Eulerian Representation of Complex Number

Leonhard Euler, a Swiss mathematician, devised Euler’s formula. Euler’s formulas are divided into two categories: It’s an important formula for solving complex exponential functions. Euler’s identity is another name for Euler’s formula. It’s used to show how trigonometric functions and complex exponential functions are related. The number of faces, vertices, and edges of any polyhedron that does not self-intersect is connected in a specific way, which is given by Euler’s formula, also known as Euler’s characteristic. 

Euler’s Formula

The following are two Euler’s formulas that are utilised in distinct situations. 

• Euler’s formula for complex analysis: 

eix = cos x + isin x

• Euler’s formula for polyhedra: 

faces + vertices – edges = 2 

Euler’s Formula for Complex Analysis

The Euler form of a complex number is significant enough to warrant its own section. It’s a really useful form that simplifies a lot of calculations. In complex analysis, Euler’s formula is used to determine the relationship between trigonometric functions and complex exponential functions. Euler’s formula can be written as follows for any real number x: 

eix = cos x + isin x 

The trigonometric functions cos and sin are used, as well as the imaginary unit i and the natural logarithm base e. This formula can be interpreted as a unit complex function eiθ tracing a unit circle in the complex plane, where θ is a real integer and is measured in radians. 

Proof: 

For example, we utilise the expansion series ex: 

ex = 1 + x + x²/2! + x³/3! + x⁴/4! + … ∞ 

We now suppose that even if x is a non-real number, this expansion holds. Even this assumption will have to be justified in a rigorous demonstration, but for now, let us take it for granted and use x = iθ.

eiθ = 1 + iθ + (iθ)²/2! + (iθ)³/3! + (iθ)⁴/4! + … ∞

= 1 + iθ – θ²/2! – iθ³/3! + θ⁴/4! + … ∞ (because i² = -1)

= (1 – θ²/2! + θ⁴/4! – … ∞) + I (θ – θ³/3! + θ⁵/5! -… ∞)

The two series are Taylor expansion series for cosθ and sinθ thus

eix = cos x + isin x

Euler’s Identity

We get eix = cos x + isin x from the formula above. When x is replaced with π, this formula yields an identity. Then there’s 

eiπ = cos π + isin π

eiπ = -1 + I (0) (as cos π = -1 and sin π = 0)

eiπ = -1 (or)

eiπ + 1 = 0

This is known as Euler’s identity. 

Euler’s Formula for Polyhedra

Polyhedra are three-dimensional solid objects with flat surfaces and straight edges. Cube, cuboid, prism, and pyramid are examples. The number of faces, vertices, and edges of every polyhedron that does not self-intersect is connected in a certain way. The number of vertices and faces together is exactly two greater than the number of edges, according to Euler’s formula for polyhedra. The formula for a polyhedron given by Euler is: 

F + V – E = 2

Here,

• F is the number of faces,

• V the number of vertices, and

• E the number of edges. 

Proof: 

It becomes a graph when we just draw dots and lines. When no lines or edges cross each other, we have a planar graph. By projecting the vertices and edges onto a plane, we can describe a cube as a planar graph. The number of dots + the number of lines + the number of regions the plane is cut into equals 2 in Euler’s graph theory. 

Uses of Euler’s Formula

The more generic formula is F + V E = X, where X is the Euler characteristic. We demonstrate that Euler’s formula can be used in any three-dimensional space, not just polyhedra. According to Euler’s graph theory, there are exactly 5 regular polyhedra. We can check for a basic polyhedron with 10 faces and 17 vertices using Euler’s formula calculator. Although the prism has 10 faces and an octagon as its basis, it has 16 vertices. 

Verification of Euler’s Formula for Solids

Solid shapes and complicated polyhedra are instances of Euler’s formulas. Let’s check the formula for a couple of simple polyhedra, like a square pyramid and a triangular prism. 

• A square pyramid has 5 faces, 5 vertices, and 8 edges.

F + V – E = 5 + 5 – 8 = 2

• A triangular prism 5 faces, 6 vertices, and 9 edges.

F + V – E = 5 + 6 – 9 = 2

Solved Examples

Q1. Express ei(π/2) in the (a + ib) form by using Euler’s formula. 

Ans. Given: θ = π/2

Using Euler’s formula,

eiθ = cosθ + isinθ

⟹ ei(π/2) = cos(π/2) + isin(π/2) = 0 + I × 1 = I 

Answer: Hence ei(π/2) in the a + ib form is i. 

Q2. Jack knows that a polyhedron has 12 vertices and 30 edges. How can he find the number of faces?

Ans. Using Euler’s formula:

F + V – E = 2

F + 12 – 30 = 2

F – 18 = 2

F = 20

Number of faces = 20. 

Q3. Sophia finds a pentagonal prism in the laboratory. What do you think the value of F + V – E is for it?

Ans. A pentagonal prism has 7 faces, 15 edges, and 10 vertices.

Let’s apply Euler’s formula here,

F + V – E = 7 + 10 – 15 = 2

F + V – E for a pentagonal prism = 2. 

Conclusion

In geometry, Euler’s formula is used to determine the relationship between polyhedra’s faces and vertices. Euler’s formula is also used to trace the unit circle in trigonometry. Euler’s approach works by approximating a solution curve with line segments to approximate solutions to certain differential equations.