Arithmetic on Complex Numbers

Addition and Subtraction of Complex Numbers

Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.

When we add complex numbers, we can visualize the addition as a shift, or translation, of a point in the complex plane.

Visualize the addition [latex]3-4i[/latex] and [latex]-1+5i[/latex].

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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 [latex]1[/latex] units to the left, and we add [latex]5[/latex] to the imaginary part, moving the point [latex]5[/latex] units vertically. This shifts the point [latex]3-4i[/latex] to [latex]2+1i[/latex].

Multiplying Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

Multiplying a Complex Number by a Real Number

Let’s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial.

Multiplying Complex Numbers Together

Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Using either the distributive property or the FOIL method, we get

[latex]\left(a+bi\right)\left(c+di\right)=ac+adi+bci+bd{i}^{2}[/latex]

Because [latex]{i}^{2}=-1[/latex], we have

[latex]\left(a+bi\right)\left(c+di\right)=ac+adi+bci-bd[/latex]

To simplify, we combine the real parts, and we combine the imaginary parts.

[latex]\left(a+bi\right)\left(c+di\right)=\left(ac-bd\right)+\left(ad+bc\right)i[/latex]

When we multiply by a complex number, it’s like we’re doing two things at once: we’re changing the size (or scaling) and turning (or rotating) the number around the starting point, or origin. To really see what’s happening, in the following examples, we’ll use the complex plane to helps us visualize these changes.

Using the complex plane, visualize the product [latex]2(1+2i)[/latex].

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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 [latex]2[/latex]. Visually, this will stretch the point outwards, away from the origin.

Using the complex plane, 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.

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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 [latex]45°[/latex].