Euler’s Formula and De Moiver’s Theorem
We know about complex numbers(z). They are of the form z=a+ib, where a and b are real numbers and ‘i’ is the solution of equation x²=-1. No real number can satisfy this equation hence its solution that is ‘i’ is called an imaginary number. When a complex exponential is written, it is written as e^iθ.
Euler’s formula explains the relationship between complex exponentials and trigonometric functions.
DeMoivers’ theorem is also a theorem used for complex numbers. This theorem is used to raise complex numbers to different powers.
State Euler’s Theorem
Euler’s law states that ‘For any real number x, e^ix = cos x + i sin x.
where,e=base of natural logarithm
x=angle in radians
This complex exponential function is sometimes denoted cis x (“cosine plus i sine”). The formula is still valid if x is a complex number.
Let z be a non zero complex number; we can write z in the polar form as,
z = r(cos θ + i sin θ) = r e^iθ, where r is the modulus and θ is argument of z.
Multiplying a complex number z with e^iα gives, zei^α = re^iθ × ei^α = rei^(α + θ).The resulting complex number re^i(α+θ) will have the same modulus r and argument (α+θ).
When x=π Euler’s formula evaluates to e^iπ+1=0, which is known as Euler’s Identity.
Image to be added soon
Euler’s Formula For Cube
Euler’s formula is related to the Faces, Edges and vertices of any polyhedron.
Euler’s formula for a cube says that in a cube, the number of vertices minus the number of edges plus the number of faces results in two.
It can be written as
Where, V=number of vertices
E=number of edges
F=number of faces
It can be proven as,
In a cube, the number of vertices = 8
number of edges= 12
number of faces= 6
Putting values into the formula, V-E+F=8-12+6
De Moiver’s Theorem
State De Moiver’s Theorem
It states that for any integer n,
(cos θ + i sin θ)^n = cos (nθ) + i sin (nθ)
We can prove this easily using Euler’s formula as given below,
We know that, (cos θ + i sin θ) = e^iθ
(cos θ + i sin θ)^n = e^i(nθ)
e^i(nθ) = cos (nθ) + i sin (nθ)
Image will be added soon
nth Roots of Unity
If any complex number satisfies the equation zn = 1, it is known as nth root of unity.
An equation of degree n will have n roots as said by the fundamental theory of algebra, there are n values of z which satisfies zn = 1.
To find the values of z, we can write,
1 = cos (2kπ) + i sin (2kπ), —(1) where k can be any integer.
z^n = 1
z = 1^(1/n)
z = [cos (2kπ) + i sin (2kπ)]^(1/n)
By De Moivre’s theorem,
z = [cos (2kπ/n) + i sin (2kπ/n)], where k = 0,1,2,3,……..,n−1
For example; if n = 3, then k = 0,1,2
We know that, z = cos (2kπ/n) + i sin (2kπ/n) = e^i(2kπ/n)
Let ω = cos (2πn) +i sin (2πn) = e^i(2πn)
nth roots of unity are found by,
When k = 0; z = 1
k = 1; z = ω
k = 2; z = ω2
k = n; z = ωn − 1
Therefore, nth roots of unity are 1,ω,ω2,ω3,…….,ωn − 1
Sum of nth roots of unity is,1 + ω + ω2 + ω3 + ⋯ + ωn − 1It is geometric series having first term 1 and common ratio ω.By using sum of n terms of a G.P,1 + ω + ω2 + ω3 + ⋯ + ωn − 1 = 1 − ωn1 − ωSince ω is nth root of unity, ωn = 1Therefore, 1 + ω + ω2 + ω3 + ⋯ + ωn − 1 = 0
Cube Roots of Unity:
We know that nth roots of unity are 1,ω,ω2,ω3,…….,ωn − 1.
Therefore, cube roots of unity are 1,ω,ω2 where,
ω = cos (2π/3) + i sin (2π/3) = −1 + √3 i2
ω2 = cos(4π/3) + i sin (4π/3) = −1 − √3 i2
Sum of the cube roots of the unity,
1 + ω + ω2 = 0
Product of cube roots of the unity,
1 × ω × ω2 = ω3 = 1
De Moiver’s Theorem Example
If z = (cosθ + i sinθ ) , show that z^n + 1/ z^n = 2 cos nθ and z^n – [1/ z^n] = 2i sin nθ .
Let z = (cosθ + i sinθ ) .
By de Moivre’s theorem ,
z^n = (cosθ + i sinθ )^n = cos nθ + i sin nθ
1/z^n=z^(-n)=cos nθ – i sin nθ
=> z^n+1/z^n = (cos nθ + i sin nθ)+(cos nθ – i sin nθ)
=> z^n+1/z^n = 2cosnθ
Also,=> z^n-1/z^n = (cos nθ + i sin nθ)-(cos nθ – i sin nθ)
=> z^n-1/z^n = 2i sin nθ
1. Why is ‘e’ Special?
e is a mathematics number which is irrational. It is one of the most important numbers which is used in a lot of places. Often it is called Euler’s number after Leonhard Euler (pronounced “Oiler”). e cannot be written as a simple fraction as it is an irrational number. e is the base of the Natural Logarithms. Though many ways of calculating the value of e are tried. none of them ever gave an exact answer, because e is irrational (not the ratio of two integers). Despite this, it is known to over 1 trillion digits of accuracy.
2. What is Rectangular and Polar Form?
In the rectangular form we a vector can be expressed in terms of its rectangular coordinates, where the horizontal axis is its real axis and the vertical axis is the imaginary axis or j-component. Whereas in polar form, these real and imaginary axes are simply represented by “A ∠θ“.