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| Meta Title | Eulerās Identity: 'The Most Beautiful Equation' | Live Science |
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| Boilerpipe Text | Euler's Equation
(Image credit: public domain)
Eulerās identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as "
the most beautiful equation
." It is a special case of a foundational equation in complex arithmetic called Eulerās Formula, which the late great physicist Richard Feynman calledĀ
in his lectures
Ā "our jewel" and "the most remarkable formula in mathematics."Ā
In an
interview with the BBC
, Prof David Percy of the Institute of Mathematics and its Applications said Euler's Identity was āa real classic and you can do no better than that ā¦ It is simple to look at and yet incredibly profound, it comprises the five most important mathematical constants.ā
Euler's Identity is written simply as:Ā
e
iĻ
Ā + 1 = 0
Article continues below
Latest Videos From Live Science
The five constants are:
TheĀ
number 0
.
The number 1.
TheĀ
numberĀ
Ļ
, an irrational number (with unending digits) that is the ratio of the circumference of a circle to its diameter. It is approximately 3.14159ā¦
The numberĀ
e
, also an irrational number. It is the base ofĀ
natural logarithms
Ā that arises naturally through study of compound interest andĀ
calculus
. The numberĀ
e
Ā pervades math, appearing seemingly from nowhere in a vast number of important equations. It is approximately 2.71828ā¦.
TheĀ
numberĀ
i
, defined as the square root of negative one: ā(-1). The most fundamental of the imaginary numbers, so called because, in reality, no number can be multiplied by itself to produce a negative number (and, therefore, negative numbers have no real square roots). But in math, there are many situations where one is forced to take the square root of a negative. The letterĀ
i
Ā is therefore used as a sort of stand-in to mark places where this was done.
Prolific mathematician
Leonhard Euler was an 18th-century Swiss-born mathematician who developed many concepts that are integral to modern mathematics. He spent most of his career in St. Petersburg, Russia. He was one of the most prolific mathematicians of all time, according to theĀ
U.S. Naval Academy
Ā (USNA), with 886 papers and books published. Much of his output came during the last two decades of his life, when he was totally blind. There was so much work that the St. Petersburg Academy continued publishing his work posthumously for more than 30 years.
Euler's important contributions include Euler's Formula and Euler's Theorem, both of which can mean different things depending on the context. According to the USNA, in mechanics, there are "Euler angles (to specify the orientation of a rigid body), Euler's theorem (that every rotation has an axis), Euler's equations for motion of fluids, and the Euler-Lagrange equation (that comes from calculus of variations)."
Multiplying complex numbers
Eulerās Identity stems naturally from interactions ofĀ
complex numbers
Ā which are numbers composed of two pieces: aĀ
real number
Ā and anĀ
imaginary number
; an example is 4+3
i
. Complex numbers appear in a multitude of applications such as wave mechanics (a study withinĀ
quantum mechanics
) and design of circuits that use alternating current (a common practice inĀ
electrical engineering
). Additionally, complex numbers (and their cousins, theĀ
hyper complex numbers
) have a property that makes them especially useful for studying computer graphics, robotics, navigation, flight dynamics, and orbital mechanics: multiplying them together causes them to rotate. This property will help us understand the reasoning behind Eulerās Identity.
Get the worldās most fascinating discoveries delivered straight to your inbox.
In the example below, five complex numbers are plotted on theĀ
complex plane
Ā and together form a āhouse shape.ā The complex plane is similar to a number line, except that itās two-dimensional. The horizontal direction represents the real numbers and the vertical axis represents imaginary numbers. Each house-shape complex number is multiplied by the complex number 4+3
i
Ā and re-plotted (green arrow). [
Related: What Are Complex Numbers?
]
As can be seen, multiplying by 4+3
i
Ā results in the house shapeĀ
dilating
Ā (increasing in area and moving away from the origin 0+0
i
Ā by the same amount) andĀ
rotating
Ā (becoming tilted by some angle). To show this is precisely the effect of multiplying by 4+3i, the effect of zooming in on the house five times and rotating by 36.9 degrees is also shown (red arrow). The exact same effect is produced.
The same effect is produced from multiplying the vertices of a figure by 4+3i and rotating the figure by 36.9 degrees and dilating it by a factor of five.
(Image credit: Robert J. Coolman)
Different amounts of dilation and rotation can produce the effects of multiplying by any number on the complex plane.
Polar form of complex numbers
The amount of rotation and dilation is determined by properties intrinsic to the number 4+3
i,
Ā which, as seen in the figure below, is five units from the origin (
r
Ā = 5) and forms an angle of 36.9 degrees with the horizontal axis (
Ļ
Ā = 36.9Ā°). These measurements are used in what is known as theĀ
polar form
Ā of a complex number (
re
iĻ
) as opposed to the normalĀ
rectangular form
Ā (
a
+
bi
).
The number 4+3i is five units from the origin and forms an angle of 36.9 degrees with the horizontal axis.
(Image credit: Robert J. Coolman)
The polar form requires thatĀ
Ļ
Ā be measured inĀ
radians
. One radian (1
rad
) is approximately 57.3 degrees; itās the measure of angle made when a circleās radius is wrapped against that circleās circumference. A measure ofĀ
Ļ
Ā radians
Ā wraps half way around a circle; a measure of 2
Ļ
Ā radians wraps a full circle.Ā
An angle measure of one radian is formed when a circleās radius is wrapped against its circumference. A half-circle is Ļ radians and a full circle is 2Ļ radians.
(Image credit: Robert J. Coolman)
The angle measure for 4+3
i
Ā is 0.644 radians (36.9Ā° = 0.644
rad
) meaning the polar form of 4+3
i
is 5
e
i
0.644
. Measures forĀ
r
Ā andĀ
Ļ
Ā can also be determined for each of the house-shape points, and yet another way of achieving the dilating/rotating effect of multiplying by 4+3
i
Ā is to multiply eachĀ
r
Ā by five, and add 36.9 degrees (or 0.644
rad
) to eachĀ
Ļ
. From this demonstration, we see that when complex numbers are multiplied together, distances multiply and angles add. This is due to a property intrinsic to exponents, which can be shown algebraically.
Using the polar form of complex numbers to show why distances multiply and angles add.
(Image credit: Robert J. Coolman)
With the polar form of complex numbers established, the matter of Eulerās Identity is merely a special case ofĀ
a
+
bi
Ā forĀ
aĀ
= -1 andĀ
bĀ
= 0. Consequently for the polar formĀ
re
iĻ
, this makesĀ
r
= 1 andĀ
Ļ
Ā =Ā
Ļ
Ā (sinceĀ
Ļ
rad
Ā = 180Ā°).
Eulerās Identity is a special case of a+bi for a = -1 and b = 0 and reiĻ for r = 1 and Ļ = Ļ.
(Image credit: Robert J. Coolman)
Derivation of polar form
Though Eulerās Identity follows from the polar form of complex numbers, it is impossible to derive the polar form (in particular the spontaneous appearance of the numberĀ
e
) withoutĀ
calculus
.
A general case of a complex number in both rectangular (a+bi) and polar (reiĻ) forms.
(Image credit: Robert J. Coolman)
We start with the rectangular form of a complex number:
aĀ
+
Ā bi
From the diagram andĀ
trigonometry
, we can make the following substitutions:
(
r
Ā·cos
Ļ
) + (
r
Ā·sin
Ļ
)
i
From here we can factor outĀ
r
:
r
Ā·(cos
Ļ
Ā +Ā
i
Ā·sin
Ļ
)
Sometimes ācos
Ļ
Ā +Ā
i
Ā·sin
Ļ
ā is named cis
Ļ
, which is shorthand for ā
c
osine plusĀ
i
maginaryĀ
s
ine
.ā
r
Ā·cis
Ļ
The function cis
Ļ
Ā turns out to be equal toĀ
e
iĻ
. This is the part thatās impossible to show without calculus. Two derivations are shown below:
Two derivations for of cisĻ = eiĻ. Both use some form of calculus.
(Image credit: Robert J. Coolman)
Thus, the equationĀ
r
Ā·cis
Ļ
Ā is written in standard polar formĀ
r
Ā·e
iĻ
.
Additional resources
ResearchGate: What is Special in Euler's Identity?
Academia.edu: Euler's Identity ā A Mathematical Proof for the Existence of God
, by Robin Robertson
Science4All: The Most Beautiful Equation of Math: Euler's Identity
Robert Coolman, PhD, is a teacher and a freelance science writer and is based in Madison, Wisconsin. He has written for Vice, Discover, Nautilus, Live Science and The Daily Beast. Robert spent his doctorate turning sawdust into gasoline-range fuels and chemicals for materials, medicine, electronics and agriculture. He is made of chemicals. |
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# Eulerās Identity: 'The Most Beautiful Equation'
[References](https://www.livescience.com/references)
By [Robert Coolman](https://www.livescience.com/author/robert-coolman)
published
July 1, 2015
When you purchase through links on our site, we may earn an affiliate commission. [Hereās how it works](https://www.livescience.com/about-live-science#section-affiliate-advertising-disclosure).
Euler's Equation (Image credit: public domain)
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Eulerās identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as "[the most beautiful equation](https://www.livescience.com/26584-5-mind-boggling-math-facts.html)." It is a special case of a foundational equation in complex arithmetic called Eulerās Formula, which the late great physicist Richard Feynman called [in his lectures](https://target.georiot.com/Proxy.ashx?tsid=74387&GR_URL=http%3A%2F%2Famazon.com%2FFeynman-Lectures-Physics-Vol-Millennium%2Fdp%2F0465024939%3Ftag%3Dhawk-future-20%26ascsubtag%3Dlivescience-us-5530728407723048703-20) "our jewel" and "the most remarkable formula in mathematics."
In an [interview with the BBC](http://www.bbc.com/news/science-environment-26151062), Prof David Percy of the Institute of Mathematics and its Applications said Euler's Identity was āa real classic and you can do no better than that ā¦ It is simple to look at and yet incredibly profound, it comprises the five most important mathematical constants.ā
Euler's Identity is written simply as: *e**iĻ* + 1 = 0
Article continues below
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The five constants are:
- The [number 0](https://www.livescience.com/27853-who-invented-zero.html).
- The number 1.
- The [number *Ļ*](https://www.livescience.com/29197-what-is-pi.html), an irrational number (with unending digits) that is the ratio of the circumference of a circle to its diameter. It is approximately 3.14159ā¦
- The number *e*, also an irrational number. It is the base of [natural logarithms](https://www.livescience.com/50940-logarithms.html) that arises naturally through study of compound interest and [calculus](https://www.livescience.com/50777-calculus.html). The number *e* pervades math, appearing seemingly from nowhere in a vast number of important equations. It is approximately 2.71828ā¦.
- The [number *i*](https://www.livescience.com/42748-imaginary-numbers.html), defined as the square root of negative one: ā(-1). The most fundamental of the imaginary numbers, so called because, in reality, no number can be multiplied by itself to produce a negative number (and, therefore, negative numbers have no real square roots). But in math, there are many situations where one is forced to take the square root of a negative. The letter *i* is therefore used as a sort of stand-in to mark places where this was done.
## Prolific mathematician
Leonhard Euler was an 18th-century Swiss-born mathematician who developed many concepts that are integral to modern mathematics. He spent most of his career in St. Petersburg, Russia. He was one of the most prolific mathematicians of all time, according to the [U.S. Naval Academy](http://www.usna.edu/Users/math/meh/euler.html) (USNA), with 886 papers and books published. Much of his output came during the last two decades of his life, when he was totally blind. There was so much work that the St. Petersburg Academy continued publishing his work posthumously for more than 30 years.
Euler's important contributions include Euler's Formula and Euler's Theorem, both of which can mean different things depending on the context. According to the USNA, in mechanics, there are "Euler angles (to specify the orientation of a rigid body), Euler's theorem (that every rotation has an axis), Euler's equations for motion of fluids, and the Euler-Lagrange equation (that comes from calculus of variations)."
## Multiplying complex numbers
Eulerās Identity stems naturally from interactions of [complex numbers](https://www.livescience.com/42966-complex-numbers.html) which are numbers composed of two pieces: a [real number](https://www.livescience.com/42619-real-numbers.html) and an [imaginary number](https://www.livescience.com/42748-imaginary-numbers.html); an example is 4+3*i*. Complex numbers appear in a multitude of applications such as wave mechanics (a study within [quantum mechanics](https://www.livescience.com/33816-quantum-mechanics-explanation.html)) and design of circuits that use alternating current (a common practice in [electrical engineering](https://www.livescience.com/47571-electrical-engineering.html)). Additionally, complex numbers (and their cousins, the [hyper complex numbers](http://mathworld.wolfram.com/HypercomplexNumber.html)) have a property that makes them especially useful for studying computer graphics, robotics, navigation, flight dynamics, and orbital mechanics: multiplying them together causes them to rotate. This property will help us understand the reasoning behind Eulerās Identity.
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In the example below, five complex numbers are plotted on the **complex plane** and together form a āhouse shape.ā The complex plane is similar to a number line, except that itās two-dimensional. The horizontal direction represents the real numbers and the vertical axis represents imaginary numbers. Each house-shape complex number is multiplied by the complex number 4+3*i* and re-plotted (green arrow). \[[Related: What Are Complex Numbers?](https://www.livescience.com/42966-complex-numbers.html)\]
As can be seen, multiplying by 4+3*i* results in the house shape **dilating** (increasing in area and moving away from the origin 0+0*i* by the same amount) and **rotating** (becoming tilted by some angle). To show this is precisely the effect of multiplying by 4+3i, the effect of zooming in on the house five times and rotating by 36.9 degrees is also shown (red arrow). The exact same effect is produced.
The same effect is produced from multiplying the vertices of a figure by 4+3i and rotating the figure by 36.9 degrees and dilating it by a factor of five. (Image credit: Robert J. Coolman)
Different amounts of dilation and rotation can produce the effects of multiplying by any number on the complex plane.
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## Polar form of complex numbers
The amount of rotation and dilation is determined by properties intrinsic to the number 4+3*i,* which, as seen in the figure below, is five units from the origin (*r* = 5) and forms an angle of 36.9 degrees with the horizontal axis (*Ļ* = 36.9Ā°). These measurements are used in what is known as the [**polar form**](http://mathworld.wolfram.com/PolarCoordinates.html) of a complex number (*re**iĻ*) as opposed to the normal **rectangular form** (*a*\+*bi*).
The number 4+3i is five units from the origin and forms an angle of 36.9 degrees with the horizontal axis. (Image credit: Robert J. Coolman)
The polar form requires that *Ļ* be measured in **radians**. One radian (1rad) is approximately 57.3 degrees; itās the measure of angle made when a circleās radius is wrapped against that circleās circumference. A measure of [*Ļ* radians](https://www.livescience.com/29197-what-is-pi.html) wraps half way around a circle; a measure of 2*Ļ* radians wraps a full circle.
An angle measure of one radian is formed when a circleās radius is wrapped against its circumference. A half-circle is Ļ radians and a full circle is 2Ļ radians. (Image credit: Robert J. Coolman)
The angle measure for 4+3*i* is 0.644 radians (36.9Ā° = 0.644rad) meaning the polar form of 4+3*i* is 5*e**i*0\.644. Measures for *r* and *Ļ* can also be determined for each of the house-shape points, and yet another way of achieving the dilating/rotating effect of multiplying by 4+3*i* is to multiply each *r* by five, and add 36.9 degrees (or 0.644rad) to each *Ļ*. From this demonstration, we see that when complex numbers are multiplied together, distances multiply and angles add. This is due to a property intrinsic to exponents, which can be shown algebraically.
Using the polar form of complex numbers to show why distances multiply and angles add. (Image credit: Robert J. Coolman)
With the polar form of complex numbers established, the matter of Eulerās Identity is merely a special case of *a*\+*bi* for *a* \= -1 and *b* \= 0. Consequently for the polar form *re**iĻ*, this makes *r*\= 1 and *Ļ* = *Ļ* (since *Ļ*rad = 180Ā°).
Eulerās Identity is a special case of a+bi for a = -1 and b = 0 and reiĻ for r = 1 and Ļ = Ļ. (Image credit: Robert J. Coolman)
## Derivation of polar form
Though Eulerās Identity follows from the polar form of complex numbers, it is impossible to derive the polar form (in particular the spontaneous appearance of the number *e*) without [calculus](https://www.livescience.com/50777-calculus.html).
A general case of a complex number in both rectangular (a+bi) and polar (reiĻ) forms. (Image credit: Robert J. Coolman)
We start with the rectangular form of a complex number:
*a* \+ *bi*
From the diagram and [trigonometry](https://www.livescience.com/51026-what-is-trigonometry.html), we can make the following substitutions:
(*r*Ā·cos*Ļ*) + (*r*Ā·sin*Ļ*)*i*
From here we can factor out *r*:
*r*Ā·(cos*Ļ* + *i*Ā·sin*Ļ*)
Sometimes ācos*Ļ* + *i*Ā·sin*Ļ*ā is named cis*Ļ*, which is shorthand for ā[**c**osine plus **i**maginary **s**ine](http://mathworld.wolfram.com/Cis.html).ā
*r*Ā·cis*Ļ*
The function cis*Ļ* turns out to be equal to *e**iĻ*. This is the part thatās impossible to show without calculus. Two derivations are shown below:
Two derivations for of cisĻ = eiĻ. Both use some form of calculus. (Image credit: Robert J. Coolman)
Thus, the equation *r*Ā·cis*Ļ* is written in standard polar form *r*Ā·e*iĻ**.*
Additional resources
- [ResearchGate: What is Special in Euler's Identity?](http://www.researchgate.net/post/What_is_Special_in_Eulers_identity_e_ipi_102)
- [Academia.edu: Euler's Identity ā A Mathematical Proof for the Existence of God](http://www.academia.edu/217151/Eulers_Identity), by Robin Robertson
- [Science4All: The Most Beautiful Equation of Math: Euler's Identity](http://www.science4all.org/le-nguyen-hoang/eulers-identity)
[Robert Coolman](https://www.livescience.com/author/robert-coolman)
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Robert Coolman, PhD, is a teacher and a freelance science writer and is based in Madison, Wisconsin. He has written for Vice, Discover, Nautilus, Live Science and The Daily Beast. Robert spent his doctorate turning sawdust into gasoline-range fuels and chemicals for materials, medicine, electronics and agriculture. He is made of chemicals.
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| Readable Markdown | 
Euler's Equation (Image credit: public domain)
Eulerās identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as "[the most beautiful equation](https://www.livescience.com/26584-5-mind-boggling-math-facts.html)." It is a special case of a foundational equation in complex arithmetic called Eulerās Formula, which the late great physicist Richard Feynman called [in his lectures](https://target.georiot.com/Proxy.ashx?tsid=74387&GR_URL=http%3A%2F%2Famazon.com%2FFeynman-Lectures-Physics-Vol-Millennium%2Fdp%2F0465024939%3Ftag%3Dhawk-future-20%26ascsubtag%3Dlivescience-us-5530728407723048703-20) "our jewel" and "the most remarkable formula in mathematics."
In an [interview with the BBC](http://www.bbc.com/news/science-environment-26151062), Prof David Percy of the Institute of Mathematics and its Applications said Euler's Identity was āa real classic and you can do no better than that ā¦ It is simple to look at and yet incredibly profound, it comprises the five most important mathematical constants.ā
Euler's Identity is written simply as: *e**iĻ* + 1 = 0
Article continues below
Latest Videos From Live Science
The five constants are:
- The [number 0](https://www.livescience.com/27853-who-invented-zero.html).
- The number 1.
- The [number *Ļ*](https://www.livescience.com/29197-what-is-pi.html), an irrational number (with unending digits) that is the ratio of the circumference of a circle to its diameter. It is approximately 3.14159ā¦
- The number *e*, also an irrational number. It is the base of [natural logarithms](https://www.livescience.com/50940-logarithms.html) that arises naturally through study of compound interest and [calculus](https://www.livescience.com/50777-calculus.html). The number *e* pervades math, appearing seemingly from nowhere in a vast number of important equations. It is approximately 2.71828ā¦.
- The [number *i*](https://www.livescience.com/42748-imaginary-numbers.html), defined as the square root of negative one: ā(-1). The most fundamental of the imaginary numbers, so called because, in reality, no number can be multiplied by itself to produce a negative number (and, therefore, negative numbers have no real square roots). But in math, there are many situations where one is forced to take the square root of a negative. The letter *i* is therefore used as a sort of stand-in to mark places where this was done.
## Prolific mathematician
Leonhard Euler was an 18th-century Swiss-born mathematician who developed many concepts that are integral to modern mathematics. He spent most of his career in St. Petersburg, Russia. He was one of the most prolific mathematicians of all time, according to the [U.S. Naval Academy](http://www.usna.edu/Users/math/meh/euler.html) (USNA), with 886 papers and books published. Much of his output came during the last two decades of his life, when he was totally blind. There was so much work that the St. Petersburg Academy continued publishing his work posthumously for more than 30 years.
Euler's important contributions include Euler's Formula and Euler's Theorem, both of which can mean different things depending on the context. According to the USNA, in mechanics, there are "Euler angles (to specify the orientation of a rigid body), Euler's theorem (that every rotation has an axis), Euler's equations for motion of fluids, and the Euler-Lagrange equation (that comes from calculus of variations)."
## Multiplying complex numbers
Eulerās Identity stems naturally from interactions of [complex numbers](https://www.livescience.com/42966-complex-numbers.html) which are numbers composed of two pieces: a [real number](https://www.livescience.com/42619-real-numbers.html) and an [imaginary number](https://www.livescience.com/42748-imaginary-numbers.html); an example is 4+3*i*. Complex numbers appear in a multitude of applications such as wave mechanics (a study within [quantum mechanics](https://www.livescience.com/33816-quantum-mechanics-explanation.html)) and design of circuits that use alternating current (a common practice in [electrical engineering](https://www.livescience.com/47571-electrical-engineering.html)). Additionally, complex numbers (and their cousins, the [hyper complex numbers](http://mathworld.wolfram.com/HypercomplexNumber.html)) have a property that makes them especially useful for studying computer graphics, robotics, navigation, flight dynamics, and orbital mechanics: multiplying them together causes them to rotate. This property will help us understand the reasoning behind Eulerās Identity.
Get the worldās most fascinating discoveries delivered straight to your inbox.
In the example below, five complex numbers are plotted on the **complex plane** and together form a āhouse shape.ā The complex plane is similar to a number line, except that itās two-dimensional. The horizontal direction represents the real numbers and the vertical axis represents imaginary numbers. Each house-shape complex number is multiplied by the complex number 4+3*i* and re-plotted (green arrow). \[[Related: What Are Complex Numbers?](https://www.livescience.com/42966-complex-numbers.html)\]
As can be seen, multiplying by 4+3*i* results in the house shape **dilating** (increasing in area and moving away from the origin 0+0*i* by the same amount) and **rotating** (becoming tilted by some angle). To show this is precisely the effect of multiplying by 4+3i, the effect of zooming in on the house five times and rotating by 36.9 degrees is also shown (red arrow). The exact same effect is produced.
The same effect is produced from multiplying the vertices of a figure by 4+3i and rotating the figure by 36.9 degrees and dilating it by a factor of five. (Image credit: Robert J. Coolman)
Different amounts of dilation and rotation can produce the effects of multiplying by any number on the complex plane.
## Polar form of complex numbers
The amount of rotation and dilation is determined by properties intrinsic to the number 4+3*i,* which, as seen in the figure below, is five units from the origin (*r* = 5) and forms an angle of 36.9 degrees with the horizontal axis (*Ļ* = 36.9Ā°). These measurements are used in what is known as the [**polar form**](http://mathworld.wolfram.com/PolarCoordinates.html) of a complex number (*re**iĻ*) as opposed to the normal **rectangular form** (*a*\+*bi*).
The number 4+3i is five units from the origin and forms an angle of 36.9 degrees with the horizontal axis. (Image credit: Robert J. Coolman)
The polar form requires that *Ļ* be measured in **radians**. One radian (1rad) is approximately 57.3 degrees; itās the measure of angle made when a circleās radius is wrapped against that circleās circumference. A measure of [*Ļ* radians](https://www.livescience.com/29197-what-is-pi.html) wraps half way around a circle; a measure of 2*Ļ* radians wraps a full circle.
An angle measure of one radian is formed when a circleās radius is wrapped against its circumference. A half-circle is Ļ radians and a full circle is 2Ļ radians. (Image credit: Robert J. Coolman)
The angle measure for 4+3*i* is 0.644 radians (36.9Ā° = 0.644rad) meaning the polar form of 4+3*i* is 5*e**i*0\.644. Measures for *r* and *Ļ* can also be determined for each of the house-shape points, and yet another way of achieving the dilating/rotating effect of multiplying by 4+3*i* is to multiply each *r* by five, and add 36.9 degrees (or 0.644rad) to each *Ļ*. From this demonstration, we see that when complex numbers are multiplied together, distances multiply and angles add. This is due to a property intrinsic to exponents, which can be shown algebraically.
Using the polar form of complex numbers to show why distances multiply and angles add. (Image credit: Robert J. Coolman)
With the polar form of complex numbers established, the matter of Eulerās Identity is merely a special case of *a*\+*bi* for *a* \= -1 and *b* \= 0. Consequently for the polar form *re**iĻ*, this makes *r*\= 1 and *Ļ* = *Ļ* (since *Ļ*rad = 180Ā°).
Eulerās Identity is a special case of a+bi for a = -1 and b = 0 and reiĻ for r = 1 and Ļ = Ļ. (Image credit: Robert J. Coolman)
## Derivation of polar form
Though Eulerās Identity follows from the polar form of complex numbers, it is impossible to derive the polar form (in particular the spontaneous appearance of the number *e*) without [calculus](https://www.livescience.com/50777-calculus.html).
A general case of a complex number in both rectangular (a+bi) and polar (reiĻ) forms. (Image credit: Robert J. Coolman)
We start with the rectangular form of a complex number:
*a* \+ *bi*
From the diagram and [trigonometry](https://www.livescience.com/51026-what-is-trigonometry.html), we can make the following substitutions:
(*r*Ā·cos*Ļ*) + (*r*Ā·sin*Ļ*)*i*
From here we can factor out *r*:
*r*Ā·(cos*Ļ* + *i*Ā·sin*Ļ*)
Sometimes ācos*Ļ* + *i*Ā·sin*Ļ*ā is named cis*Ļ*, which is shorthand for ā[**c**osine plus **i**maginary **s**ine](http://mathworld.wolfram.com/Cis.html).ā
*r*Ā·cis*Ļ*
The function cis*Ļ* turns out to be equal to *e**iĻ*. This is the part thatās impossible to show without calculus. Two derivations are shown below:
Two derivations for of cisĻ = eiĻ. Both use some form of calculus. (Image credit: Robert J. Coolman)
Thus, the equation *r*Ā·cis*Ļ* is written in standard polar form *r*Ā·e*iĻ**.*
Additional resources
- [ResearchGate: What is Special in Euler's Identity?](http://www.researchgate.net/post/What_is_Special_in_Eulers_identity_e_ipi_102)
- [Academia.edu: Euler's Identity ā A Mathematical Proof for the Existence of God](http://www.academia.edu/217151/Eulers_Identity), by Robin Robertson
- [Science4All: The Most Beautiful Equation of Math: Euler's Identity](http://www.science4all.org/le-nguyen-hoang/eulers-identity)
Robert Coolman, PhD, is a teacher and a freelance science writer and is based in Madison, Wisconsin. He has written for Vice, Discover, Nautilus, Live Science and The Daily Beast. Robert spent his doctorate turning sawdust into gasoline-range fuels and chemicals for materials, medicine, electronics and agriculture. He is made of chemicals. |
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