Euler's Formula

Euler's formula was given by Leonhard Euler, a Swiss mathematician. There are two types of Euler's formulas:

• For complex analysis: It 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.
• 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 these formulas along with a few solved examples.

What is Euler's Formula?

The following are two different Euler's formulas used in different contexts.

• Euler's formula for complex analysis: e^ix = cos x + isin x
• Euler's formula for polyhedra: faces + vertices - edges = 2

Let us learn each of these formulas in detail.

Euler's Formula For complex analysis

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:

e^ix = cos x + isin x

Here, cos and sin are trigonometric functions, i 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 e^iθ 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.

Proof

We use the following expansion series for e^x :

e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ... ∞

Now, we assume that this expansion holds true even if x is a non-real number. In a rigorous proof, even this assumption will have to be justified, but for now, let us take its truth to be granted, and use x = iθ.

e^iθ = 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

e^ix = cos x + isin x

Euler's Identity

From the above formula, we have e^ix = cos x + isin x. This formula leads to an identity when x is replaced with π. Then we get

e^iπ = cos π + isin π

e^iπ = -1 + i (0) (as cos π = -1 and sin π = 0)

e^iπ = -1 (or)

e^iπ + 1 = 0

This is known as Euler's identity.

Euler's Formula 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:

F + V - E = 2

Here,

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

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 (H1, H2, and H3) are to be connected to the 3 utility gas (G), water (W), and electricity (E). 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.

SOLID	F	V	E	Euler's Formula: F + V - E
Tetrahedron	4	4	6	2
Cube	6	8	12	2
Octahedron	8	6	12	2
Dodecahedron	12	20	30	2

Let's take a quick look at a couple of examples to understand Euler's formula, better.

Solved Examples Using Euler's Formula

Example 1: Express e^i(π/2) in the (a + ib) form by using Euler's formula.

Solution:

Given: θ = π/2

Using Euler's formula,

e^iθ = cosθ + isinθ

⟹ e^i(π/2) = cos(π/2) + isin(π/2) = 0 + i × 1 = i

Answer: Hence e^i(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,

e^iθ = 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.