Euler’s Form of a Complex Number

The Euler form of the complex number is one of the important concepts of the Complex number. For many students, solving complex numbers may be quite difficult, especially when it has to be computed in a polar form. The importance of the Euler form cut across complex numbers, trigonometric functions, and hyperbolic functions. The scope of this article is on how it relates to complex numbers.

Generally, computing the polar form of complex numbers may be a bit complex. The multiplication and division may seem a bit difficult. Expressing the Euler form of complex numbers simply requires the use of the exponential symbol e, which is known as the base of the natural logarithm ln. The basic characteristic of the Euler form for complex numbers is its simple expression.

What’s Euler’s Formula?

The Euler formula is an expression given as e^ix. This equation is highly connected to the trigonometric function cosx + isinx. That is;

e^ix = cosx + isinx

where;

e is the natural logarithm i is the imaginary unit which is given as

x is a real number which could be a negative integer, zero, fraction or natural number.

From the above formula, the value of e^ix is the same as the addition of the trigonometric function. The above formula is the condition on which Euler’s formula calculator is based.

We will consider some common examples to prove that e^ix = cosx + isinx is true. It is important students leave their calculator in Radians when dealing with π

Now let x = 0,

We will try to see if the value of eⁱ⁰ will be equal to cos0 + isin0

Now eⁱ⁰ = e⁰ = 1

While;

cos0 + isin0 = 1 + 0 = 1

clearly, eⁱ⁰ = cos0 + isin0 = 1

Based on the above, it means that

For x = 1;

eⁱ¹ = cos1 + isin1

for x = 2,

eⁱ² = cos2 + isin2

for x = π;

e^iπ = cos + isinπ

for x = 2π;

e^i2π = cos2 + isin2π = 1

The above result establishes that whatever the logarithm expression is no different from its trigonometric equivalent.

The Euler Formula Identity

The Euler Identity is the ability to get the expression

e^ix + 1 = 0

now for x = π,

e^iπ = cosπ + isinπ

e^iπ = – 1 + 0

The Euler’s formula for complex numbers is represented as;

e^iπ + 1 = 0

so it is the application of the radian as x can the Euler identity be achieved. That being said, we will consider how the Euler formula can be used to solve the polar form of complex numbers easily

Euler’s Form of a Complex Number

The Euler form of a complex number helps mathematicians calculate complex numbers more easily.

Now recall that the polar form of a complex number is;

Z = cosrϴ + isinrϴ

Z = r(cosϴ + isinϴ)

Now we can see that cosϴ + isinϴ = cosx + isinx. Where x = ϴ

So that means

cosϴ + isinϴ = cosx + isinx = e^ix = e^iϴ where x = ϴ

with this being the case we will have our polar form of complex number which is given as Z = r(cosϴ + isinϴ) will become

z = re^iϴ

now we will consider an example to show what this expression.

Example 1

Express the complex number z =7(cos60⁰ + isin60⁰)

in the Euler form

solution

the Euler form of the complex number is given as

z = re^iϴ

clearly, ϴ = 60⁰, r = 7

Therefore, z = 7eⁱ⁶⁰

While the above example is easy, the importance of the Euler formula comes in place when computing the division and multiplication of the complex polar number.

How to Multiply and Divide the complex number

Example 2

Find the multiplication of the complex numbers in both the polar form and Euler form

Z1 = 5(cos30⁰ + isin30⁰)

Z2 = 5(cos30⁰ + isin30⁰)

We have computed this particular example before and the solution derived is

Z1 X Z2 = 5 X 5(cos(30 + 30) + isin(30 +30)) = 25(cos60⁰ + isin60⁰)

Now the expression of the Euler form in multiplication format is given as

So with this being the case, we will have the multiplication of the two complex numbers Z1Z2

Example 3

Divide the complex numbers Z1 = 6(cos45⁰ + isin45⁰) by Z2 = 2(cos30⁰ + isin30⁰)

Now for the polar form division it would be

Z1/Z2 = 6/2(cos(45 – 30) + isin(45 – 30))

Z1/Z2 = 3(cos15⁰ + isin15⁰)

Now the Euler form of the complex number division

The difference between the formula of division

Example 4

Find the multiplication of the two complex numbers Z1 = 5(cos50⁰ + isin50⁰) and Z2= 7(cos25⁰ + isin25⁰). Leave your answer in the Euler formula form. Also, find their division.

Solution

The Euler formula is given as

Conclusion

THE Euler form of the complex number is one of the easiest concepts in the Complex number series. Its importance is tied to its easier method for most students to find the multiplication and division of complex numbers. However, its richness goes beyond that when considered from a holistic point of view. The formula is extremely important in showing the relationship of complex numbers to trigonometry. It expresses the importance of the cosine and sines and how they help in establishing the complex number theory.

This article strategically considered some important examples to drive home the concept of the Euler formula for complex numbers. The aim is to make it easy for students to find complex numbers and rearrange the solution in the expression they want. The Euler formula is very important in establishing the De Moivre theorem o the complex number. The De Moivre theorem help students find the multiplication of a complex number by itself up to the nth term.