Euler’s Formula

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Euler’s formula, either of two important mathematical theorems of Leonhard Euler. The first formula, used in trigonometry and also called the Euler identity, says eix = cos x + isin x, where e is the base of the natural logarithm and i is the square root of −1 (see irrational number). When x is equal to π or 2π, the formula yields two elegant expressions relating π, e, and ieiπ = −1 and e2iπ = 1, respectively. The second, also called the Euler polyhedra formula, is a topological invariance (see topology) relating the number of faces, vertices, and edges of any polyhedron. It is written F + V = E + 2, where F is the number of faces, V the number of vertices, and E the number of edges. A cube, for example, has 6 faces, 8 vertices, and 12 edges and satisfies this formula.

Euler’s Formula for Complex Numbers

(There is another “Euler’s Formula” about Geometry,
this page is about the one used in Complex Numbers)

First, you may have seen the famous “Euler’s Identity”:

eiπ + 1 = 0

It seems absolutely magical that such a neat equation combines:

  • e (Euler’s Number)
  • i (the unit imaginary number)
  • π (the famous number pi that turns up in many interesting areas)
  • 1 (the first counting number)
  • 0 (zero)

And also has the basic operations of add, multiply, and an exponent too!

But if you want to take an interesting trip through mathematics, you will discover how it comes about.

Interested? Read on!

Discovery

It was around 1740, and mathematicians were interested in imaginary numbers.

He must have been so happy when he discovered this!

And it is now called Euler’s Formula.

Let’s give it a try:

The answer is a combination of a Real and an Imaginary Number, which together is called a Complex Number.

We can plot such a number on the complex plane (the real numbers go left-right, and the imaginary numbers go up-down):

Here we show the number 0.45 + 0.89 i

Which is the same as e1.1i

Let’s plot some more!

A Circle!

Yes, putting Euler’s Formula on that graph produces a circle:

And when we include a radius of r we can turn any point (such as 3 + 4i) into reix form by finding the correct value of x and r:

It is Another Form

It is basically another way of having a complex number.

This turns out to very useful, as there are many cases (such as multiplication) where it is easier to use the reix form rather than the a+bi form.

Euler’s Formula

(There is another “Euler’s Formula” about complex numbers,
this page is about the one used in Geometry and Graphs)

Euler’s Formula

For any polyhedron that doesn’t intersect itself, the

  • Number of Faces
  • plus the Number of Vertices (corner points)
  • minus the Number of Edges

always equals 2

This can be written: F + V − E = 2

Example With Platonic Solids

Let’s try with the 5 Platonic Solids:

It is still an icosahedron (but no longer convex).

In fact it looks a bit like a drum where someone has stitched the top and bottom together.

Now, there are the same number of edges and faces … but one less vertex!

So:

F + V − E = 1

Oh No! It doesn’t always add to 2.

The reason it didn’t work was that this new shape is basically different … that joined bit in the middle means that two vertices get reduced to 1.

Euler Characteristic

So, F+V−E can equal 2, or 1, and maybe other values, so the more general formula is

F + V − E = χ

Where χ is called the “Euler Characteristic“.

Here are a few examples:

Donut and Coffee Cup

Lastly, this discussion would be incomplete without showing that a Donut and a Coffee Cup are really the same!

Well, they can be deformed into one another.

We say the two objects are “homeomorphic” (from Greek homoios = identical and morphe = shape)

Just like the platonic solids are homeomorphic to the sphere.

And your body is homeomorphic to a torus if you pinch your nose closed.

Euler’s Formula

Euler’s formula was given by Leonhard Euler, a Swiss mathematician. There are two types of Euler’s formulas: a) For complex analysis, b) For polyhedra. a) Euler’s formula used in complex analysis: Euler’s formula is a key formula used to solve complex exponential functions. Euler’s formula is also sometimes known as Euler’s identity. It is used to establish the relationship between trigonometric functions and complex exponential functions. b) Euler’s formula for polyhedra: For any polyhedron that does not self-intersect, the number of faces, vertices, and edges is related in a particular way, and that is given by Euler’s formula or also known as Euler’s characteristic. Let us learn this formula along with a few solved examples.

What is Euler’s Formula?

a) For complex analysis: The Euler’s form of a complex number is important enough to deserve a separate section. It is an extremely convenient representation that leads to simplifications in a lot of calculations. Euler’s formula in complex analysis is used for establishing the relationship between trigonometric functions and complex exponential functions. Euler’s formula is defined for any real number x and can be written as:

eiix = cos x + iisin x

Here, cos and sin are trigonometric functions, ii is the imaginary unit, and e is the base of the natural logarithm. The interpretation of this formula can be taken in a complex plane, as a unit complex function eiiθ tracing a unit circle, where θ is a real number and is measured in radians.  

This representation might seem confusing at first. What sense does it make to raise a real number to an imaginary number? However, you may rest assured that a valid justification for this relation exists.  Although we will not discuss rigorous proof for this, you may observe the following approximate proof to see why it should be true.

We use the following expansion series for  ex :

b) For polyhedra

Polyhedra are 3D solid shapes whose surfaces are flat and edges are straight.  For example cube, cuboid, prism, and pyramid. For any polyhedron that does not self-intersect, the number of faces, vertices, and edges are related in a particular way.

Euler’s formula for polyhedra tells us that the number of vertices and faces together is exactly two more than the number of edges. Euler’s formula for a polyhedron can be written as:

Faces + Vertices – Edges = 2

Here,

  • F is the number of faces,
  • V the number of vertices, and
  • E the number of edges.


Euler’s Formula Proof

When we draw dots and lines alone, it becomes a graph. We obtain a planar graph when no lines or edges cross each other. We can represent a cube as a planar graph by projecting the vertices and edges onto a plane.

According to Euler’s formula graph theory, the number of dots − the number of lines + the number of regions the plane is cut into = 2.

Solution for the Utilities Problem

Euler’s formula is proved using the utility problem: The three houses are to be connected to the 3 utility gas, water, and electricity. They are to be connected in such a way that no pipe passes over the other pipe. To get a complete cycle with no intersection in any planar graph, we remove an edge to create a tree. This brings down both edges and faces by one, leaving vertices − edges + faces = a constant. We repeat this process until the remaining graph is a tree. Finally, we obtain vertices − edges + faces = 2, i.e. the Euler characteristic. Consider our utility graph and apply Euler’s formula graph theory.

In order to prove that we can’t represent this graph in the form without any intersecting edges, we need to use Euler’s Formula graph theory on this. We find that there are 6 vertices and 9 edges. We need to verify Euler’s formula and check for the number of faces.

F + V – E = 2

F + 6 – 9 = 2

F = 5

If each of the 5 faces had 4 edges bounding them, we get the graph as below.

We notice that we need 10 edges. However, the problem has only 9 edges. By this contradiction, we obtain Euler’s formula proof. This two-dimensional planar graph when inflated into a solid becomes an octahedron. An octahedron has 8 faces, 6 vertices, and 12 edges. Thus with the help of Euler’s formula proof, it is impossible to make the utility connections.

Faces + Vertices − Edges = 28 + 6 − 12 = 2

What Is Euler’s Formula Used For?

F + V − E can equal 2 or 1 and have other values, so the more generic formula is F + V − E = X, where X is the Euler characteristic. We verify Euler’s formula to study any three-dimensional space and not just polyhedra. Euler’s graph theory proves that there are exactly 5 regular polyhedra. We can use Euler’s formula calculator and verify if there is a simple polyhedron with 10 faces and 17 vertices. The prism, which has an octagon as its base, has 10 faces, but the number of vertices is 16.

Verification of Euler’s Formula for Solids

Euler’s formula examples include solid shapes and complex polyhedra. Let’s verify the formula for a few simple polyhedra such as 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

Euler’s Formula Explanation

There are 5 platonic solids for which Euler’s formula can be proved. They are cube, tetrahedron, octahedron, dodecahedron, and icosahedron. Let’s verify Euler’s formula on these complex polyhedra which serve as Euler’s formula examples.

SOLIDFVEF + V – E
Tetrahedron4462
Cube68122
Octahedron86122
Dodecahedron1220302

Solved Examples Using Euler’s Formula

Example 1: Express ei(π/2) in the (a + ib) form by using  Euler’s formula.

Solution:

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.

Example 2: Express 3e5i in the (a + ib) form by using Euler’s formula.
Solution:

Given: θ = 5
Using Euler’s formula,

eiθ = cosθ + isinθ

⟹ e5i = cos5 + i sin5 = 0.284 + i(−0.959) = 0.284 − 0.959i

Now, 
3e5i = 0.852 – 2.877i 

Answer: Hence, 3e5i in the a + ib form is 3e5i = 0.852 – 2.877i.

Example 3: Jack knows that a polyhedron has 12 vertices and 30 edges. How can he find the number of faces?
Solution:

Using Euler’s formula:

F + V − E = 2

F + 12 − 30 = 2

F − 18 = 2

F = 20

Answer: Number of faces = 20.

Example 4: Sophia finds a pentagonal prism in the laboratory. What do you think the value of F + V − E is for it?
Solution:

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

Answer: F + V − E for a pentagonal prism = 2.

FAQs on Euler’s Formula

What is Euler’s Formula for Solids?

For solid shapes, especially polyhedra, the sum of the faces and vertices will be 2 more than their edges. Faces + vertices = edges + 2

What is Euler’s Formula for the Cube?

A cube, also known as a hexahedron has 6 faces, 8 vertices, and 12 edges, and satisfies Euler’s formula. According to Euler’s formula, F + V − E = 6 + 8 − 12 = 2

What is Euler’s formula for Complex Numbers?

Euler’s formula for complex numbers is eiθ = icosθ + isinθ where i is an imaginary number. Many trigonometric identities are derived from this formula.

What is Euler’s Number?

Euler’s number or ‘e’, is an important constant, used across different branches of mathematics has a value of 2.71828.

What Is Euler’s Formula Used For?

Euler’s formula in geometry is used for determining the relation between the faces and vertices of polyhedra. And in trigonometry, Euler’s formula is used for tracing the unit circle.  

What Are the Limitations of Euler’s Formula?

In the field of civil engineering, the crippling stress increases with the decrease in the slenderness ratio. In case it reaches zero, the crippling stress will touch infinity which isn’t practically possible. 

What Is the Purpose of Euler’s Formula?

The purpose of Euler’s formula in a polyhedron is to find the relationship between the number of vertices and edges. This further helps in solving problems related to this property.

What Does Euler’s Formula Mean?

When talking in terms of complex numbers, Euler’s formula states that an imaginary or exponential growth will trace out a circle.

Euler’s Formula and Trigonometry

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