Eulers Formula

What is Euler’s Formula?

Eulers Formula, named after Leonhard Euler, is a mathematical formula in complex analysis that provides the essential link between trigonometric functions and complex exponential functions. The essential link between trigonometric functions and complex exponential functions is established by Euler’s formula. Another name for Eulers Formula is Euler’s identity. For polyhedra, Eulers Formula states that the number of faces, vertices, and edges is connected in a certain way for every polyhedron that does not self-intersect.

This article comprehensively covers both the Euler’s Formulas in detail.

Euler’s Formula Equation

Here are two possible Eulers Formula that are applied in various situations.

1. e^(ix) = cos x + i sin x, which is Eulers Formula for complex analysis.
2. faces + vertices – edges = 2, which is Eulers Formula for polyhedra.

According to Euler’s identity or formula, the complex analysis result for any real number x is represented by:

e^(ix) = cos x + i sin x

where,

• x is an actual number.
• e is the natural logarithm’s base.
• sin x and cos x are trigonometric functions.
• i is an imaginary unit

According to Euler’s polyhedral formula, there are precisely two more faces and vertices combined than there are edges. The polyhedron’s Eulers Formula is as follows:

Edges + 2 = Faces + Vertices

F + V = E + 2

Or

F + V – E = 2

where,

• F stands for “faces”
• V is the number of vertices.
• E is the number of edges.

Euler’s Formula for Complex Analysis

Eulers Formula of a complex number is significant enough in Mathematics. It is a very practical model that allows many calculations to be simplified. The link between complex exponential functions and trigonometric functions is established using Euler’s formula in complex analysis. For each real number x, Eulers Formula is defined and can be expressed as:

e^(ix) = cos x + i sin x

Here, i is the imaginary unit, cos and sin are trigonometric functions, and e is the natural logarithm’s base.

This equation can be understood as a unit complex function tracing a unit circle in a complex plane, where is a real value expressed in radians.

A particularly elegant consequence of Euler’s formula is Euler’s identity, which occurs when
x = π
e^(iπ) + 1=0

Euler’s Formula for Polyhedrons

The three-dimensional solid objects known as polyhedra have smooth surfaces and straight edges. Examples include the cube, cuboid, prism, and pyramid. There is a certain relationship between the number of faces, vertices, and edges for every polyhedron that does not self-intersect. The number of vertices and faces combined is exactly two greater than the number of edges, according to Euler’s formula for polyhedra. One can write the Eulers Formula for a polyhedron as follows:

F + V – E = 2

Here, the number of faces, vertices, and edges are denoted by the letters F, V, and E respectively.

When one only draws dots and lines, it turns into a graph. When there are no intersections of any lines or edges, one can have a planar graph. By transferring the vertices and edges of a cube onto a plane, one can represent it as a planar graph. The number of dots minus the number of lines plus the number of regions the plane is divided into equals two, according to Euler’s formula for graph theory.

What is Euler’s Formula Used For?

Eulers Formula, one of the most significant mathematical equations, has a number of fascinating applications in a variety of fields.

• Among them are the well-known Euler’s identity, exponential representation of complex numbers, and alternative trigonometric and hyperbolic function definitions.
• Eulers Formula is also used in complex number generalisation of exponential and logarithmic functions, De Moivre’s Theorem alternative proofs and trigonometric additive identities.
• The more general formula is F + V –  E = X, where X is the Euler characteristic. F + V – E can equal 2 or 1 and have other values.
• In order to examine any three-dimensional space, not just polyhedra, students can validate Euler’s formula.
• Euler’s formula can be used in the phasor analysis of circuits to express the impedance of a capacitor or an inductor.
• There is a sphere of imaginary units in the four-dimensional quaternion space. Euler’s formula is applicable for any point r on this sphere. The element is known as a versor in quaternions and is exp xr=cos x + r sin x.

Verification of Euler’s Formula for Solids

Examples of Euler’s formulas include complex polyhedra and solid forms. One can check the formula for a few basic polyhedra, like a triangle and a square pyramid.

There are 5 faces, 5 vertices, and 8 edges in a square pyramid.

F + V − E = 5 + 5 − 8 = 2

For a triangle with five faces, six vertices, and nine edges.

F + V − E = 5 + 6 − 9 = 2

Solved Examples Using Euler’s Formula

Example 1: A polyhedron has 8 vertices and 12 edges. Determine the number of faces it has

Solution:

V=8 (number of vertices)
E=12 (number of edges)
Using Euler’s formula:
V−E+F=2
8−12+F=2
F=12−8+2
F=6

Example 2: A polyhedron has 10 faces and 15 edges. How many vertices does it have?

Solution:

F=10 (number of faces)
E=15 (number of edges)
Using Euler’s formula:
V−E+F=2
V−15+10=2
V=15−10+2
V=7

Example 3: A convex polyhedron has 6 faces and 12 vertices. How many edges does it have?

Solution:

F=6 (number of faces)
V=12 (number of vertices)
Using Euler’s formula:
V – E + F=2
12−E+6=2
E=12+6−2
E=16