Arithmetic With Complex Numbers
Learning Outcomes
- Identify the difference between an imaginary number and a complex number
- Identify the real and imaginary parts of a complex number
- Plot a complex number on the complex plane
- Perform arithmetic operations on complex numbers
- Graph physical representations of arithmetic operations on complex numbers as scaling or rotation
Complex Numbers
[footnote]Portions of this section are remixed from Precalculus: An Investigation of Functions by David Lippman and Melonie Rasmussen. CC-BY-SA[/footnote] The numbers you are most familiar with are called real numbers. These include numbers like 4, 275, -200, 10.7, ½, π, and so forth. All these real numbers can be plotted on a number line. For example, if we wanted to show the number 3, we plot a point on the number line equidistant between 2 and 4. To solve certain problems like [latex]x^{2}=–4[/latex], it became necessary to introduce imaginary numbers.Recall Square roots
Recall that the square root of a number [latex]\sqrt{n}[/latex] is another way of asking the question what number when multiplied by itself results in the number [latex]n[/latex]? Ex. [latex]\sqrt{9}=3[/latex] because [latex]3 \ast 3 = 9[/latex]. It is also true that [latex](-3)\ast (-3) = 9[/latex] although we agreed that using the radical symbol requests only the principle root, the positive one. But, there is no number that when multiplied by itself results in a negative number. The property of integer multiplication states that both a negative number squared and a positive number squared result in a positive number. Indeed, you saw in the review section to this module that the square root of a negative number does not exist in the set of real numbers. Mathematicians realized the helpfulness of being to do calculations with such numbers though, so they assigned a value to [latex]\sqrt{-1}[/latex], calling it the imaginary unit [latex]i[/latex]. In this section, you'll use what you know about simplifying and doing operations on square roots and binomials to perform arithmetic on complex numbers (numbers that are the sum of a real and an imaginary number).Imaginary Number i
The imaginary number [latex]i[/latex] is defined to be [latex]i=\sqrt{-1}[/latex]. Any real multiple of [latex]i[/latex], like 5[latex]i[/latex], is also an imaginary number.Example
Simplify [latex]\sqrt{-9}[/latex]. We can separate [latex]\sqrt{-9}[/latex] as [latex]\sqrt{9}\sqrt{-1}[/latex]. We can take the square root of 9, and write the square root of [latex]-1[/latex] as i.[latex]\sqrt{-9}=\sqrt{9}\sqrt{-1}=3i[/latex]
The form of a complex number
The form of a complex number may look familiar to you. It looks a little like a binomial in one variable, doesn't it? Ex. [latex]a + bi[/latex] is a complex number. And [latex](a + bx)[/latex] is a binomial. You may be wondering about arithmetic on complex numbers -- does it behave similarly to arithmetic on binomial expressions? Good question! The answer is yes, it does.Complex Number
A complex number is a number [latex]z=a+bi[/latex], where- [latex]a[/latex] and [latex]b[/latex] are real numbers
- [latex]a[/latex] is the real part of the complex number
- [latex]b[/latex] is the imaginary part of the complex number
Complex Plane
In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.
Example
Plot the number [latex]3-4i[/latex] on the complex plane. The real part of this number is 3, and the imaginary part is [latex]-4[/latex]. To plot this, we draw a point 3 units to the right of the origin in the horizontal direction and 4 units down in the vertical direction.
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[ohm_question]65079[/ohm_question]Arithmetic on Complex Numbers
Before we dive into the more complicated uses of complex numbers, let’s make sure we remember the basic arithmetic involved. To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.Multiplying complex numbers
Multiplying complex numbers is algebraically similar to multiplying binomial expressions. Can you recognize the FOIL process in the example below?Example
Add [latex]3-4i[/latex] and [latex]2+5i[/latex]. Adding [latex](3-4i)+(2+5i)[/latex], we add the real parts and the imaginary parts.[latex]3+2-4i+5i[/latex]
[latex]5+i[/latex]
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Subtract [latex]2+5i[/latex] from [latex]3-4i[/latex].Answer: [latex](3-4i)-(2+5i)=1-9i[/latex]
[ohm_question]61710[/ohm_question]Example
Visualize the addition [latex]3-4i[/latex] and [latex]-1+5i[/latex]. The initial point is [latex]3-4i[/latex]. When we add [latex]-1+3i[/latex], we add [latex]-1[/latex] to the real part, moving the point 1 units to the left, and we add 5 to the imaginary part, moving the point 5 units vertically. This shifts the point [latex]3-4i[/latex] to [latex]2+1i[/latex].
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[ohm_question]13121[/ohm_question]Example
Multiply: [latex]4\left(2+5i\right)[/latex] To multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials. Distribute and simplify.[latex]4(2+5i)\\\,\,\,= 4\cdot2+4\cdot5i\\\,\,\,=8+20i[/latex]
Example
Multiply: [latex](2+5i)(4+i)[/latex].[latex]\begin{array}{l}\left(2+5i\right)\left(4+i\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Expand.}\\=8+20i+2i+5i^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Since }i=\sqrt{-1},i^{2}=-1\\=8+20i+2i+5\left(-1\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Simplify.}\\=3+22i\end{array}[/latex]
Try It
Multiply [latex]3-4i[/latex] and [latex]2+3i[/latex].Answer: Multiply [latex](3-4i)(2+3i)=6+9i-8i-12{{i}^{2}}=6+i-12(-1)=18+i[/latex]
[ohm_question]1418[/ohm_question]Example
Visualize the product [latex]2(1+2i)[/latex]. Multiplying we’d get[latex]\begin{align}&2\cdot1+2\cdot2i\\&=2+4i\\\end{align}[/latex]
Notice both the real and imaginary parts have been scaled by 2. Visually, this will stretch the point outwards, away from the origin.
Example
Visualize the product [latex]i\left(l+2i\right)[/latex]. Multiplying, we’d get[latex]\begin{align}&i\cdot1+i\cdot2i\\&=i+2{{i}^{2}}\\&=i+2(-1)\\&=-2+i\\\end{align}[/latex]
In this case, the distance from the origin has not changed, but the point has been rotated about the origin, 90° counter-clockwise.
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Multiply [latex]3-4i[/latex] and [latex]2+3i[/latex].Answer: Multiply [latex](3-4i)(2+3i)=6+9i-8i-12{{i}^{2}}=6+i-12(-1)=18+i[/latex]
[ohm_question]1418[/ohm_question]Example
Visualize the result of multiplying [latex]1+2i[/latex] by [latex]1+i[/latex]. Then show the result of multiplying by [latex]1+i[/latex] again. Multiplying [latex]1+2i[/latex] by [latex]1+i[/latex],[latex]-4+2i[/latex]
[latex]\begin{align}&(1+2i)(1+i)\\&=1+i+2i+2{{i}^{2}}\\&=1+3i+2(-1)\\&=-1+3i\\\end{align}[/latex]
Multiplying by [latex]1+i[/latex] again,[latex]\begin{align}&(-1+3i)(1+i)\\&=-1-i+3i+3{{i}^{2}}\\&=-1+2i+3(-1)\\&=-4+2i\\\end{align}[/latex]
If we multiplied by [latex]1+i[/latex] again, we’d get [latex]–6–2i[/latex]. Plotting these numbers in the complex plane, you may notice that each point gets both further from the origin, and rotates counterclockwise, in this case by 45°.
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[ohm_question]131223[/ohm_question]Licenses & Attributions
CC licensed content, Original
- Question ID 131223, 131218. Authored by: Lumen Learning. License: CC BY: Attribution.
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CC licensed content, Shared previously
- Complex arithmetic. Authored by: OCLPhase2. License: CC BY: Attribution.
- Visualizing complex arithmetic. Authored by: OCLPhase2. License: CC BY: Attribution.
- Question 1418. Authored by: WebWork Rochester. License: CC BY: Attribution.
