The Euler form of a complex number is a powerful way to represent complex numbers using the exponential function. It expresses a complex number in the form $re^{i\theta}$, where $r$ is the modulus and $\theta$ is the argument of the number. This form simplifies calculations like multiplication, division, powers, and roots of complex numbers, making it widely used in mathematics, engineering, and physics. In this article, we will explore the Euler form of complex numbers, its formulas, practice questions, properties, and examples.
Euler’s Formula: Definition and Significance
Euler’s formula, introduced by Leonhard Euler, connects complex numbers with exponential and trigonometric functions. It states:
$e^{i\theta} = \cos \theta + i \sin \theta$
This formula simplifies operations on complex numbers such as multiplication, division, and finding powers or roots.
Euler form of a complex number
![Euler form of complex number]()
Euler’s formula establishes a fundamental relationship between trigonometric functions and exponential functions. Geometrically, it bridges two representations of the same unit complex number in the complex plane. This representation is extremely convenient and simplifies many calculations.
The polar form of a complex number is:
$z = r(\cos \theta + i \sin \theta)$
In Euler form, the $(\cos \theta + i \sin \theta)$ part of the polar form is represented using the exponential function:
$z = r e^{i \theta}$
Here, $r = |z|$ is the modulus and $\theta = \arg(z)$ is the argument of the complex number.
Euler's Formula: Derivation
We know the expansion of $e^x$ is:
$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$
Replacing $x$ by $ix$, we get:
$\begin{aligned} e^{ix} &= 1 + \frac{ix}{1!} + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \ldots \\ &= 1 + \frac{ix}{1!} - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \ldots \end{aligned}$
Rearranging real and imaginary terms:
$e^{ix} = \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots \right) + i \left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots \right)$
We notice that the first bracket is the expansion of $\cos x$ and the second bracket is the expansion of $\sin x$, so:
$e^{ix} = \cos x + i \sin x$
Similarly:
$e^{-i\theta} = \cos \theta - i \sin \theta$
Euler forms simplify algebra for complex numbers, especially for multiplication, division, and powers. Any complex number can be expressed in different forms:
Cartesian form: $z = x + iy$
Polar form: $z = r(\cos \theta + i \sin \theta)$
Euler form: $z = |z| e^{i\theta}$
Conversion Between Rectangular, Polar, and Euler Form
This section explains how to convert a complex number between rectangular, polar, and Euler forms. Understanding these conversions helps simplify calculations and makes it easier to work with complex numbers in different mathematical contexts.
Converting Rectangular Form $a + bi$ to Euler Form $re^{i\theta}$
A complex number in rectangular form $z = a + bi$ can be converted to Euler form by first finding the modulus and argument:
$r = \sqrt{a^2 + b^2}, \quad \theta = \tan^{-1}\left(\frac{b}{a}\right)$
Then, the Euler form is: $z = re^{i\theta}$
This conversion allows easier handling of multiplication, division, and powers.
Converting Polar Form $r(\cos\theta + i\sin\theta)$ to Euler Form
The polar form of a complex number:
$z = r(\cos\theta + i \sin\theta)$
can be expressed in Euler form using Euler’s formula:
$z = r e^{i\theta}$
This is useful because Euler form simplifies algebraic operations involving complex numbers.
Converting Euler Form Back to Rectangular or Polar Form
To convert Euler form $z = re^{i\theta}$ back:
Rectangular form: $z = r\cos\theta + i r\sin\theta$
Polar form: $z = r(\cos\theta + i\sin\theta)$
This flexibility allows seamless transition between all three forms depending on the calculation.
Operations on Complex Numbers in Euler Form
In this section, we will discuss how to perform key operations like multiplication, division, powers, and roots on complex numbers using their Euler form. These methods make calculations faster and more straightforward compared to traditional forms.
Multiplication Using Euler Form
For two complex numbers $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$: $z_1 \cdot z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)}$
The moduli multiply and the arguments add, which is simpler than working in rectangular form.
Division Using Euler Form
For division: $\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1-\theta_2)}$
Here, the moduli divide and the arguments subtract, reducing complexity in calculations.
Powers of Complex Numbers – De Moivre’s Theorem in Euler Form
Using Euler form, raising a complex number to the $n^{th}$ power is straightforward: $(z)^n = (re^{i\theta})^n = r^n e^{in\theta}$
This is a direct application of De Moivre’s theorem.
Roots of Complex Numbers Using Euler Form
Finding $n^{th}$ roots becomes easy: $z^{1/n} = r^{1/n} e^{i(\theta + 2k\pi)/n}, \quad k = 0,1,2,\dots,n-1$
This generates all $n$ distinct roots efficiently.
Properties of Euler Form of Complex Number
This section covers how to perform key operations like multiplication, division, powers, and roots on complex numbers using their Euler form.
1. Multiplication of Two Complex Numbers
Let $z = |z| e^{i\theta_1} \quad \text{and} \quad w = |w| e^{i\theta_2}$
Multiplying these two complex numbers:
$z \cdot w = |z| e^{i\theta_1} \cdot |w| e^{i\theta_2} = |z| \cdot |w| \, e^{i(\theta_1+\theta_2)}$
This shows that in Euler form, the moduli multiply and the arguments add.
2. Division of Two Complex Numbers
For two complex numbers in Euler form:
$z = |z| e^{i\theta_1}, \quad w = |w| e^{i\theta_2}$
the division is given by:
$\frac{z}{w} = \frac{|z|}{|w|} \, e^{i(\theta_1 - \theta_2)}$
Here, the moduli divide and the arguments subtract, simplifying division significantly.
3. Logarithm of a Complex Number
For a complex number in Euler form:
$z = |z| e^{i\theta}$
the natural logarithm is:
$\begin{aligned} \log_e(z) &= \log_e\left(|z| e^{i\theta}\right) \\ &= \log_e(|z|) + \log_e\left(e^{i\theta}\right) \\ &= \log_e(|z|) + i \arg(z) \end{aligned}$
This property makes handling powers and roots of complex numbers much easier.
4. Euler’s Identity
Euler’s identity is one of the most elegant equations in mathematics:
$e^{i \pi} + 1 = 0 \quad \text{or equivalently} \quad e^{i\pi} = -1$
It beautifully connects five fundamental mathematical constants: $e$, $i$, $\pi$, $1$, and $0$, and demonstrates the deep relationship between exponential and trigonometric functions.
Euler’s identity is derived by substituting $x = \pi$ in Euler’s formula:
$e^{ix} = \cos x + i \sin x$
and simplifying, showing its connection to the unit circle in the complex plane.
Relationship Between Modulus, Argument, and Exponential Form
In Euler form $z = re^{i\theta}$, the modulus $r$ gives the distance from the origin, and $\theta$ gives the angle with the positive real axis. This geometric connection is key in physics and engineering applications.
Comparison with Polar and Rectangular Forms
Rectangular form $a+bi$: easy for addition and subtraction.
Polar form $r(\cos\theta + i \sin\theta)$: better for multiplication/division, but slightly longer.
Euler form $re^{i\theta}$: most compact and efficient for algebraic operations, exponentiation, and root calculations.
Solved Examples Based On the Euler Form of a Complex Number
Example 1: Euler's form of $z=-\sqrt{3}-i$ is
Solution:
Euler's Form of a Complex Number -
$z=r e^{i \theta}$
where r denotes the modulus of z and $\theta$ denotes the argument of z .
Now, $z=-\sqrt{3}-i$
$\therefore r=|z|=\sqrt{3+1}=2$
and Z being in 3rd quadrant, its arg(z) will be :
$-\pi+\tan ^{-1}\left|\frac{-1}{-\sqrt{3}}\right|=-\pi+\frac{\pi}{6}=-\frac{5 \pi}{6}$
$Z=2 e^{-i \frac{5 \pi}{6}}$
Example 2: $\mathrm{z}=\frac{16}{1+i \sqrt{3}}$ , its Euler form is?
Solution:
We simplify z, and for that, we normalize the denominator
$\mathrm{z}=\frac{16}{1+\mathrm{i} \sqrt{3}} \cdot \frac{1-\mathrm{i} \sqrt{3}}{1-\mathrm{i} \sqrt{3}}=4(1-\mathrm{i} \sqrt{3})$
Now we see it lies in the 4th quadrant, so the argument is going to be -ve.
First we find r = |z| = 4·2=8
$\theta=\arg (\mathrm{z})=\tan ^{-1}(-\sqrt{3})=\frac{-\pi}{3}$
So euler form $=r e^{\mathrm{i} \theta}=8 \mathrm{e}^{-\frac{\pi}{3} \mathrm{i}}$
Hence, the answer is $8 e^{-\frac{\pi}{3}}$.
Example 3: Real part of $e^{e^{10}}$ is equal to:
Solution:
$e^{e^{i \theta}} = e^{\cos \theta + i \sin \theta} = e^{\cos \theta} \cdot e^{i \sin \theta}$
$= e^{\cos \theta} \cdot [\cos (\sin \theta) + i \sin (\sin \theta)]$
$= e^{\cos \theta} \cdot \cos (\sin \theta) + i , e^{\cos \theta} \cdot \sin (\sin \theta)$
$\text{(Real part)} \quad \text{(Imaginary part)}$
Example 4: Euler's form of $z=\frac{1-7 i}{(2+i)^2}$ will be
Solution:
Euler's Form of a Complex Number -
$z=r e^{i \theta}$
where $r$ denotes the modulus of $z$ and $\theta$ denotes the argument of $z$.
Now,
$Z=\frac{1-7 i}{(2+i)^2}=\frac{1-7 i}{4-1+4 i}=\frac{1-7 i}{3+4 i} \times \frac{3-4 i}{3-4 i}=\frac{3-28-25 i}{25}$
$\therefore z=-1-i$
$\therefore r=|z|=\sqrt{1+1}=\sqrt{2} \text { and } \arg (\mathrm{z})=\tan ^{-1}\left|\frac{-1}{-1}\right|-\pi$
$r=\sqrt{2} \text { and } \arg (\mathrm{z})=\frac{-3 \pi}{4} \Rightarrow z=\sqrt{2} e^{-i \frac{3 \pi}{4}}$
Example 5: If $z$ and $w$ are two complex numbers such that $|z w|=1$ and $\arg (z)-\arg (w)=\frac{\pi}{2}$ then $\bar{z} w=-i$ equals to:
Solution:
Euler's Form of a Complex Number -
$z=r e^{i \theta}$
- wherein
r denotes the modulus of $z$ and $\theta$ denotes the argument of $z$.
Polar Form of a Complex Number -
$z=r(\cos \theta+i \sin \theta)$
- wherein
$r=$ modulus of $z$ and $\theta$ is the argument of $z$
Now,
$|z w|=1$ and $\arg (z)-\arg (w)=\frac{\pi}{2}$
Let $|z|=r$ $\Rightarrow z=r e^{i \theta}$
$|\omega|=\frac{1}{r}$ $=>\omega=\frac{1}{r} e^{i \phi}$
$\arg (z)-\arg (w)=\frac{\pi}{2}$
$\theta-\phi=\frac{\pi}{2}$
$\theta=\frac{\pi}{2}+\phi$
$z \bar{\omega}=r e^{i \theta} \cdot \frac{1}{r} e^{-i \phi}$
$=r e^{i(\theta-\phi)}$
$=r e^{i\left(\frac{\pi}{2}+\phi-\phi\right)}$
$=r e^{i\left(\frac{\pi}{2}\right)}$
$=\cos \left(\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{2}\right)$
$=0+i .1 = i$
