined as $$i = \sqrt{-1}$$. These roots appear when the discriminant of a quadratic equation is negative, indicating that there are no real solutions. Complex roots always occur in conjugate pairs, meaning if 'a + bi' is a root, then 'a - bi' is also a root, which plays a crucial role in understanding polynomial functions and their graphs.
Conjugate: In mathematics, a conjugate refers to a pair of complex numbers that have the same real part but opposite imaginary parts. The concept of conjugates is crucial when performing operations involving complex numbers, particularly in simplifying expressions and rationalizing denominators. Understanding how to use conjugates can help streamline calculations and reveal important properties of complex numbers.
De Moivre's Theorem: De Moivre's Theorem states that for any real number $$ heta$$ and integer $$n$$, the complex number in trigonometric form can be expressed as \( (r( ext{cos} \theta + i \text{sin} \theta))^n = r^n (\text{cos}(n\theta) + i \text{sin}(n\theta)) \). This theorem provides a powerful tool for raising complex numbers to powers and extracting roots, connecting it closely to operations with complex numbers and polar coordinates.
Discriminant: The discriminant is a key component of the quadratic formula, represented as $$D = b^2 - 4ac$$, which helps determine the nature of the roots of a quadratic equation. It indicates whether the roots are real and distinct, real and repeated, or complex. Understanding the discriminant is essential for solving quadratic equations and analyzing their graphs, as it directly relates to the behavior of the parabola formed by these equations.
Division of complex numbers: Division of complex numbers refers to the process of finding the quotient when one complex number is divided by another. This operation involves multiplying both the numerator and denominator by the conjugate of the denominator to eliminate any imaginary parts in the denominator, resulting in a complex number in standard form, which is 'a + bi'. Understanding how to divide complex numbers also helps in grasping other operations such as addition and multiplication, and is essential for solving equations involving complex variables.
Electrical Engineering: Electrical engineering is a branch of engineering that deals with the study, design, and application of systems and devices that utilize electricity, electromagnetism, and electronics. It plays a crucial role in the development of technologies that are essential for modern living, such as communication systems, power generation, and electronic devices, often utilizing complex numbers for analysis and design purposes.
Euler's Formula: Euler's Formula states that for any real number $x$, the equation $e^{ix} = ext{cos}(x) + i ext{sin}(x)$ holds true. This remarkable formula establishes a deep connection between complex exponential functions and trigonometric functions, allowing complex numbers to be represented in a polar form. The beauty of Euler's Formula lies in its ability to simplify calculations involving complex numbers and to reveal the geometric interpretations of these numbers on the complex plane.
Geometric transformations: Geometric transformations are operations that alter the position, size, or shape of geometric figures in a coordinate plane. These transformations include translations, rotations, reflections, and dilations, each impacting the figures in distinct ways. They are fundamental in understanding how shapes interact with one another and help illustrate the concepts of symmetry and congruence in mathematics.
Imaginary part: The imaginary part of a complex number is the component that is multiplied by the imaginary unit, denoted as 'i', which is defined as the square root of -1. In a complex number written in the form 'a + bi', 'a' represents the real part while 'bi' represents the imaginary part, where 'b' is a real number. Understanding the imaginary part is crucial in working with complex numbers, as it helps in performing operations and visualizing them in the complex plane.
Imaginary Unit: The imaginary unit is defined as the square root of -1, represented by the symbol 'i'. This concept allows for the extension of real numbers into complex numbers, which are expressed in the form a + bi, where a and b are real numbers. The imaginary unit is crucial for performing operations involving complex numbers and provides a way to solve equations that do not have real solutions.
Modulus: In mathematics, the modulus of a complex number is the distance from the origin to the point represented by that complex number in the complex plane. It is a crucial concept for understanding how to perform operations with complex numbers, particularly in converting between rectangular and polar forms. The modulus provides insight into the size of the complex number and is calculated using the Pythagorean theorem.
Multiplication of complex numbers: Multiplication of complex numbers involves combining two complex numbers to produce another complex number, using the distributive property and the fact that the imaginary unit $i$ satisfies $i^2 = -1$. This operation can be performed in rectangular form or in trigonometric form, providing different insights into the behavior and properties of complex numbers. Understanding this multiplication is crucial for further topics like polar coordinates and analyzing functions in the complex plane.
Polar form: Polar form is a way of expressing complex numbers in terms of their magnitude (or modulus) and angle (or argument) instead of the traditional rectangular coordinate system. This representation connects the complex number's position on the complex plane to trigonometric concepts, making it easier to perform multiplication and division operations. Polar form is particularly useful in various applications, including electrical engineering and signal processing, where rotations and oscillations are involved.
Quadratic equation with complex solutions: A quadratic equation with complex solutions is a second-degree polynomial equation that does not have real number solutions, typically expressed in the form $ax^2 + bx + c = 0$ where the discriminant $b^2 - 4ac$ is negative. These equations yield solutions that involve imaginary numbers, represented in the form $x = \frac{-b \pm \sqrt{D}}{2a}$, where $D$ is the negative discriminant. Understanding these complex solutions is essential when working with polynomials that do not intersect the x-axis.
Real Part: The real part refers to the component of a complex number that is a real number, distinguishing it from the imaginary part. In the context of complex numbers, a number is typically expressed in the form $$a + bi$$, where $$a$$ represents the real part and $$bi$$ represents the imaginary part. Understanding the real part is essential for performing operations with complex numbers, as it helps in visualizing them on the complex plane and simplifying calculations.
Rectangular Form: Rectangular form refers to the way complex numbers are expressed in the format $$a + bi$$, where $$a$$ is the real part and $$b$$ is the imaginary part. This representation allows for straightforward arithmetic operations, such as addition and subtraction, and facilitates a clear visualization of complex numbers on a Cartesian coordinate system. Understanding rectangular form is essential as it serves as a foundation for converting complex numbers into other forms, like polar form, which can simplify multiplication and division.
Signal processing: Signal processing is the analysis, interpretation, and manipulation of signals, which can include audio, video, or any form of data representation. It involves techniques that transform or enhance these signals to improve their quality or extract important information. Signal processing is closely related to complex numbers and operations as it often utilizes mathematical representations of signals in the form of complex numbers for efficient computation and analysis.
Subtraction of complex numbers: Subtraction of complex numbers involves taking one complex number and removing the corresponding parts of another complex number, resulting in a new complex number. This process utilizes the structure of complex numbers, which consist of a real part and an imaginary part, allowing for straightforward calculations. Understanding this operation is essential as it lays the groundwork for more advanced concepts and operations within the realm of complex numbers.
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