Euler’s Number

Euler's number, written as e, is an irrational and transcendental constant, with its value being approximately 2.71828. This value is approximate because the decimal expansion goes infinitely without repeating.

We see this value appear naturally in many areas of math and science while describing continuous growth or decay. The Euler's number helps describe any process that requires continuous and smooth change. 

Euler’s number, e is defined by the following equation: 

Euler’s formula for compound interest,

\(A = P^{rt}\)

Where, 

FV stands for future value

PV represents present value of balance or sum

e is the mathematical constant

r  is Interest rate being compounded, and 

t is time in years

Euler’s form connects trigonometry and exponential functions helping in complex analysis. it provides an efficient framework unifying exponential and trigonometric expressions helping simplify mathematical computations. 

For any value of x, the formula is given by:

\(e^ix = cos x + isin x\)

Here,
cos and sin represent the trigonometric ratios functions.
i is the imaginary unit, and
e is the base of the natural logarithm.

Geometrically, this formula can be visualized on a complex plane where \(e^{i𝜃} \) traces a unit circle as the angle θ is measured in radians.

Let us look at the approximate proof of this formula for better understanding.

Let’s start with the taylor series expansion of the exponential function \(e^x\):

\(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \)

This expansion  holds true for real as well as complex numbers.

Substituting x = iθ, where i is the imaginary unit (i² = -1)

\(e^{i\theta} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \cdots \)

Now, we group real and imaginary terms separately:

\(e^{i\theta} = \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots \right) + i\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots \right) \)
 

The trigonometric series is 

\(\cos \theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots \)

\(\sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots \)

Now, substituting these expansions back to our expression for \(e^iθ\), we get

\(e^iθ = cosθ + isinθ\)

Hence, proved \(e^iθ = cosθ + isinθ\)

Euler's identity: From the above formula, we get \(e^ix = cosx + isinx\)

when x =  π, this formula give the identity 

\(e^{i π} = cos π + isin π\)

\(e^{i π} = -1 + i (0) \), because cos π = -1 and sin π = 0

\(e^{i π} = -1\) or  \(e^{iπ} + 1 = 0\)

This is Euler's identity.

A polyhedron is a three-dimensional solid shape. It consists of flat faces and straight edges. Cubes, cuboids, prisms and pyramids are some examples of polyhedra. If a polyhedron does not intersect itself, its vertices, faces and edges follow a specific relationship.

According to Euler's formula, the sum of the number of vertices and faces is exactly two more than the number of edges. Mathematically, it can be expressed as:

\(F + V - E = 2\)

Here, F, V and E represent the number of faces, vertices and edges.

The value of e is approximately 2.718. Euler's number is mostly used to calculate the rate of change or growth, such as in finance, radioactive decay, and so on. Here are some examples

Example 1: Calculate the final amount when $100 is invested for 5 years at a 4% interest rate compounded continuously.

Solution: Euler's formula for compounding interest is
A = Pe^rt

Given, \(P = 100\)
\( r = 0.04\)
\(t = 5\)

\(A = 100e^{0.04 \times 5} \)
\(= 100 × 1.2214\)
\(  = 122.14\)

Therefore, the money in the account after 5 years is $122.14.
 

Example 2: Find the value of e when n = 3 

Solution: Given n = 3,

\(\lim\limits_{x \to \infty} ( 1 + \frac1n)^n \)

\(e =\) \((1 + \frac13)^3 \) \(= 2.37037\)

This is an approximation; Euler's number e ≈ 2.71828.

Given below are a few tips and tricks, that help students better understand and apply the Euler number.
 

• The approximate value of Euler's number is 2.718. For most calculations, you 2.72 can be used for a quick estimate. 
   
• e naturally appears when calculating growth or decay, so understanding its connection to compound interest is important. Remember \(A = P \times e^{rt} \)
   
• Use \(e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \) . this helps you remember that e is related to repeated growth.
   
• The function \(f(x) = e^x\) is unique because its derivative and integral are both the same as the function itself. This makes calculus problems simpler.
   
• For rough calculations, you can remember: \(e \approx 2 + \frac{7}{25} \quad \text{or} \quad e \approx 2.718 \) this helps quickly estimate \(e^x\) without a calculator.

1. Biology: It is used to calculate the exponential growth and decay of organisms

2.  Physics: Radioactive decay follows an exponential pattern modeled using Euler’s number. 

3. Finance: Compound interest calculations in finance reveal growth and decline patterns, which support better risk management

4. Computer Science: It helps study complex algorithms in fields such as machine learning, computer graphics, optimization, and many more.

5. Weather: Euler’s number is used in studying weather changes, such as temperature changes over time, which involves exponential functions.