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Course: Algebra 2 (Eureka Math/EngageNY) > Unit 1

Lesson 15: Topic D: Lesson 37: Complex numbers intro
Math
Algebra 2 (Eureka Math/EngageNY)
Module 1: Polynomial, rational, and radical relationships
Topic D: Lesson 37: Complex numbers intro
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The complex plane

Google Classroom
Learn what the complex plane is and how it is used to represent complex numbers.
The Imaginary unit, or i, is the number with the following equivalent properties:
  • i2=−1
  • −1=i
A complex number is any number that can be written as a+bi, where i is the imaginary unit and a and b are real numbers.
a is called the real part of the number, and b is called the imaginary part of the number.

The complex plane

Just like we can use the number line to visualize the set of real numbers, we can use the complex plane to visualize the set of complex numbers.
A coordinate plane where the x-axis is labeled the real axis and the y-axis is labeled the imaginary-axis. Both axes are scaled by one.
The complex plane consists of two number lines that intersect in a right angle at the point (0,0).
The horizontal number line (what we know as the x-axis on a Cartesian plane) is the real axis.
The vertical number line (the y-axis on a Cartesian plane) is the imaginary axis.

Plotting a complex number

Every complex number can be represented by a point in the complex plane.
For example, consider the number 3−5i. This number, also expressed as 3+(−5)i, has a real part of 3 and an imaginary part of −5.
The location of this number on the complex plane is the point that corresponds to 3 on the real axis and −5 on the imaginary axis.
A coordinate plane where the x-axis is labeled the real axis and the y-axis is labeled the imaginary-axis. Both axes are scaled by one. A point is at (three, negative five i). A vertical dashed line extends from the three on the real axis to the point and a horizontal line extends from negative five on the imaginary-axis to the point.
So the number 3+(−5)i is associated with the point (3,−5). In general, the complex number a+bi corresponds to the point (a,b) in the complex plane.

Check your understanding

Problem 1
Plot the complex number −4+7i.

Problem 2
Plot the complex number 6i+1.

Problem 3
Plot the complex number −i−3.

Problem 4
Plot the complex number 4i.

Problem 5
Plot the complex number −7.

Connections to the real number line

In Pythagoras's days, the existence of irrational numbers was a surprising discovery! They wondered how something like 2 could exist without an accurate complete decimal expansion.
The real number line, however, helps rectify this dilemma. Why? Because 2 has a specific location on the real number line, showing that it is indeed a real number. (If you take the diagonal of a unit square and place one end on 0, the other end corresponds to the number 2.)
Likewise, every complex number does indeed exist because it corresponds to an exact location on the complex plane! Perhaps by being able to visualize these numbers, we can understand that calling these numbers "imaginary" was an unfortunate misnomer.
Complex numbers exist and are very much a part of mathematics. The real number line is simply the real axis on the complex plane, but there is so much beyond that single line!

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  • spunky sam blue style avatar for user g1y
    Posted 8 years ago. Direct link to g1y's post “Are complex numbers inclu...”
    Are complex numbers included in real numbers? If so, are they irrational?
    (14 votes)
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    • duskpin ultimate style avatar for user Hugo Barbosa
      Posted 4 years ago. Direct link to Hugo Barbosa's post “It is actually the other ...”
      Incredible Answer
      Great Answer
      Good Answer
      It is actually the other way around. The Real numbers are a subset of the set that contains all of Complex numbers, so are the Imaginary numbers. Imagine a big circle with 2 small circles inside it that don't intersect with each other, that would be the set of the Complex number (big circle) and the Real and Imaginary sets (small circles).
      (91 votes)
  • old spice man blue style avatar for user Muramalla Lasya Gayathri
    Posted 7 years ago. Direct link to Muramalla Lasya Gayathri's post “"If you take the diagonal...”
    "If you take the diagonal of a unit square and place one end on 0, the other end corresponds to the number √​2​​​." explain please !!
    (27 votes)
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    • leaf green style avatar for user kubleeka
      Posted 7 years ago. Direct link to kubleeka's post “A unit square is a square...”
      Incredible Answer
      Great Answer
      Good Answer
      A unit square is a square of side length 1. The diagonal of this square has length √2, which you can prove with Pythagorean theorem.

      Now look at the number line. If we take that diagonal of length √2 and place one end at 0, then the other end will fall a distance of √2 from 0. So it will fall on the point √2.
      (51 votes)
  • aqualine seed style avatar for user kyalopatrick24
    Posted 7 years ago. Direct link to kyalopatrick24's post “Would it be correct to sa...”
    Would it be correct to say that the number 0 is a real number, pure imaginary number and a complex number all at the same time?
    (10 votes)
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    • starky ultimate style avatar for user Ron Joniak
      Posted 7 years ago. Direct link to Ron Joniak's post “Logically, one could make...”
      Great Answer
      Good Answer
      Logically, one could make an argument that 0 is neither real nor
      imaginary, since is has neither an imaginary nor a real part. But
      excluding it from either the real or the imaginary axis would be
      extremely awkward; so we define "purely imaginary" in a negative way,
      not as a number that HAS only an imaginary part, but as one that DOES
      NOT have any (non-zero) real part:

      So, yes, 0 is a real number, pure imaginary number (0i) and a complex number (0i) all at the same time.
      (32 votes)
  • blobby green style avatar for user rylan.wetsell
    Posted a year ago. Direct link to rylan.wetsell's post “What does it mean by "√2,...”
    What does it mean by "√2, has a specific location on the real number line, showing that it is indeed a real number. (If you take the diagonal of a unit square and place one end on 0, the other end corresponds to the number √2.)"?
    (10 votes)
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  • blobby green style avatar for user nikhilsivajiss
    Posted 10 months ago. Direct link to nikhilsivajiss's post “Where can you apply the c...”
    Where can you apply the complex plane in real life?
    (11 votes)
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    • piceratops tree style avatar for user Pяιмє
      Posted a day ago. Direct link to Pяιмє's post “Great Question! Here's wh...”
      Great Question! Here's what I found:


      The complex plane has numerous applications in various fields, including mathematics, physics, engineering, and many others. Here are some real-life applications:

      Electrical Engineering: In electrical engineering, complex numbers are used to represent the amplitudes and phases of alternating current (AC) signals. Techniques like phasor analysis make extensive use of complex numbers to analyze circuits and systems.
      Control Systems: In control theory, complex numbers are used to analyze the behavior of dynamic systems. Techniques such as the Laplace transform and frequency domain analysis involve complex numbers extensively.
      Signal Processing: Complex numbers play a crucial role in signal processing, particularly in fields like digital communications, audio processing, and image processing. They are used to represent signals in both the time and frequency domains.
      Quantum Mechanics: In quantum mechanics, complex numbers are used to represent wavefunctions, which describe the behavior of particles at the quantum level. Complex analysis is essential for understanding the mathematical foundations of quantum mechanics.
      Fluid Dynamics: Complex numbers are used in fluid dynamics to analyze the behavior of fluid flow, particularly in problems involving potential flow and aerodynamics.
      Control Theory: Complex numbers are used to analyze and design control systems for various applications, including robotics, aerospace systems, and industrial automation.
      Mathematical Analysis: Complex analysis is a branch of mathematics that studies functions of complex numbers. It has applications in diverse areas such as number theory, geometry, and applied mathematics.
      Finance: Complex numbers are used in financial modeling to represent interest rates, exchange rates, and other financial quantities that exhibit periodic behavior.
      These are just a few examples, but the applications of the complex plane extend far beyond these areas. Essentially, any field that involves oscillatory behavior, waves, or periodic phenomena can benefit from the use of complex numbers and the complex plane.
      (2 votes)
  • aqualine seedling style avatar for user kitty-chan
    Posted 8 months ago. Direct link to kitty-chan's post “What is the complex plane...”
    What is the complex plane for?
    (4 votes)
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    • leaf green style avatar for user Tanner P
      Posted 8 months ago. Direct link to Tanner P's post “The complex plane is used...”
      Good Answer
      The complex plane is used to visualize complex numbers. You’ll learn later on that multiplying complex numbers can be thought of as rotations around the plane, similar to how negative numbers are like reflections on the number line. So, the complex plane helps visualize complex numbers and certain operations you can do with them.

      Hope this helps!
      (20 votes)
  • blobby green style avatar for user 2027nayakneal
    Posted a year ago. Direct link to 2027nayakneal's post “for something complex it ...”
    for something complex it isn’t as complex as i thought
    (9 votes)
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  • blobby green style avatar for user Arpan Agarwala
    Posted 10 months ago. Direct link to Arpan Agarwala 's post “Why do imaginary numbers ...”
    Why do imaginary numbers exist on the y axis?
    (6 votes)
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  • blobby green style avatar for user Naveen Giri
    Posted 7 years ago. Direct link to Naveen Giri's post “Can we use the why axis i...”
    Can we use the why axis instead of the x axis as the real number plane?
    (0 votes)
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    • mr pink red style avatar for user andrewp18
      Posted 7 years ago. Direct link to andrewp18's post “The real axis is by conve...”
      Good Answer
      The real axis is by convention the horizontal axis of the complex plane. There's nothing really stopping you from doing it the other way around, but that just complicates things unnecessarily and makes it harder to communicate your ideas with others.
      (23 votes)
  • blobby green style avatar for user Christian Miller
    Posted 5 years ago. Direct link to Christian Miller's post “So numbers are always com...”
    So numbers are always complex plus whether or not it has the imaginary or real part?
    (2 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      Posted 5 years ago. Direct link to Polina Vitić's post “Yes, you are right! All ...”
      Good Answer
      Yes, you are right!

      All complex numbers have a real part and an imaginary part. Because either part can be 0, all real numbers and all imaginary numbers are also complex numbers.

      Examples of real numbers and their complex forms:
      4 = 4 + 0i
      87 = 87 + 0i
      -5 = -5 + 0i
      √2 = √2 + 0i

      Examples of imaginary numbers and their complex forms:
      3i = 0 + 3i
      99i = 0 + 99i
      -12i = 0 - 12i
      √6i = 0 + √6i

      Hope this helps!
      (13 votes)