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I’ll show you. It starts with the Roots of Unity. ## What is unity? Unity just means the number 1. Let’s start with the square roots of 1. *x*² = 1 Or, what number multiplied by itself gives you a result of 1? The answers are straightforward: *x* = 1, -1 ## Cube roots of unity Sometimes interesting answers lie behind deceptively simple questions. Let’s ask a similar question. What are cube roots of 1? Or what number multiplied by itself three times yields 1? *x³ =* 1 Well, 1 works for sure, but -1 does not work anymore (-1 x -1 x -1 = -1) If you remember some math you might think it’s *i —* the square root of negative one. *i* x *i* x *i* = -*i.* That’s not 1. What about -*i*? -*i* x -*i* x -*i* = -*i.* Not 1 either. How to find the other roots? Algebra of course\! ## Algebra to the rescue Rewrite *x³* \= 1to *x³-*1³ = 0 and solve for *x*. We know *x*\=1 is a root already so: (*x* -1)(*x*² + *x* + 1) = 0 To solve the second term, use the quadratic formula. Yes, the one you were forced to memorize in high school. ![]() Quadratic formula Here, a, b, c are just 1 and after substituting in: ![]() Which reduces to: ![]() What strange numbers are these? How did we go from such a simple question on to the cube roots of 1 to numbers involving fractions, radicals, and imaginary numbers? Sometimes by shifting your frame of reference, things become clearer. We’ll take that shift now to the complex plane for the fourth roots of 1. ## The fourth roots of unity on the complex plane Take x⁴ = 1. We know 1 is a root. When we plot 1 on the complex plane, it looks like this: ![]() 1 on the complex plane We can also write 1 in complex notation as 1 + 0*i*. 0*i* means there’s no imaginary component. How to raise a number to a power in the complex plane? A key insight is that raising to a power in the complex plane involves rotation. Since we’re talking about the fourth root, that’s four rotations. Here are the other three: ![]() *i* on the complex plane ![]() \-1 on the complex plane ![]() \-i on the complex plane Four times around the origin in the complex plane is how you get the four roots of 1. If you look at the labels on the graphs, you already know the 4 roots: { 1, *i*, -1 , -*i* } ## Another representation of -1 Let’s study this one a bit more: Press enter or click to view image in full size ![]() We know this number as -1. In the complex plane it can be written as -1 + 0*i*. Another way to write complex numbers is the following formula: a + b*i* = r *cos* (θ) + *i* *sin* (θ) Anywhere sine and cosine come into play there must be a circle involved. And we already see that with the rotation around the unit circle. Before you tap out, let’s simplify a bit. In this case we’re using the unit circle so *r* = 1.What is the angle θ? It’s on the graph — 180°. Another way to write that is π. You may remember the formula for the circumference of a circle is 2π*r* and with *r=*1 that simplifies to 2π*.*We’ve only gone halfway around the circle to get to -1, so we’ve gone a distance of π. With all this in mind, let’s rewrite it: \-1 + 0*i* \= 1 *cos* π+ *i* *sin* π Simplifying to: \-1 = *cos* π+ *i* *sin* π Let’s appreciate this for a second. A formula with sine, cosine, *i*, and π equals the humble integer -1. Cool. ## Enter Euler — the home stretch There is a formula we can use here for a neat transformation. It is called Euler’s formula after the famous 18th century mathematician Leonhard Euler: ![]() Euler’s formula We already have something that matches the right-hand side. In our case *x* = π. Let’s substitute that in. ![]() We’re almost done. We already know the right-hand side is -1. Therefore: ![]() Which can be re-written as: ![]() Euler’s identity This is also known as Euler’s identity. It is widely considered the most beautiful equation in math. It relates five of the most fundamental numbers in mathematics: 0, 1, *e*, *i*, π ## Connection to Geometry There is a fascinating connection to geometry in the Roots of Unity. If you plot the roots on the complex plane, each root draws a polygon. The cube roots trace a triangle, the fourth roots a square, the fifth roots a pentagon, and so on. Press enter or click to view image in full size ![]() Roots of unity plotted on the complex plane ## Namaste Who could have thought something as simple as the roots of the number one could reveal all this mathematic depth? Imaginary numbers, *e*, π, sine, cosine, polygons, symmetry — all revealed by the Roots of Unity. I hope you found the universe a tiny bit more fascinating today. ## More Reading [Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the equality where Euler's identity is named after… en.wikipedia.org](https://en.wikipedia.org/wiki/Euler%27s_identity?source=post_page-----8d431802bbd4---------------------------------------) [Numberphile Videos about numbers - it's that simple. Videos by Brady Haran www.youtube.com](https://www.youtube.com/user/numberphile/?source=post_page-----8d431802bbd4---------------------------------------) [Mathologer Enter the world of the Mathologer for really accessible explanations of hard and beautiful math(s). In real life the… www.youtube.com](https://www.youtube.com/c/Mathologer/?source=post_page-----8d431802bbd4---------------------------------------) [blackpenredpen I am a teacher. I do math for fun. I miss Kobe. www.youtube.com](https://www.youtube.com/c/blackpenredpen?source=post_page-----8d431802bbd4---------------------------------------) [3Blue1Brown 3Blue1Brown, by Grant Sanderson, is some combination of math and entertainment, depending on your disposition. The goal… www.youtube.com](https://www.youtube.com/c/3blue1brown/?source=post_page-----8d431802bbd4---------------------------------------) [Michael Penn Math videos covering Calculus, Differential Equations, Number Theory and more www.youtube.com](https://www.youtube.com/c/MichaelPennMath/?source=post_page-----8d431802bbd4---------------------------------------) [Math](https://medium.com/tag/math?source=post_page-----8d431802bbd4---------------------------------------) [Mathematics](https://medium.com/tag/mathematics?source=post_page-----8d431802bbd4---------------------------------------) [Algebra](https://medium.com/tag/algebra?source=post_page-----8d431802bbd4---------------------------------------) [STEM](https://medium.com/tag/stem?source=post_page-----8d431802bbd4---------------------------------------) [Science](https://medium.com/tag/science?source=post_page-----8d431802bbd4---------------------------------------) \-- \-- [](https://medium.com/@johnlatwc?source=post_page---post_author_info--8d431802bbd4---------------------------------------) [](https://medium.com/@johnlatwc?source=post_page---post_author_info--8d431802bbd4---------------------------------------) [Written by John Lambert](https://medium.com/@johnlatwc?source=post_page---post_author_info--8d431802bbd4---------------------------------------) [572 followers](https://medium.com/@johnlatwc/followers?source=post_page---post_author_info--8d431802bbd4---------------------------------------) ·[9 following](https://medium.com/@johnlatwc/following?source=post_page---post_author_info--8d431802bbd4---------------------------------------) Distinguished Engineer, Microsoft Threat Intelligence Center, @JohnLaTwC ## No responses yet [Help](https://help.medium.com/hc/en-us?source=post_page-----8d431802bbd4---------------------------------------) [Status](https://status.medium.com/?source=post_page-----8d431802bbd4---------------------------------------) [About](https://medium.com/about?autoplay=1&source=post_page-----8d431802bbd4---------------------------------------) [Careers](https://medium.com/jobs-at-medium/work-at-medium-959d1a85284e?source=post_page-----8d431802bbd4---------------------------------------) [Press](mailto:pressinquiries@medium.com) [Blog](https://blog.medium.com/?source=post_page-----8d431802bbd4---------------------------------------) [Privacy](https://policy.medium.com/medium-privacy-policy-f03bf92035c9?source=post_page-----8d431802bbd4---------------------------------------) [Rules](https://policy.medium.com/medium-rules-30e5502c4eb4?source=post_page-----8d431802bbd4---------------------------------------) [Terms](https://policy.medium.com/medium-terms-of-service-9db0094a1e0f?source=post_page-----8d431802bbd4---------------------------------------) [Text to speech](https://speechify.com/medium?source=post_page-----8d431802bbd4---------------------------------------)

[](https://medium.com/@johnlatwc?source=post_page---byline--8d431802bbd4---------------------------------------) 5 min read Sep 16, 2020 \-- Want to see the most beautiful equation in math? I’ll show you. It starts with the Roots of Unity. ## What is unity? Unity just means the number 1. Let’s start with the square roots of 1. *x*² = 1 Or, what number multiplied by itself gives you a result of 1? The answers are straightforward: *x* = 1, -1 ## Cube roots of unity Sometimes interesting answers lie behind deceptively simple questions. Let’s ask a similar question. What are cube roots of 1? Or what number multiplied by itself three times yields 1? *x³ =* 1 Well, 1 works for sure, but -1 does not work anymore (-1 x -1 x -1 = -1) If you remember some math you might think it’s *i —* the square root of negative one. *i* x *i* x *i* = -*i.* That’s not 1. What about -*i*? -*i* x -*i* x -*i* = -*i.* Not 1 either. How to find the other roots? Algebra of course\! ## Algebra to the rescue Rewrite *x³* \= 1to *x³-*1³ = 0 and solve for *x*. We know *x*\=1 is a root already so: (*x* -1)(*x*² + *x* + 1) = 0 To solve the second term, use the quadratic formula. Yes, the one you were forced to memorize in high school. Quadratic formula Here, a, b, c are just 1 and after substituting in: Which reduces to: What strange numbers are these? How did we go from such a simple question on to the cube roots of 1 to numbers involving fractions, radicals, and imaginary numbers? Sometimes by shifting your frame of reference, things become clearer. We’ll take that shift now to the complex plane for the fourth roots of 1. ## The fourth roots of unity on the complex plane Take x⁴ = 1. We know 1 is a root. When we plot 1 on the complex plane, it looks like this: 1 on the complex plane We can also write 1 in complex notation as 1 + 0*i*. 0*i* means there’s no imaginary component. How to raise a number to a power in the complex plane? A key insight is that raising to a power in the complex plane involves rotation. Since we’re talking about the fourth root, that’s four rotations. Here are the other three: *i* on the complex plane \-1 on the complex plane \-i on the complex plane Four times around the origin in the complex plane is how you get the four roots of 1. If you look at the labels on the graphs, you already know the 4 roots: { 1, *i*, -1 , -*i* } ## Another representation of -1 Let’s study this one a bit more: Press enter or click to view image in full size We know this number as -1. In the complex plane it can be written as -1 + 0*i*. Another way to write complex numbers is the following formula: a + b*i* = r *cos* (θ) + *i* *sin* (θ) Anywhere sine and cosine come into play there must be a circle involved. And we already see that with the rotation around the unit circle. Before you tap out, let’s simplify a bit. In this case we’re using the unit circle so *r* = 1.What is the angle θ? It’s on the graph — 180°. Another way to write that is π. You may remember the formula for the circumference of a circle is 2π*r* and with *r=*1 that simplifies to 2π*.*We’ve only gone halfway around the circle to get to -1, so we’ve gone a distance of π. With all this in mind, let’s rewrite it: \-1 + 0*i* \= 1 *cos* π+ *i* *sin* π Simplifying to: \-1 = *cos* π+ *i* *sin* π Let’s appreciate this for a second. A formula with sine, cosine, *i*, and π equals the humble integer -1. Cool. ## Enter Euler — the home stretch There is a formula we can use here for a neat transformation. It is called Euler’s formula after the famous 18th century mathematician Leonhard Euler: Euler’s formula We already have something that matches the right-hand side. In our case *x* = π. Let’s substitute that in. We’re almost done. We already know the right-hand side is -1. Therefore: Which can be re-written as: Euler’s identity This is also known as Euler’s identity. It is widely considered the most beautiful equation in math. It relates five of the most fundamental numbers in mathematics: 0, 1, *e*, *i*, π ## Connection to Geometry There is a fascinating connection to geometry in the Roots of Unity. If you plot the roots on the complex plane, each root draws a polygon. The cube roots trace a triangle, the fourth roots a square, the fifth roots a pentagon, and so on. Press enter or click to view image in full size Roots of unity plotted on the complex plane ## Namaste Who could have thought something as simple as the roots of the number one could reveal all this mathematic depth? Imaginary numbers, *e*, π, sine, cosine, polygons, symmetry — all revealed by the Roots of Unity. I hope you found the universe a tiny bit more fascinating today. ## More Reading