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
i=imaginary unit
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).
Euler's Identity
When x=π Euler’s formula evaluates to e^iπ+1=0, which is known as Euler's Identity.
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Euler's Formula
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
V-E+F=2
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
=2
Hence proved.
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)
Therefore,
e^i(nθ) = cos (nθ) + i sin (nθ)
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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.
We have,
z^n = 1
z = 1^(1/n)
From (1),
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 .
Solution
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
FAQs on Euler's Formula and De Moiver’s Theorem
1. What is Euler's formula in the context of complex numbers?
Euler's formula establishes a fundamental relationship between trigonometric functions and the complex exponential function. For any real number θ (in radians), the formula is stated as: eiθ = cos(θ) + i sin(θ). Here, 'e' is the base of the natural logarithm, 'i' is the imaginary unit (√-1), and cos(θ) and sin(θ) are the trigonometric functions which represent the coordinates on a complex plane.
2. What is De Moivre's theorem, and what is it used for?
De Moivre's theorem provides a straightforward method for calculating powers of complex numbers. The theorem states that for any complex number in the polar form (cos θ + i sin θ) and any integer 'n', the following holds true: (cos θ + i sin θ)n = cos(nθ) + i sin(nθ). Its primary applications in Class 11 Maths are to simplify the process of finding powers and roots of complex numbers.
3. How do you represent a complex number in polar and rectangular forms?
A complex number can be represented in two main forms:
- Rectangular Form: Expressed as z = a + ib, where 'a' is the real part and 'b' is the imaginary part. This corresponds to the coordinates (a, b) on the Argand plane.
- Polar Form: Expressed as z = r(cos θ + i sin θ), where 'r' is the modulus (or distance from the origin) and 'θ' is the argument (or angle with the positive real axis). This form is particularly useful for multiplication, division, and finding roots or powers of complex numbers.
4. What are the main applications of De Moivre's theorem for a Class 11 student?
For a Class 11 student following the CBSE syllabus, De Moivre's theorem is primarily applied in two key areas:
- Finding Powers of Complex Numbers: It simplifies raising a complex number to a high power without performing tedious repeated multiplications. For example, calculating (1 + i)10 becomes much easier.
- Finding the nth Roots of a Complex Number: It forms the basis for finding all 'n' distinct roots of a complex number, including the nth roots of unity, which is a crucial concept.
5. How does Euler's formula help in understanding De Moivre's theorem?
Euler's formula provides one of the most elegant proofs for De Moivre's theorem. We start with Euler's formula: eiθ = cos(θ) + i sin(θ). If we raise both sides to the power of 'n', we get (eiθ)n = (cos θ + i sin θ)n. Using the laws of exponents, the left side becomes ei(nθ). Applying Euler's formula again to this result gives cos(nθ) + i sin(nθ). By equating the results, we directly arrive at (cos θ + i sin θ)n = cos(nθ) + i sin(nθ), thereby proving the theorem.
6. Why is the number 'e' so important in Euler's formula for complex numbers?
The number 'e' (Euler's number) is crucial because it connects exponential functions, which describe growth and decay, with trigonometric functions, which describe periodic motion or rotations. In the context of complex numbers, the expression eiθ represents a point on the unit circle in the Argand plane. This creates a powerful link, allowing rotational operations (trigonometry) to be handled with the simpler rules of exponents, which is the core reason it simplifies so many calculations in complex analysis.
7. What is the geometric interpretation of multiplying two complex numbers using Euler's formula?
Using Euler's formula, two complex numbers z1 = r1eiθ₁ and z2 = r2eiθ₂ can be multiplied easily: z1z2 = r1r2ei(θ₁+θ₂). The geometric interpretation of this is profound:
- The magnitudes (lengths) of the vectors are multiplied (r1r2).
- The arguments (angles) of the vectors are added (θ₁ + θ₂).
Essentially, multiplying by a complex number corresponds to a rotation and a scaling in the complex plane.
8. Can De Moivre's theorem be applied to non-integer exponents?
The standard statement of De Moivre's theorem is for integer values of 'n'. When 'n' is a rational number (like 1/2 or 3/4), the formula cos(nθ) + i sin(nθ) gives one of the possible values or roots. However, for a fractional exponent like 1/q, there are 'q' distinct roots. While the theorem is the foundation for finding these roots, a more generalised formula is required to find all of them, not just the principal value.
