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2.4 Complex Numbers

3 min read•june 24, 2024

expand our number system, combining real and imaginary parts. They're written as , where is the square root of -1. This powerful concept allows us to solve equations that were previously impossible.

We can add, subtract, multiply, and divide complex numbers using specific rules. They can be represented in different forms like algebraic and polar, and graphed on a . Understanding complex numbers opens up new mathematical possibilities.

Complex Numbers

Arithmetic with complex numbers

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  • Addition and subtraction
    • Add or subtract real parts and imaginary parts separately (like combining like terms)
    • (2+3i)+(4+5i)=(2+4)+(3+5)i=6+8i(2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5)i = 6 + 8i
    • (7−2i)−(3+4i)=(7−3)+(−2−4)i=4−6i(7 - 2i) - (3 + 4i) = (7 - 3) + (-2 - 4)i = 4 - 6i
  • Multiplication
    • Multiply terms using distributive property and i2=−1i^2 = -1
    • (2+3i)(4+5i)=8+10i+12i+15i2=8+22i−15=−7+22i(2 + 3i)(4 + 5i) = 8 + 10i + 12i + 15i^2 = 8 + 22i - 15 = -7 + 22i
    • (1−i)(2+3i)=2+3i−2i−3i2=2+i+3=5+i(1 - i)(2 + 3i) = 2 + 3i - 2i - 3i^2 = 2 + i + 3 = 5 + i
  • Division
    • Multiply numerator and denominator by of denominator to rationalize
    • Complex of a+bia + bi is a−bia - bi
    • 2+3i4+5i=(2+3i)(4−5i)(4+5i)(4−5i)=8−10i+12i−15i216+20i−20i−25i2=23+2i41=2341+241i\frac{2 + 3i}{4 + 5i} = \frac{(2 + 3i)(4 - 5i)}{(4 + 5i)(4 - 5i)} = \frac{8 - 10i + 12i - 15i^2}{16 + 20i - 20i - 25i^2} = \frac{23 + 2i}{41} = \frac{23}{41} + \frac{2}{41}i

Powers of imaginary unit

  • Cyclic pattern repeats every four powers of ii
    • i1=ii^1 = i, i2=−1i^2 = -1, i3=−ii^3 = -i, i4=1i^4 = 1, then pattern repeats (i5=ii^5 = i, i6=−1i^6 = -1, etc.)
  • Simplify powers by breaking down into multiples of four
    • i7=i4⋅i3=1⋅(−i)=−ii^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i
    • i15=(i4)3⋅i3=13⋅(−i)=−ii^{15} = (i^4)^3 \cdot i^3 = 1^3 \cdot (-i) = -i
    • i−5=1i5=1i=−ii^{-5} = \frac{1}{i^5} = \frac{1}{i} = -i

Forms of complex numbers

  • ()
    • a+bia + bi where aa is real part and bb is imaginary part (3+2i3 + 2i, −4−7i-4 - 7i)
    • r(cos⁡θ+isin⁡θ)r(\cos\theta + i\sin\theta) or reiθre^{i\theta} where rr is (magnitude) and θ\theta is (angle)
    • r=a2+b2r = \sqrt{a^2 + b^2}, argument θ=tan⁡−1(ba)\theta = \tan^{-1}(\frac{b}{a})
    • 2+2i2 + 2i in polar form: r=22+22=22r = \sqrt{2^2 + 2^2} = 2\sqrt{2}, θ=tan⁡−1(22)=45∘\theta = \tan^{-1}(\frac{2}{2}) = 45^\circ or π4\frac{\pi}{4} radians
    • ###'s_Formula_0### relates exponential and trigonometric functions: eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta
  • Converting between algebraic and polar forms
    • Algebraic to polar: use modulus and argument formulas
    • Polar to algebraic: a=rcos⁡θa = r\cos\theta, b=rsin⁡θb = r\sin\theta
    • 2(cos⁡30∘+isin⁡30∘)=2(32+12i)=3+i2(\cos 30^\circ + i \sin 30^\circ) = 2(\frac{\sqrt{3}}{2} + \frac{1}{2}i) = \sqrt{3} + i
  • Graphical representation on
    • Real part on horizontal axis, imaginary part on vertical axis
    • Modulus is distance from origin, argument is angle with positive real axis
    • 3−4i3 - 4i plotted 3 units right and 4 units down from origin
    • (a, b) correspond to the a + bi

Advanced Complex Number Concepts

  • are complex numbers with zero imaginary part (e.g., 3 + 0i)
  • are complex numbers with zero real part (e.g., 0 + 2i)
  • : (r(cos⁡θ+isin⁡θ))n=rn(cos⁡(nθ)+isin⁡(nθ))(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta)), useful for finding powers of complex numbers

Key Terms to Review (33)

A + bi: The term a + bi represents a complex number, where 'a' is the real part and 'b' is the imaginary part. Complex numbers are an extension of the real number system and are used to represent quantities that have both a magnitude and a direction.
Algebraic Form: Algebraic form refers to the way an expression or equation is written using variables, constants, and mathematical operations. It provides a structured and symbolic representation of mathematical relationships, allowing for manipulation and analysis of quantities.
Argand Diagram: The Argand diagram, also known as the complex plane, is a graphical representation of complex numbers. It provides a visual way to understand and work with complex numbers, which are essential in various areas of mathematics and physics.
Argument: An argument is a set of statements or premises that are used to support or justify a particular conclusion or claim. It is a logical structure that connects various pieces of information to make a coherent and persuasive case.
Complex Addition: Complex addition is the process of adding two complex numbers together by adding their real parts and imaginary parts separately. This operation is a fundamental concept in the study of complex numbers, which extend the real number system to include both real and imaginary components.
Complex conjugate: A complex conjugate of a complex number is obtained by changing the sign of its imaginary part. If the complex number is $a + bi$, its complex conjugate is $a - bi$.
Complex Division: Complex division is the process of dividing one complex number by another complex number. It involves representing the complex numbers in the form $a + bi$, where $a$ and $b$ are real numbers, and then applying specific steps to perform the division operation.
Complex Multiplication: Complex multiplication is a fundamental operation in the field of complex numbers, which extends the basic arithmetic of real numbers to the complex number system. It involves the multiplication of two complex numbers, resulting in a new complex number that captures both the magnitude and direction of the original operands.
Complex number: A complex number is a number that has both a real part and an imaginary part, typically expressed in the form $a + bi$ where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with the property that $i^2 = -1$. Complex numbers extend the concept of one-dimensional real numbers to the two-dimensional complex plane.
Complex Numbers: Complex numbers are a mathematical concept that extend the real number system by including the imaginary unit, denoted as $i$, which is defined as the square root of -1. They are used to represent quantities that have both magnitude and direction, and are essential in various areas of mathematics, including algebra, calculus, and physics.
Complex plane: The complex plane is a two-dimensional space where each point represents a complex number. The horizontal axis is the real part, and the vertical axis is the imaginary part.
Complex Plane: The complex plane, also known as the Argand plane or Gaussian plane, is a two-dimensional coordinate system used to represent and visualize complex numbers. It provides a geometric interpretation of complex numbers, where the real and imaginary components are plotted on perpendicular axes, allowing for a deeper understanding of complex number operations and properties.
Complex Plane Coordinates: Complex plane coordinates refer to the graphical representation of complex numbers on a two-dimensional coordinate system, known as the complex plane. This system allows for the visualization and analysis of complex number operations and relationships.
Conjugate: A conjugate is a pair of complex numbers or algebraic expressions that have the same magnitude but opposite signs in their imaginary parts. This concept is crucial in the understanding of real numbers and complex numbers, as conjugates are used to simplify calculations and perform operations such as division.
De Moivre's theorem: De Moivre's theorem states that for any complex number expressed in polar form, the $n$th power of the complex number can be found using the formula $(r( ext{cos} \theta + i \text{sin} \theta))^n = r^n (\text{cos}(n\theta) + i\text{sin}(n\theta))$. This theorem connects complex numbers to trigonometric functions and provides a powerful tool for calculating powers and roots of complex numbers, particularly when they are represented in polar form.
Euler: Euler is a fundamental mathematical constant named after the renowned Swiss mathematician Leonhard Euler. It is a transcendental number that represents the base of the natural logarithm and is essential in the study of complex numbers and polar coordinate systems.
Euler's Formula: Euler's formula is a fundamental equation in complex analysis that connects the exponential function with the trigonometric functions of sine and cosine. It is a powerful tool for representing and manipulating complex numbers in both algebraic and geometric forms.
Fundamental Theorem of Algebra: The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex number coefficients has at least one complex number solution. It is a foundational result in algebra that connects the properties of polynomials to the nature of complex numbers.
Gauss: Gauss, also known as Carl Friedrich Gauss, was a renowned German mathematician, astronomer, and physicist who made significant contributions to various fields, including complex numbers. Gauss is often referred to as the 'Prince of Mathematics' for his groundbreaking work and the profound impact he had on the development of mathematical concepts and theories.
I: The imaginary unit, denoted as 'i', is a fundamental concept in the study of complex numbers. It is defined as the square root of -1, and it represents a number that, when multiplied by itself, results in a negative value.
Imaginary number: An imaginary number is a complex number in which the real part is zero and the imaginary part is non-zero. It is typically written in the form $bi$, where $b$ is a real number and $i$ denotes the imaginary unit, defined as $i = \sqrt{-1}$.
Imaginary Numbers: Imaginary numbers are a special type of number that cannot be represented on the real number line. They are used in mathematics, physics, and engineering to solve problems that involve quantities that cannot be expressed using only real numbers.
Imaginary Unit: The imaginary unit, denoted as $i$, is a mathematical construct that represents the square root of -1. It is a fundamental concept in the study of complex numbers, which are an extension of the real number system.
Modulus: The modulus (or absolute value) of a real number is its distance from zero on the number line, denoted as $|x|$. It is always a non-negative value.
Modulus: The modulus of a complex number is the absolute value or magnitude of the number, representing its distance from the origin on the complex plane. It is a fundamental concept in the study of complex numbers and their polar form.
Polar Form: Polar form is a way of representing complex numbers and graphing equations in a polar coordinate system. It involves expressing a complex number or a curve in terms of its magnitude (or modulus) and angle (or argument) rather than its rectangular (Cartesian) coordinates.
Polar form of a conic: The polar form of a conic is an equation representing conic sections (ellipse, parabola, hyperbola) using polar coordinates $(r, \theta)$. It often involves parameters like the eccentricity $e$ and the directrix.
Pure Imaginary Number: A pure imaginary number is a complex number that has a real part of zero and only a non-zero imaginary part. It is represented in the form $bi$, where $b$ is a real number and $i$ represents the imaginary unit, defined as the square root of -1.
R(cos θ + i sin θ): The expression r(cos θ + i sin θ) is a fundamental representation of complex numbers in polar form. It allows for the concise expression of a complex number using a magnitude (r) and an angle (θ), providing an alternative to the more familiar Cartesian form (a + bi).
Re^(iθ): The expression $re^{i\theta}$ is a fundamental representation of complex numbers in polar form. It combines the modulus (r) and argument (θ) of a complex number into a single, compact expression, allowing for easy manipulation and visualization of complex number operations.
Real numbers: Real numbers include all rational and irrational numbers, encompassing the entire set of numbers on the number line. They can be positive, negative, or zero.
Real Numbers: Real numbers are the set of all numbers that include both rational numbers (fractions and integers) and irrational numbers (numbers that cannot be expressed as a simple fraction). Real numbers are the foundation for algebra and are essential for understanding and working with complex numbers.
Rectangular Form: Rectangular form is a way of representing complex numbers, where a complex number is expressed as the sum of a real part and an imaginary part. This representation provides a clear and concise way to work with complex numbers in various mathematical contexts.
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