Euler’s Formula: Complex Numbers, Equations & Applications

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Euler’s Formula is a fundamental mathematical formula specifying the relationship between the complex exponential and trigonometric functions.

Euler’s Formula consists of two different equations, including one for complex analysis and the other for polyhedrons.
It has a wide range of applications in mathematics, physics, chemistry, and engineering.
The equation is named after the legendary mathematician Leonhard Euler.
It describes a counterclockwise turn along the unit circle in the complex plane.
A complex number is any number in the form a+ib, where i is an imaginary number and a and b are real numbers.
Euler’s formula helps in solving de Moivre’s theorem and trigonometric additive identities.
It helps in solving complex mathematical problems.
The formula also helps in raising complex numbers to different powers.

Read More: Trigonometry Table

Table of Content

Euler’s Formula Equations
Solved Examples of Euler’s Formula Equations
What are Polyhedrons?
Euler’s Formula for Polyhedron
Solved Examples of Euler’s Formula for Polyhedron
Euler’s Formula for Cube
Applications of Euler’s Formula
Things to Remember
Sample Questions

Key Terms: Euler’s Formula, Complex Numbers, Trignometric Function, Exponential Functions, Euler’s Identity, Polyhedron, Faces, Vertices

Euler’s Formula Equations

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Euler’s formula helps state a relationship between trigonometric functions and complex exponential functions.

Euler’s Formula for any real number x, in complex exponential analysis is given by: e^ix = cos x + i sin x

where, x = real number, e = base of natural logarithm, sin x & cos x = trigonometric functions and i = imaginary unit

The expression cos x + i sin x is also known as cis x.
It is often referred to as the Euler’s identity and has significant application in the field of Mathematics and Engineering.
The relationship between the imaginary complex number and exponential growth is represented by a circle.
It establishes important relationships in Mathematics and thus reduces the complications of certain mathematical calculations.
In the Euler’s formula when x = 0 then e⁰= cos 0 + i sin 0 which gives us 1 = 1.

Read More: Trapezoid Formula

Solved Examples of Euler’s Formula Equations

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Given below are some examples of calculating the complex equation using the euler’s formula.

Example 1: Find the value of e^iπ/2.

Ans. Given e^iπ/2

Using Euler’s formula: e^ix = cos x + i sin x

e ^iπ/2 = cos π/2 + i sin π/2
e ^iπ/2 = 0 + i × 1
e ^iπ/2 = i

Example 2: Find the value of e^iπ.

Ans. Given e^iπ

Using Euler’s formula: e^ix = cos x + i sin x

e^iπ = cos π + i sin π
e^iπ = -1

Read More: Eccentricity

What are Polyhedrons?

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Polyhedron refers to the three-dimensional figures that are formed by the combination of polygons. A polygon is a figure that has many sides. Polyhedrons include different forms like a tetrahedron, pentahedrons, etc. It is classified based on the number of faces.

Euler’s characteristics are noted provable in all polyhedrons.
Polyhedrons that have no gaps in the structure, which must not intersect itself and have distinct vertices, are required for Euler’s characteristics.
With the figures becoming complex, Euler’s characteristics will provide us with complex numbers.
This makes Euler’s formula provable with the help of polyhedrons.

Read More: Bayes Theorem Formula

Euler’s Formula for Polyhedrons

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Euler’s formula is used to define the relationship among faces, vertices, and edges of a Polyhedron. This is often referred to as Euler’s characteristics.

Euler’s formula states that the number of vertices and faces together is exactly two more than the number of edges.
The formula is as follows:

Number of faces + number of vertices - number of edges = 2

F + V = E + 2

Where, F = number of faces
V = number of vertices
E = number of edges

Solved Examples on Euler’s Formula for Polyhedrons

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Given below are some examples of calculating the polyhedra equation using the euler’s formula.

Example 1: If a polyhedron contains 10 faces and 28 edges, then identify the name of the polyhedron.

Ans. Given, Number of faces = F = 10

Number of edges = 28
Using Euler’s formula of solids,
F + V = E + 2
10 + V = 28 + 2
V = 30 – 10
V = 20

Example 2: If a polyhedron contains 12 faces and 20 edges, then identify the name of the polyhedron.

Ans. Given, Number of faces = F = 12

Number of edges = 20
Using Euler’s formula of solids,
F + V = E + 2
12 + V = 20 + 2
V = 22 – 12
V = 10

Euler's Formula For Cube

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Euler’s characteristics are often proved with the help of polyhedrons like tetrahedrons, cubes, octahedrons, and dodecahedrons. To make our proof look livelier and easier to understand, we can take a cube as an example,

Formula to be proved: Number of faces + number of vertices - number of edges = 2

L.H.S.

Number of faces of a cube = 6
Number of Vertices of a cube = 8
Number of edges of a cube = 12
Applying the above data into the formula, we have
6 + 8 - 12 = 14 -12 = 2 = R.H.S

Here, L.H.S = R.H.S

Hence, Euler’s characteristics were proved with the help of a cube.

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Applications of Euler’s formula

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Euler’s formula has a wide application in both engineering and mathematical fields.

Euler’s formula (Euler’s identity) is applicable in reducing the complication of certain mathematical calculations that include exponential complex numbers.
In the field of engineering, Euler’s formula works on finding the credentials of a polyhedron.
It helps in solving the Pythagoras theorem.
By applying the value of (number of) faces, vertices, or edges (any two), we can find the single missing value.
The value would be the number of faces, vertices, or edges.
Euler’s formula works on the AC principle and is used in electronic devices.

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Things to Remember

Euler's Formula is defined as a formula specifying the relationship between the exponential and trigonometric equations.
It has been proved right in the case of polyhedrons such as cubes, tetrahedrons, octahedrons, etc.
Euler's formula bridges two complex numbers having the same unit in a complex plane.
It is applicable in reducing the complication of certain engineering equations.
Euler's formula for complex equations is given by e^ix = cos x + i sin x
The formula for polyhedra: Number of faces + number of vertices - number of edges = 2.

Read More: Limit Formula

Sample Questions

Ques. What is a polyhedron? (2 marks)

Ans. Polyhedron refers to the 3-dimension figures formed by the combination of polygons (figures with many sides). Polyhedrons include different forms like a tetrahedron, pentahedrons, etc. This is classified based on the number of faces. Decahedron has ten faces, whereas Hexahedron has six faces, and it goes on.

Ques. State the different formulas stated by Euler? (3 marks)

Ans. Euler has two different formulas, one for the complex numbers and the other for Polyhedrons. The formula for complex numbers is: e^ix= cos x + i sin x. The formula for Polyhedrons is: Number of faces + number of vertices - number of edges = 2

For example in case of polyhedron, If a polyhedron contains 20 faces and 48 edges, then identify the name of the polyhedron.

Solution: Given the number of faces = F = 20

Number of edges = 48
Using Euler’s formula for solids,
F + V = E + 2
20 + V = 48 + 2
V = 50 – 20
V = 30

Ques. Prove Euler’s formula on Octahedron? (3 marks)

Ans. We know that Octahedron has eight faces, six vertices, and 12 edges .......... (1)

Also, we have Euler’s formula,
Number of faces + number of vertices - number of edges = 2
Applying (1) to Euler’s formula, we get
8 + 6 -12 = 2, which gives,
L.H.S. = R.H.S.
Hence, Euler’s formula is proved with the help of Octahedron.

Ques. With the help of Euler’s formula, find the number of edges in Tetrahedron, where the number of faces and number of vertices is 4 for each, respectively? (3 marks)

Ans. We have Euler’s formula, Number of faces + number of vertices - number of edges = 2

We are also given that, Number of faces in Tetrahedron = 4
Number of vertices in Tetrahedron = 4
Applying this to Euler’s formula, we get,
4 + 4 -number of edges in Tetrahedron = 2
i.e., 8 – 2 = number of edges in Tetrahedron.
Therefore, the number of edges in Tetrahedron = 6.

Read More: Sin2x Formula

Ques. How can we depict Euler’s formula with the help of a graph? (3 marks)

Ans. For depicting Euler’s formula with the help of a graph, we have to first draw non-intersecting lines to form a three-dimensional image of a Polyhedron on the graph paper. According to the graph theory stated by Euler, the sum of the number of dots of the figure and the number of regions the plain is cut into when reduced from the number of lines in the figure will give you two as the answer.

Euler’s Formula

Ques. Using Euler’s formula (Euler’s identity), solve e^ix, where a= 30? (3 marks)

Ans. We have Euler’s formula,

e^ix= cos x + i sin x
Applying x = a = 30, we get,
eⁱ³⁰ = cos 30 + i sin 30
eⁱ³⁰ = √3/2 + i 0.5

Ques. Prove Euler’s formula for a polyhedron with 12 faces, 20 vertices, and 30 edges? (2 marks)

Ans. We have Euler’s formula, Number of faces + number of vertices - number of edges = 2

Here we have,
12 + 20 – 30 = 32-20
Thus giving us value equal to 2
Hence, Euler’s formula is proved.

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Ques: Raju was asked to draw a Polyhedron on his Mathematics exam. He draws a figure that has 10 faces, 20 edges, and 15 vertices. Did Raju draw a Polyhedron? Answer with the help of Euler’s formula? (3 marks)

Ans: According to Euler’s formula, in a Polyhedron,

Number of faces + number of vertices - number of edges = 2

Here the given figure has 10 faces, 20 edges, and 15 vertices.
Applying this to Euler’s formula, we get
L.H.S. = Number of faces + number of vertices - number of edges
10 + 15 – 20 ‡ 2
Since L.H.S ‡ R.H.S
We can say that such a Polyhedron doesn’t exist at all.

Ques. Express 3e⁶ⁱ in the (a + ib) form by using Euler's formula? (3 marks)

Ans. Given: θ = 6

Using Euler's formula,e^iθ = cosθ + isinθ
e⁶ⁱ = cos 6 + i sin 6
This gives us value equal to 0.96 – 0.279 i
Now,
3e⁶ⁱ = 2.852 - 0.837i

Ques. Express e¹⁰ⁱ in the general form using Euler's formula? (2 marks)

Ans. We have, x = 10

Using the formula, we get,
e^ix= cos x + i sin x
cos 10 + i sin 10
-0.839+ i (-0.544)
-0.839 -0.544i

Ques. Jack knows that a polyhedron has 16 vertices and 30 edges. How can he find the number of faces? (2 marks)

Ans. Using Euler's formula:

F + V − E = 2
F + 16 − 30 = 2
F − 14 = 2
F = 16
Number of faces = 16.

Ques. Sophia finds an octahedral prism in the laboratory. What do you think the value of F + V − E is for it? (2 marks)

Ans. A octahedral prism has 8 faces, 12 edges, and 6 vertices.

Let's apply Euler's formula here,
F + V − E
8 + 6 − 12 = 2
F + V − E for an octahedral prism = 2.

Ques. Find the value of e^i2π? (2 marks)

Ans. Given e^i2π

Using Euler’s formula,
e^ix = cos x + i sin x
e^i2π = cos 2π + i sin 2π
e^_i2π = 1

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Euler’s Formula: Complex Numbers, Equations & Applications

Euler’s Formula Equations

Solved Examples of Euler’s Formula Equations

What are Polyhedrons?

Euler’s Formula for Polyhedrons

Solved Examples on Euler’s Formula for Polyhedrons

Euler's Formula For Cube

Applications of Euler’s formula

Things to Remember

Sample Questions

CBSE X Related Questions

1.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

2.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

3.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

4.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

5.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

6.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

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Euler’s Formula: Complex Numbers, Equations & Applications

Euler’s Formula Equations

Solved Examples of Euler’s Formula Equations

What are Polyhedrons?

Euler’s Formula for Polyhedrons

Solved Examples on Euler’s Formula for Polyhedrons

Euler's Formula For Cube

Applications of Euler’s formula

Things to Remember

Sample Questions

CBSE X Related Questions

1.The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

2.Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

3.In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

4.PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

5.Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent. Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

6.An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

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SUBSCRIBE TO OUR NEWS LETTER

1.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

2.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

3.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

4.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

5.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

6.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is