Named after the Swiss mathematician Leonhard Euler – “Euler’s Identity” is often described as an example of deep mathematical beauty. Some see it almost as proof of the existence of some hidden form of order in the universe that we can only just catch a glimpse of…
- “e” is Euler’s number, the base of the natural logarithms,
- “i” is the imaginary unit, defined as the square root of -1 which means that i2 = −1
- “π” is pi, the ratio of the circumference of a circle to its diameter.
This equation is considered amazing because it seems to sets up a relationship between three apparently completely unrelated subjects.
“e” originated from the study of compound interest; “π” from Geometry and Trigonometry; and “i” was “invented” to help with solving equations involving negative numbers.
Compound Interest, Trigonometry, and Imaginary Numbers! The question that has intrigued mathematicians and physicists since Euler first published his famous formula in 1748, is “Why in God’s name do these (apparently completely unrelated) areas of mathematics fit together so neatly in this beautiful equation?”
What is “e”?
Euler’s number is defined as “e is the base of natural logarithms”
Okay, so what are the natural logarithms?
The natural logarithm (ln) is defined as “a logarithm to the base e”
Great! So that clears that up then!…
Part 1 – Micro-e
So what is “e” really?
It is often, considered amazing that “e” appears to crop up virtually everywhere in science, (from physics to biology, from economics to climate change), but in truth it is not amazing at all! The one constant in this universe, is “change”; and the reason “e” appears everywhere is that “e” is all about change!
What’s important about “e” however, is not “e” itself, but “e” as it is used in the “exponential function”
y = ex
For practical purposes we can think of exponential functions as a mathematical tool that allows us to define a multiplicative step-size of change (i.e. they allows us to define a macroscopic step-size of change using a base microscopic unit.). So “ex ” is a mathematically definable “step-size of incremental change”, and “e” itself just happens to be the value of (y = ex) when x =1.
In the everyday speak we could say that “e” is a number we converge to when we take an infinitesimally small growth rate and applies it an infinite number of times.
Mathematically however “e” is defined as a limit: “e” is equal to the limit of (1 + 1/n)n as n goes to infinity.
e = Limit as n → ∞ (1 + 1/n)n = 2.718281828459045……
For practical purposes, “e” can be difficult to pin down because the question arises as to what size “n” do we want to use to calculate “e”?
If, for Instance, we set n=1000 we find that the exact value of (1+ (1/1000))1000 is 2.71692393223552. So even with only 1000 steps of 1.001, e is correct to 2 decimal places, with 10,000 steps of 1.0001 it is correct to 3 decimal places, and with 100,000 steps of 1.00001 “e” is correct to 4 decimal places, and so on to infinity.
Making the number of steps smaller and smaller, will give more trailing digits of “e”, but since there is no limit to how much a number can be divided, there is therefore no limit to the number of trailing digits we can add to “e”. This is why “e” is transcendental!…
That “e” is transcendental is simply a consequence of the fact that “e” is a manufactured quantity, defined using limits to infinity, and infinity is an idea not a number! However although the concept of infinity might be vague, “the concept of “e””, on the other hand, is actually all about very finely-grained precision…
“e” without limits
The concept and formula for “e” is neat and very precise because the Number of Steps (n) must be the exact inverse of the tiny Step-Size (1/n).
In order to discuss the preciseness of “e”, it is better, for the purposes of this discussion, to limit the influence of “infinity” and define “e” with very precise numbers.
To do so, we will use both a very large number and a very small number which are exact inverses of each other. We will use the symbol Ħ for the very large number, and we will use the symbol ħ for the very small number.
Ħ = 10,000,000,000,000,000,000,000,000,000,000,000 = 10+34
ħ = 0.000,000,000,000,000,000,000,000,000,000,000,1 = 10-34
Since ħ = 1/ Ħ we can write a precision equation for “e” as
e = ( 1 + ħ )Ħ = 2.718281828459045……
Thus the macro-number “e” or “Macro-e” is simply the step-size (1+ħ) applied Ħ number of times. Using Ħ we can also define “Micro-e”
e1 = ( 1 + ħ )Ħ
e1/Ħ = ( 1 + ħ )
eħ = ( 1 + ħ )
It is here that we get to what is so important about “e” – micro-e is what makes the exponential function (ex) so useful to “the calculus of infinitesimals” (as calculus was historically known)!
“Micro-e” is a unit of fine-grained precision, we could think of (1+ħ) as a “Unit Step-Size” — a “Fundamental Unit of Linear Change”.
Defining the “Step-Size of Linear Change”
So now that we have defined a unit step-size of linear change; we will use this very fine-grained unit to define other, more coarsely-grained mathematical steps. Thus (1+ħ) applied xĦ number of times yields the exponential function (ex).
( 1 + ħ )xĦ = (eħ)xĦ = (eħĦ)x = ex
So (y = ex) is a function that allows us to define, with precision, any macro step-size as being equivalent to a certain number of steps (xĦ) of the smallest, almost infinitesimal, unit of change (1+ħ). In effect we can define any step-size we like from infinitesimally small to infinitely large using an appropriate power (positive for growth or negative for decay) of (ex); and when we set x=1 we get Ħ micro-steps of (1+ħ) is equivalent to one single macro-step of (e1).
( 1 + ħ )1Ħ = (eħ)1Ħ = (eħĦ)1 = e1 = 2.718281828459
So (y = ex) is a definable linear step-size – it is a defined amount of linear change that occurs in a defined amount of time.
[Click here for a history and more complete analysis of “What is e?” ]
Part 2 – Micro-ei
Step-Size of Oscillation
Euler discovered that that there is a relationship between “e” raised to the power of “ix”, and the trigonometric functions, cosine and sine. Euler discovered that “e” raised to the power of “ix”, can be written as
eix = cos(x) + i sin(x)
This expression is known as “Euler’s Formula”, and it appears to relate an “imaginary step-size” to a complex number.
Complex numbers are a combination of real and imaginary numbers. The combination can be used to define the co-ordinates of a 2-dimensional vector (i.e. a mathematical objects that incorporate both a size/magnitude and an angle of direction). Euler’s formula, (eix) is a unit vector (i.e. a vector whose magnitude = 1). So this imaginary step-size (eix) is a step-size vector with angle of direction (ix) only.
Furthermore we know that “e” is simply (1+ħ)Ħ therefore we can write
eix = ((1+ħ)Ħ)ix = cos(x) + i sin(x)
ei = ((1+ħ)Ħ)i = cos(1) + i sin(1)
eiħ = (1+ħ)i = cos(ħ) + i sin(ħ)
(1+ħ)i = (0.99999999 + iħ)
“Micro-ei” is a unit of fine-grained precision, we could think of (1+iħ) as a “Fundamental Unit of Angular Change”.
Defining the “Step-Size of Angular Change”
So now that we have defined a unit step-size of angular change; we will use this very fine-grained unit to define other, more coarsely-grained mathematical steps. Thus (1 + iħ) applied xĦ number of times yields the exponential function (eix).
(1 + iħ)xĦ = (eiħ)xĦ = (eħĦ)ix = eix
So (y = eix) is a function that allows us to define, with precision, any macro step-size as being equivalent to a certain number of steps (xĦ) of the smallest, almost infinitesimal, unit of change (1+iħ). When we set x=1 we get Ħ micro-steps of (1 + iħ) is equivalent to one single macro-step of (ei1) – which is an angular step of 1 Radian, and is equal to the complex number (0.54 + i 0.84).
(1 + iħ)1Ħ = (eiħ)Ħ = (eħĦ)i = ei
And so, although Euler, nor anyone else for that matter, ever specifically saw it as such; what Euler discovered was a way to define a precise “angular step-size”. Thus (y = eix) is a definable angular step-size – it is a defined amount of angular change that occurs in a defined amount of time.
Euler’s identity – “Maximum Step-Size of Angular Change”
Angles were “born” in the domain of geometry, and consequently angles measured in radians are often described as fractions of “π”. Unlike a linear step-size which has no maximum, an angular step-size has a maximum of “π”. (This is of course because an angular step-size of say “2π” is exactly the same as an angular step-size of zero…)
Thus if we set x=3.14159 (i.e. x=π), then πĦ micro-steps of (1 + iħ) is equivalent to one single macro-step of (eiπ) – which is an angular step of π Radians, and is equal to the complex number (-1.00 + i0.00).
(1 + iħ)3.14159Ħ = (eiħ)πĦ = (eħĦ)iπ = eiπ
eiπ = cos(π) + i sin(π)
eiπ = -1 + i (0)
So Euler’s Identity is the special “maximum” case of the generalized formula for defining the coarseness (i.e. the step-size) of angular change. When we set “x = π” we are defining a macro step-size which is the “maximum angular change, in one discrete unit of time”.
Part 3 – Micro-eiπ
Another Fundamental Unit of Change
Even though “i” might have started life as a humble “imaginary number”, much belittled because it was not really “real”, merely “invented” for the purposes of solving algebraic equations; it turns out that “i” is in fact the ideal mathematical tool to define the step-size of angular change.
But using (1+iħ) means that we are defining the step-size as a percentage of 1 radian; and this means that a maximum step-size of angular change is 314.159% of 1 Radian. Obviously it would seem preferable to define the angular step-size as a percentage of the maximum angular change (i.e. as a percentage of π radians).
So let’s define a new fundamental unit of angular change. As before, we know that “e” is simply “(1+ħ)Ħ ” therefore we can write
eix = ((1+ħ)Ħ)ix = cos(x) + i sin(x)
eiπ = ((1+ħ)Ħ)iπ = cos(π) + i sin(π)
eiπħ = (1+ħ)iπ = cos(πħ) + i sin(πħ)
(-1)ħ = (1+ħ)iπ = (0.99999999 + iπħ)
“Micro-eiπ” is a unit of fine-grained precision, we could think of (1+iπħ) as another “Fundamental Unit of Angular Change”.
Another “Definable Step-Size”
So once again, we can use this very fine-grained step-size to define other, more coarsely-grained angular steps.
Thus the micro-step (1 + iπħ) applied xĦ number of times, yields the exponential function (-1)x.
(1 + iπħ)xĦ = (eiπħ)xĦ = (eħĦ)iπx = (eiπ)x = (-1+i0)x
So when we set x=ħ we get 1 micro-step of (1 + iπħ) is equivalent to one single micro-step of (-1)ħ
(1 + iπħ)ħĦ = (1 + iπħ)1 = (-1)ħ
And when we set x=1 we get Ħ micro-steps of (1 + iπħ) is equivalent to one single macro-step of (-1)1
(1 + iπħ)1Ħ = (1 + iπħ)Ħ = (-1)1
And so, although it is not generally seen as such what Euler really discovered was a way to define a precise angular step-size in terms of the maximum angular change. Thus (y = eiπx) or (y = (-1+i0)x) is a definable angular step-size – it is a defined amount of angular change that occurs in a defined amount of time.
Part 4 – The Meaning of “i”
Defining the “Step-Size of Oscillation”
Euler’s Formula is usually thought of as a mathematical formula that describes how a “vector rotates” through an angle in the complex plane. But back in 1581 Galileo realized that “rotation” secret identity is that he is also an “oscillation”.
So although angles are most often associated with geometry and trigonometry; to an engineer angles are usually associated with the amount of energy in an “oscillation” — which means that our rotation in the complex-plane y=(-1+i0)x can actually be thought of as a definable angular step-size of oscillation in terms of a “single half-cycle (-1+i0)”.
Furthermore (-1+i0)ħ can be thought of as a fundamental unit “step-size of angular oscillation”.
When we set x=0.5 we get “i”
(1 + iπħ).5Ħ = (-1)x = (-1).5 = i
But in setting x=0.5 we are in effect setting an angular frequency of π/2 radians; and so, although it is not generally seen as such, we can say that
“i” is a Quarter-Cycle
So, a “quarter-rotation in the complex plane” is equivalent to the “quarter-cycle on the real line”, and “i” is one-quarter of a coarse oscillation.
Furthermore since (1 + iπħ)xĦ is equivalent to angular frequency, we see that the coarser the step-size, the higher the frequency. Thus we can rephrase the statement above and say that Euler nearly discovered a way to define a “coarse angular frequency” in terms of a “fundamental angular unit of oscillation”.
(-1)ħ = (1 + iπħ)
Conclusion – The Beauty of Mathematics
Euler’s identity is a beautiful example of the “quantized mathematics of change” . This quantization of mathematics allows us to define very precise “coarse” macro step-sizes (both linear and angular) using integer amounts of the quantized micro units (1+ħ) and (1+iπħ).
- (y = ex) defines the coarseness of linear growth and decay
- (y = eiπx) defines the coarseness of the frequency of oscillation
Euler’s Identity is simple the value of (y = eiπx) when x=1. So despite what is generally believed, Euler identity is not telling us something special about “rotations”; it is telling us something special about “oscillations”. For even though “i” might have started life as a humble “imaginary number”, much belittled because it was not really “real”, merely “invented” for the purposes of solving algebraic equations; it turns out that
“i” is equivalent to the “Amplitude of an Oscillation” on the real line, and consequently it is the ideal mathematical tool to help define “frequency” or “step-size of oscillation”
And thus we can say that
Taking any root of (-1+i0) will yield a frequency of oscillation.
And the interesting thing about a “discrete” step-size of oscillation, is that not all “coarse” oscillations produce nice, regular, tick-tock-like symmetry – as we can see from an examination of “Coarse Harmonic Motion”.