Glossary

1.1 The imaginary unit i and imaginary numbers

4 min read•august 13, 2024

expand our mathematical toolkit beyond real numbers. The imaginary unit 'i' is defined as the square root of -1, allowing us to solve equations that were previously unsolvable. This concept opens up new possibilities in algebra and geometry.

Imaginary numbers are represented on the complex plane, with the on the horizontal axis and the imaginary part on the vertical axis. This visualization helps us understand their properties and perform arithmetic operations, laying the foundation for further study of complex analysis.

The imaginary unit 'i'

Definition and properties of the imaginary unit 'i'

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  • The imaginary unit 'i' is defined as a number that satisfies the equation [i^2 = -1](https://www.fiveableKeyTerm:i^2_=_-1)
  • 'i' is not a real number and cannot be represented on the real number line
  • The square of 'i' equals -1, which is a real number
  • When 'i' is multiplied by itself an odd number of times, the result is either 'i' or '-i'
    • i1=ii^1 = i
    • i3=i2⋅i=−1⋅i=−ii^3 = i^2 \cdot i = -1 \cdot i = -i
  • When 'i' is multiplied by itself an even number of times, the result is either 1 or -1
    • i2=−1i^2 = -1
    • i4=(i2)2=(−1)2=1i^4 = (i^2)^2 = (-1)^2 = 1

Imaginary numbers in the complex plane

Representation of imaginary numbers in the complex plane

  • An imaginary number is a number that can be written as a real number multiplied by 'i'
    • 3i3i, −2i-2i, and 2i\sqrt{2}i are examples of imaginary numbers
  • The complex plane is a two-dimensional representation of complex numbers
    • The real part is plotted on the horizontal axis
    • The imaginary part is plotted on the vertical axis
  • Pure imaginary numbers are complex numbers with a real part equal to zero
    • Pure imaginary numbers are represented on the vertical axis of the complex plane
    • Examples of pure imaginary numbers: 2i2i, −5i-5i, and −12i-\frac{1}{2}i
  • The distance of an imaginary number from the origin in the complex plane represents its magnitude or absolute value
  • The direction of an imaginary number in the complex plane is always along the vertical axis
    • Positive imaginary numbers are directed upward
    • Negative imaginary numbers are directed downward

Arithmetic with imaginary numbers

Addition and subtraction of imaginary numbers

  • Addition and subtraction of imaginary numbers are performed by adding or subtracting their coefficients while keeping 'i'
    • (2i)+(3i)=5i(2i) + (3i) = 5i
    • (−4i)−(7i)=−11i(-4i) - (7i) = -11i
  • The real parts and imaginary parts are added or subtracted separately
    • (3+2i)+(4−5i)=(3+4)+(2−5)i=7−3i(3 + 2i) + (4 - 5i) = (3 + 4) + (2 - 5)i = 7 - 3i

Multiplication and division of imaginary numbers

  • Multiplication of imaginary numbers is performed by multiplying their coefficients and applying the property i2=−1i^2 = -1 when necessary
    • (2i)⋅(3i)=2⋅3⋅i2=6⋅(−1)=−6(2i) \cdot (3i) = 2 \cdot 3 \cdot i^2 = 6 \cdot (-1) = -6
    • (4i)⋅(−2i)=4⋅(−2)⋅i2=−8⋅(−1)=8(4i) \cdot (-2i) = 4 \cdot (-2) \cdot i^2 = -8 \cdot (-1) = 8
  • Division of imaginary numbers is performed by multiplying the numerator and denominator by the of the denominator
    • The complex conjugate of a+bia + bi is a−bia - bi, obtained by changing the sign of the imaginary part
    • 2i3i=2i3i⋅3i3i=2⋅3⋅i232⋅i2=−69=−23\frac{2i}{3i} = \frac{2i}{3i} \cdot \frac{3i}{3i} = \frac{2 \cdot 3 \cdot i^2}{3^2 \cdot i^2} = \frac{-6}{9} = -\frac{2}{3}
  • The absolute value (modulus) of an imaginary number a+bia + bi is given by the formula a2+b2\sqrt{a^2 + b^2}
    • The absolute value represents the distance from the origin in the complex plane
    • ∣3i∣=02+32=9=3|3i| = \sqrt{0^2 + 3^2} = \sqrt{9} = 3

Solving equations with imaginary numbers

Solving techniques for equations involving imaginary numbers

  • Equations involving imaginary numbers can be solved by applying the same algebraic techniques used for solving equations with real numbers
  • When solving equations with imaginary numbers, keep the real and imaginary parts separate and equate the corresponding parts on both sides of the equation
    • 2x+3i=5+7i2x + 3i = 5 + 7i becomes 2x=52x = 5 (real part) and 3i=7i3i = 7i (imaginary part)
  • Equations involving higher powers of 'i' can be simplified by applying the properties of 'i'
    • i2=−1i^2 = -1, i3=−ii^3 = -i, and i4=1i^4 = 1 can be used to reduce the powers of 'i'
    • x2+2ix−3=0x^2 + 2ix - 3 = 0 can be simplified to x2−3+2ix=0x^2 - 3 + 2ix = 0

Solving quadratic equations with imaginary solutions

  • Quadratic equations with negative discriminants (b2−4ac<0b^2 - 4ac < 0) have complex solutions involving imaginary numbers
  • The quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} can be used to find the complex solutions of quadratic equations
    • When the discriminant (b2−4acb^2 - 4ac) is negative, the solutions will involve imaginary numbers
    • Example: x2+4x+5=0x^2 + 4x + 5 = 0 has solutions x=−2±ix = -2 \pm i, as the discriminant is 42−4⋅1⋅5=−44^2 - 4 \cdot 1 \cdot 5 = -4

Key Terms to Review (16)

Addition of Complex Numbers: The addition of complex numbers involves combining two or more complex numbers to produce another complex number. Each complex number is expressed in the form $$a + bi$$, where $$a$$ is the real part and $$b$$ is the imaginary part. When adding complex numbers, the real parts are added together and the imaginary parts are added together, resulting in a new complex number.
Argand Plane: The Argand plane is a two-dimensional graphical representation of complex numbers, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This plane provides a visual way to understand and operate with complex numbers, facilitating the addition, subtraction, multiplication, and division of these numbers as geometric transformations in the plane.
Carl Friedrich Gauss: Carl Friedrich Gauss was a renowned German mathematician and physicist known for his contributions to various fields including number theory, statistics, and complex analysis. His work laid the foundation for many concepts in mathematics, particularly regarding the imaginary unit and the properties of complex numbers, which have profound implications in various mathematical mappings and transformations.
Complex Conjugate: A complex conjugate is a number that has the same real part as a given complex number but an opposite imaginary part. This concept is essential in understanding the properties of complex numbers, their algebraic operations, and their geometric representations. Complex conjugates are commonly used in calculations involving complex division and provide insights into the behavior of functions involving imaginary numbers.
Complex Numbers: Complex numbers are numbers that consist of a real part and an imaginary part, typically expressed in the form $a + bi$, where $a$ is the real part, $b$ is the imaginary part, and $i$ is the imaginary unit defined as $ ext{sqrt}(-1)$. These numbers expand our understanding of the number system by allowing solutions to equations that do not have real solutions, such as $x^2 + 1 = 0$. The incorporation of complex numbers is crucial for many mathematical concepts and applications, including solving polynomial equations and analyzing oscillatory behavior in engineering and physics.
Gottfried Wilhelm Leibniz: Gottfried Wilhelm Leibniz was a prominent German mathematician and philosopher who co-developed calculus independently of Isaac Newton in the late 17th century. His work laid the groundwork for modern mathematics and introduced important concepts such as the formal notation for derivatives, which are essential in understanding functions, including those involving the imaginary unit i and imaginary numbers.
I^2 = -1: The expression i^2 = -1 defines the imaginary unit 'i', which is fundamental in the study of complex numbers. This equation indicates that when 'i' is squared, the result is negative one, fundamentally changing the way we understand numbers by allowing solutions to equations that have no real solutions. This concept opens up a new dimension in mathematics, leading to the development of complex analysis and its many applications in various fields.
Imaginary unit i: The imaginary unit i is defined as the square root of -1, a fundamental concept in mathematics that allows for the extension of the real number system to include imaginary numbers. This concept is crucial for defining complex numbers, which take the form a + bi, where a and b are real numbers. The introduction of i enables mathematicians to solve equations that would otherwise have no solutions in the realm of real numbers.
Magnitude of a complex number: The magnitude of a complex number, often denoted as |z|, is the distance of that number from the origin in the complex plane. It is calculated using the formula |z| = $$ ext{sqrt}(a^2 + b^2)$$ for a complex number z = a + bi, where a is the real part and b is the imaginary part. Understanding magnitude is essential for working with both imaginary numbers and complex exponentials, as it helps in visualizing complex numbers geometrically and plays a role in defining operations like multiplication and division in the complex plane.
Multiplication of complex numbers: Multiplication of complex numbers is an operation that combines two complex numbers to produce another complex number. This process involves multiplying the real and imaginary parts, following the rule that the square of the imaginary unit i is -1, which leads to unique interactions between these components. Understanding this multiplication is essential for working with complex numbers and is foundational for more advanced topics in analysis and applications.
Polar Form: Polar form is a way to express complex numbers using a magnitude (or modulus) and an angle (or argument), making it easier to perform multiplication and division of complex numbers. This representation connects deeply with the concepts of imaginary numbers and their geometric visualization in the complex plane, as well as with the mathematical operations involving complex exponents and logarithms. In polar form, a complex number is written as $$r( ext{cos} \theta + i\text{sin} \theta)$$ or simply as $$re^{i\theta}$$, where $$r$$ is the distance from the origin and $$\theta$$ is the angle measured from the positive real axis.
Purely Imaginary Numbers: Purely imaginary numbers are complex numbers that have a real part equal to zero and a non-zero imaginary part, expressed in the form of $$bi$$, where $$b$$ is a real number. These numbers play a crucial role in complex analysis, as they can be visualized on the complex plane along the imaginary axis, showing how they differ from real numbers. Understanding purely imaginary numbers is essential for grasping the broader concepts of imaginary and complex numbers.
Real part: The real part of a complex number is the component that lies along the horizontal axis of the complex plane, representing a value without an imaginary component. It plays a crucial role in understanding the structure of complex numbers, where a complex number can be expressed as a sum of its real and imaginary parts, typically written in the form $$a + bi$$, with $$a$$ being the real part and $$b$$ being the coefficient of the imaginary unit $$i$$.
Rectangular Form: Rectangular form is a way of representing complex numbers in the standard format of a + bi, where 'a' is the real part and 'bi' is the imaginary part. This representation allows for easy addition, subtraction, and visualization of complex numbers on the Cartesian plane, linking the real and imaginary components seamlessly. It is foundational in understanding complex numbers, as it establishes a direct connection to operations involving the imaginary unit i and serves as a basis for converting to other forms like polar representation.
X and y as real and imaginary parts: In the context of complex numbers, 'x' represents the real part and 'y' denotes the imaginary part, where a complex number is expressed in the form $$z = x + yi$$. This representation allows for a clear understanding of how complex numbers can be manipulated and visualized in the complex plane, where the x-axis corresponds to the real part and the y-axis corresponds to the imaginary part.
Z = x + iy: The expression z = x + iy represents a complex number, where 'z' is the complex number itself, 'x' is the real part, and 'iy' is the imaginary part, with 'i' being the imaginary unit. This notation captures the essence of complex numbers by combining real and imaginary components into a single entity that can be used in various mathematical contexts, particularly in complex analysis.
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