Glossary

8.8 Use the Complex Number System

3 min read•june 25, 2024

expand our mathematical toolkit, allowing us to solve equations that were previously impossible. They introduce the , where i² = -1, and combine real and imaginary parts in the form .

Operations with complex numbers involve adding, subtracting, multiplying, and dividing. These operations follow specific rules, like using the for division. Complex numbers have practical applications in fields like electrical engineering and quantum mechanics.

Complex Numbers

Square roots of negative numbers

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  • are the written in the form [bi](https://www.fiveableKeyTerm:Bi)[bi](https://www.fiveableKeyTerm:Bi) where bb is a real number
    • The imaginary unit ii is defined as [i2](https://www.fiveableKeyTerm:i2)=−1[i^2](https://www.fiveableKeyTerm:i^2) = -1 (e.g., i=−1i = \sqrt{-1})
  • To evaluate the square root of a negative number factor out −1-1 from the , simplify the square root of the positive number, and multiply the result by ii
    • Example: −16=−1⋅16=−1⋅16=i⋅4=4i\sqrt{-16} = \sqrt{-1 \cdot 16} = \sqrt{-1} \cdot \sqrt{16} = i \cdot 4 = 4i

Operations with complex numbers

  • Complex numbers are written in the form a+bia + bi where aa and bb are real numbers (aa is the , bb is the )
  • To add or subtract complex numbers add or subtract the real parts and imaginary parts separately
    • (a+bi)+(c+di)=(a+c)+(b+d)i(a + bi) + (c + di) = (a + c) + (b + d)i
    • (a+bi)−(c+di)=(a−c)+(b−d)i(a + bi) - (c + di) = (a - c) + (b - d)i
    • Example: (3+2i)+(4−5i)=7−3i(3 + 2i) + (4 - 5i) = 7 - 3i
  • To multiply complex numbers use the and the fact that i2=−1i^2 = -1
    • (a+bi)(c+di)=ac+adi+bci+bdi2=(ac−bd)+(ad+bc)i(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i
    • Example: (2+3i)(4−i)=8−2i+12i−3i2=11+10i(2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2 = 11 + 10i

Division of complex numbers

  • To divide complex numbers multiply the numerator and denominator by the complex conjugate of the denominator
    • The complex conjugate of a+bia + bi is a−bia - bi
    • Multiplying by the complex conjugate eliminates the imaginary part in the denominator resulting in a real number
    • a+bic+di=(a+bi)(c−di)(c+di)(c−di)=(ac+bd)+(bc−ad)ic2+d2\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}
    • Example: 2+3i4−5i=(2+3i)(4+5i)(4−5i)(4+5i)=−7+22i41=−741+2241i\frac{2 + 3i}{4 - 5i} = \frac{(2 + 3i)(4 + 5i)}{(4 - 5i)(4 + 5i)} = \frac{-7 + 22i}{41} = -\frac{7}{41} + \frac{22}{41}i

Powers of i

  • Powers of ii follow a : i1=ii^1 = i, i2=−1i^2 = -1, i3=−ii^3 = -i, i4=1i^4 = 1, and the pattern repeats
  • To simplify expressions with powers of ii divide the exponent by 4 and use the remainder to determine the simplified form
    • Remainder 0 results in 1, remainder 1 results in ii, remainder 2 results in −1-1, remainder 3 results in −i-i
    • Example: i13=i4⋅3+1=(i4)3⋅i1=13⋅i=ii^{13} = i^{4 \cdot 3 + 1} = (i^4)^3 \cdot i^1 = 1^3 \cdot i = i

Geometric representation of complex numbers

  • Complex numbers can be represented on a (also known as an )
    • The horizontal axis represents the real part, and the vertical axis represents the imaginary part
  • The of a complex number z=a+biz = a + bi is given by ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}
  • The of a complex number is the angle it makes with the positive real axis
  • Complex numbers can be expressed in , which relates to their geometric representation
  • connects complex exponentials to trigonometric functions: eix=cos⁡x+isin⁡xe^{ix} = \cos x + i \sin x

Applications of complex numbers

  • Complex numbers model various real-world situations
    • Electrical engineering represents , voltage, and current as complex numbers
    • Quantum mechanics describes the state of a quantum system using complex numbers
    • Fluid dynamics models two-dimensional fluid flow with complex numbers
  • When solving real-world problems with complex numbers
    • Represent relevant quantities as complex numbers
    • Apply appropriate complex number operations to solve the problem
    • Interpret the results in the context of the real-world situation
    • Example: In electrical engineering, impedance Z=R+XiZ = R + Xi where RR is , XX is , and ii is the imaginary unit

Key Terms to Review (25)

A + bi: The term a + bi represents a complex number, which is a number that has both a real part (a) and an imaginary part (bi). Complex numbers are used to represent quantities that cannot be fully described by real numbers alone, and they play a crucial role in various mathematical and scientific applications, including the study of 8.8 Use the Complex Number System.
Absolute Value (Modulus): The absolute value, also known as the modulus, of a real number is the distance of that number from zero on the number line. It represents the magnitude or size of a number, regardless of its sign. The absolute value of a number is always a non-negative real number.
Argand Diagram: An Argand diagram is a graphical representation of the complex number system, where complex numbers are plotted on a two-dimensional coordinate plane. It provides a visual aid for understanding and working with complex numbers.
Argument (Phase): The argument, or phase, of a complex number refers to the angle or direction of the number on the complex plane. It represents the orientation of the complex number relative to the positive real axis. The argument is a crucial concept in understanding and working with complex numbers, particularly in the context of their geometric representation and applications in various mathematical and engineering domains.
Bi: The prefix 'bi' is a numerical prefix that means 'two' or 'twice'. In the context of complex numbers, the term 'bi' is used to describe the imaginary part of a complex number, which is represented by the variable 'i'.
Complex Conjugate: A complex conjugate is a pair of complex numbers that have the same real part but the opposite sign of the imaginary part. It is a fundamental concept in the complex number system and is essential for understanding various operations and properties involving complex numbers.
Complex Numbers: Complex numbers are a numerical system that extends the real number system to include imaginary numbers. They are represented in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part, and $i$ represents the square root of -1.
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 parts of a complex number are plotted on the horizontal and vertical axes, respectively.
Cyclic Pattern: A cyclic pattern refers to a repetitive sequence or trend that occurs within a set of data or a system over time. This pattern is characterized by the regular and predictable recurrence of specific events, behaviors, or numerical values, often observed in various natural and man-made phenomena.
Distributive Property: The distributive property is a fundamental algebraic rule that states that the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of expressions involving multiplication and addition or subtraction.
Euler's Formula: Euler's formula is a fundamental relationship in complex number theory that connects the exponential function with the trigonometric functions. It provides a way to represent complex numbers using trigonometric functions and is widely used in various fields of mathematics and physics.
I: The symbol 'i' represents the imaginary unit, a fundamental concept in the complex number system. It is defined as the square root of -1, written mathematically as $i = \sqrt{-1}$. The complex number system extends the real number system by introducing this imaginary unit, allowing for the representation of a wider range of mathematical concepts and applications.
I^2: The term $i^2$ refers to the square of the imaginary unit $i$, which is defined as the square root of -1. In the context of the complex number system, understanding the properties and behavior of $i^2$ is crucial for working with and manipulating complex numbers.
Imaginary Numbers: Imaginary numbers are a special set of numbers that are used to represent quantities that cannot be expressed using real numbers alone. They are an essential part of the complex number system, which extends the real number line to include both real and imaginary components.
Imaginary Part: The imaginary part of a complex number is the component that represents the imaginary or non-real portion of the number. It is the part of the complex number that is multiplied by the imaginary unit, $i$, where $i^2 = -1$. The imaginary part, along with the real part, defines the complete complex number.
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 complex number system, which extends the real number line to include numbers with both real and imaginary components.
Impedance: Impedance is a measure of the opposition to the flow of alternating current (AC) in an electrical circuit. It is a complex quantity that combines resistance and reactance, and it determines the relationship between voltage and current in an AC circuit.
Polar Form: Polar form is a way of representing complex numbers that uses two values: the magnitude (or modulus) and the angle (or argument). This representation provides an alternative to the more common Cartesian form, which uses real and imaginary components. Polar form is particularly useful for performing operations on complex numbers, such as multiplication, division, and exponentiation.
Powers of i: The powers of the imaginary unit, i, are a fundamental concept in the complex number system. The imaginary unit, i, is defined as the square root of -1, and its powers exhibit a cyclical pattern that is crucial in understanding complex number operations and applications.
Pure Imaginary: A pure imaginary number is a complex number that has a real part of zero and only an imaginary part. It is represented in the form $bi$, where $b$ is a real number and $i$ is the imaginary unit, defined as $i^2 = -1$. Pure imaginary numbers are an essential component of the complex number system, which is used to extend the real number system and provide solutions to certain mathematical problems.
Radicand: The radicand is the quantity or expression under the radical sign in a radical expression. It represents the value or number that is to be operated on by the radical symbol, such as the square root or cube root.
Reactance: Reactance is a measure of the opposition to the flow of alternating current (AC) in an electrical circuit, caused by the inductive or capacitive properties of the circuit components. It is a key concept in understanding the behavior of AC circuits and the complex number system.
Real Part: The real part of a complex number is the component of the number that represents the real number portion, as opposed to the imaginary number portion. It is a crucial concept in understanding and working with complex numbers.
Resistance: Resistance is a fundamental concept in the study of complex number systems. It refers to the opposition or hindrance to the flow of electrical current or the transmission of electromagnetic waves. In the context of complex numbers, resistance is a crucial factor in understanding the behavior and properties of these numbers.
Square Roots of Negative Numbers: The square root of a negative number is a complex number, as it cannot be expressed as a real number. This concept is an important part of the complex number system, which extends the real number system to include imaginary numbers.
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