Complex numbers

Definition

Complex numbers are numbers that consist of a real part and an imaginary part, 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 $$i = \sqrt{-1}$$. They extend the concept of one-dimensional number lines to a two-dimensional plane, allowing for solutions to equations that have no real solutions. This two-dimensional nature is crucial for understanding their algebraic properties and operations.

5 Must Know Facts For Your Next Test

1. Complex numbers can be added, subtracted, multiplied, and divided using specific rules that extend the operations of real numbers.
2. The addition of complex numbers involves combining their real parts and their imaginary parts separately.
3. Multiplication of complex numbers requires the use of the distributive property and the fact that $$i^2 = -1$$.
4. Complex numbers can be represented graphically on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
5. The absolute value or modulus of a complex number $$z = a + bi$$ is given by $$|z| = \sqrt{a^2 + b^2}$$, representing its distance from the origin in the complex plane.

Review Questions

• How do you perform addition and subtraction of complex numbers, and what should you keep in mind when doing so?
  • When adding or subtracting complex numbers, you treat the real parts and imaginary parts separately. For example, if you have two complex numbers $$z_1 = a + bi$$ and $$z_2 = c + di$$, their sum is given by $$z_1 + z_2 = (a + c) + (b + d)i$$. It's important to ensure you are correctly combining both parts; if either part has a negative component, it should be correctly accounted for in your final answer.
• Explain how to multiply two complex numbers and highlight any unique aspects of this operation.
  • To multiply two complex numbers, you apply the distributive property as you would with binomials. For example, multiplying $$z_1 = a + bi$$ and $$z_2 = c + di$$ involves expanding it to get $$z_1 z_2 = ac + adi + bci + bdi^2$$. Remember that since $$i^2 = -1$$, you need to substitute this back into your result to combine like terms correctly. This highlights how multiplication involves not only combining parts but also applying the special property of the imaginary unit.
• Analyze how representing complex numbers on a graph enhances your understanding of their properties and operations.
  • Representing complex numbers on a graph allows you to visualize them in a two-dimensional space known as the complex plane. The horizontal axis represents the real part while the vertical axis represents the imaginary part. This visualization helps in understanding operations such as addition and multiplication geometrically; for instance, adding complex numbers corresponds to vector addition in this plane. It also aids in grasping concepts like modulus and argument, giving deeper insights into their behavior beyond mere algebraic manipulation.

Related terms

real numbers: Real numbers include all the rational and irrational numbers that can be found on the number line, excluding imaginary numbers.

imaginary unit: The imaginary unit, denoted as $$i$$, is defined as $$i = \sqrt{-1}$$ and is used to express the imaginary part of complex numbers.

complex conjugate: The complex conjugate of a complex number $$a + bi$$ is $$a - bi$$ and has important properties in complex arithmetic, particularly in division.