{"id":955,"date":"2025-07-17T18:23:59","date_gmt":"2025-07-17T18:23:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=955"},"modified":"2026-01-13T19:16:12","modified_gmt":"2026-01-13T19:16:12","slug":"complex-numbers-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/complex-numbers-learn-it-3\/","title":{"raw":"Complex Numbers: Learn It 3","rendered":"Complex Numbers: Learn It 3"},"content":{"raw":"

Arithmetic on Complex Numbers<\/h2>\r\n

Addition and Subtraction of Complex Numbers<\/h3>\r\nJust 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.\r\n\r\n
\r\n

addition and subtraction of complex numbers<\/h3>\r\nAdding complex numbers:\r\n

[latex]\\left(a+bi\\right)+\\left(c+di\\right)=\\left(a+c\\right)+\\left(b+d\\right)i[\/latex]<\/p>\r\nSubtracting complex numbers:\r\n

[latex]\\left(a+bi\\right)-\\left(c+di\\right)=\\left(a-c\\right)+\\left(b-d\\right)i[\/latex]<\/p>\r\n\r\n<\/section>

Add [latex]3-4i[\/latex] and [latex]2+5i[\/latex].\r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703380\"]\r\nAdding [latex](3-4i)+(2+5i)[\/latex], we add the like terms by combining the real parts and the imaginary parts.\r\n

[latex]3+2-4i+5i[\/latex]<\/p>\r\n

[latex]5+i[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

[ohm_question hide_question_numbers=1]318868[\/ohm_question]<\/section>When we add complex numbers, we can visualize the addition as a shift, or translation<\/strong>, of a point in the complex plane.\r\n\r\n
Visualize the addition [latex]3-4i[\/latex] and [latex]-1+5i[\/latex].\r\n[reveal-answer q=\"703381\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703381\"]\r\nThe 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].
\"A<\/center>[\/hidden-answer]<\/section>\r\n

Multiplying Complex Numbers<\/h3>\r\nMultiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.\r\n

Multiplying a Complex Number by a Real Number<\/h4>\r\nLet\u2019s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial.\r\n\r\n
So, for example,\r\n

[latex]\\begin{align}3(6+2i)&=(3\\cdot6)+(3\\cdot2i)&&\\text{Distribute.}\\\\&=18+6i&&\\text{Simplify.}\\end{align}[\/latex]<\/p>\r\n\r\n<\/section>

[ohm_question hide_question_numbers=1]317770[\/ohm_question]<\/section>\r\n

Multiplying Complex Numbers Together<\/h4>\r\nNow, let\u2019s 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\r\n

[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci+bd{i}^{2}[\/latex]<\/p>\r\nBecause [latex]{i}^{2}=-1[\/latex], we have\r\n

[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci-bd[\/latex]<\/p>\r\nTo simplify, we combine the real parts, and we combine the imaginary parts.\r\n

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

Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together.<\/section>
Multiply: [latex](2+5i)(4+i)[\/latex].\r\n[reveal-answer q=\"703385\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703385\"]\r\n

[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]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

[ohm_question hide_question_numbers=1]318869[\/ohm_question]<\/section>
[ohm_question hide_question_numbers=1]317772[\/ohm_question]<\/section><\/section>When we multiply by a complex number, it's like we're doing two things at once: we're changing the size (or scaling<\/strong>) and turning (or rotating<\/strong>) 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.\r\n\r\n
Using the complex plane, visualize the product [latex]2(1+2i)[\/latex].\r\n[reveal-answer q=\"703386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703386\"]\r\nMultiplying we\u2019d get\r\n

[latex]\\begin{align}&2\\cdot1+2\\cdot2i\\\\&=2+4i\\\\\\end{align}[\/latex]<\/p>\r\nNotice both the real and imaginary parts have been scaled by [latex]2[\/latex]. Visually, this will stretch the point outwards, away from the origin.\r\n\r\n

\"Graph<\/center>[\/hidden-answer]\r\n\r\n<\/section>
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.\r\n[reveal-answer q=\"703387\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703387\"]\r\nMultiplying [latex]1+2i[\/latex] by [latex]1+i[\/latex],\r\n

[latex]-4+2i[\/latex]<\/p>\r\n

[latex]\\begin{align}&(1+2i)(1+i)\\\\&=1+i+2i+2{{i}^{2}}\\\\&=1+3i+2(-1)\\\\&=-1+3i\\\\\\end{align}[\/latex]<\/p>\r\nMultiplying by [latex]1+i[\/latex] again,\r\n

[latex]\\begin{align}&(-1+3i)(1+i)\\\\&=-1-i+3i+3{{i}^{2}}\\\\&=-1+2i+3(-1)\\\\&=-4+2i\\\\\\end{align}[\/latex]<\/p>\r\nIf we multiplied by [latex]1+i[\/latex] again, we\u2019d get [latex]\u20136\u20132i[\/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\u00b0[\/latex].\r\n\r\n

\"The<\/center>[\/hidden-answer]\r\n\r\n<\/section>
\r\n
\r\n
\r\n

Simplifying Powers of i<\/em><\/h2>\r\n

The powers of i<\/em>\u00a0are cyclic. Let\u2019s look at what happens when we raise i<\/em>\u00a0to increasing powers.<\/p>\r\n\r\n

[latex]\\begin{align}&{i}^{1}=i\\\\ &{i}^{2}=-1\\\\ &{i}^{3}={i}^{2}\\cdot i=-1\\cdot i=-i\\\\ &{i}^{4}={i}^{3}\\cdot i=-i\\cdot i=-{i}^{2}=-\\left(-1\\right)=1\\\\ &{i}^{5}={i}^{4}\\cdot i=1\\cdot i=i\\end{align}[\/latex]<\/div>\r\n

We can see that when we get to the fifth power of i<\/em>, it is equal to the first power. As we continue to multiply i<\/em>\u00a0by itself for increasing powers, we will see a cycle of 4. Let\u2019s examine the next 4 powers of i<\/em>.<\/p>\r\n\r\n

[latex]\\begin{align}&{i}^{6}={i}^{5}\\cdot i=i\\cdot i={i}^{2}=-1\\\\ &{i}^{7}={i}^{6}\\cdot i={i}^{2}\\cdot i={i}^{3}=-i\\\\ &{i}^{8}={i}^{7}\\cdot i={i}^{3}\\cdot i={i}^{4}=1\\\\ &{i}^{9}={i}^{8}\\cdot i={i}^{4}\\cdot i={i}^{5}=i\\end{align}[\/latex]<\/div>\r\n
\r\n
\r\n

Evaluate [latex]{i}^{35}[\/latex].<\/p>\r\n[reveal-answer q=\"604528\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604528\"]\r\n

Since [latex]{i}^{4}=1[\/latex], we can simplify the problem by factoring out as many factors of [latex]{i}^{4}[\/latex] as possible. To do so, first determine how many times 4 goes into 35: [latex]35=4\\cdot 8+3[\/latex].<\/p>\r\n

[latex]{i}^{35}={i}^{4\\cdot 8+3}={i}^{4\\cdot 8}\\cdot {i}^{3}={\\left({i}^{4}\\right)}^{8}\\cdot {i}^{3}={1}^{8}\\cdot {i}^{3}={i}^{3}=-i[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/section>","rendered":"

Arithmetic on Complex Numbers<\/h2>\n

Addition and Subtraction of Complex Numbers<\/h3>\n

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.<\/p>\n

\n

addition and subtraction of complex numbers<\/h3>\n

Adding complex numbers:<\/p>\n

[latex]\\left(a+bi\\right)+\\left(c+di\\right)=\\left(a+c\\right)+\\left(b+d\\right)i[\/latex]<\/p>\n

Subtracting complex numbers:<\/p>\n

[latex]\\left(a+bi\\right)-\\left(c+di\\right)=\\left(a-c\\right)+\\left(b-d\\right)i[\/latex]<\/p>\n<\/section>\n

Add [latex]3-4i[\/latex] and [latex]2+5i[\/latex].<\/p>\n