Diagram 1.0: Euler’s number

Diagram 2.0: Real Number Line

Euler’s number, denoted as e (approximately 2.71828), is one of the most profound and indispensable constants in mathematics. Much like its famous cousin π (pi), e is irrational and non-repeating, yet it possesses unique properties that make it central to modelling exponential growth and continuous change in the natural and scientific world.

This article explores the origins of e, its mathematical significance, and its real-world applications in fields ranging from finance to epidemiology and artificial intelligence.

Diagram 3.0: Limit Definition of e

 

 

What exactly is e?

 It is emerged from early studies in compound interest. The constant arises from the expression: (refer diagram 3.0). Diagram 4.0 below shows that as n increases ( ,  the value of the expression close to 𝑒≈2.71828.

Diagram 4.0: Table of simulations as , Limits approaches e

 

This limit was first encountered by Jacob Bernoulli during his exploration of interest compounding. However, it was Leonhard Euler who formalized the constant, introduced the notation e, and explored its deeper mathematical properties in the 18th century.

Unlike π, which is rooted in geometry, e finds its power in growth, decay, and change. It forms is the base of natural logarithms and underpins continuous processes in nature and society.

Why is e so Special?

The most defining property of e that seen in calculus is 

  

This means that the derivative of  is the same as the original function.

Diagram 5.0: Exponential Graph and Its Derivative

 

In simpler terms, the rate of growth of at any point is equal to its current value—this makes e the only number with this property. It models processes where growth is continuously proportional to the present amount, a phenomenon found across nature and science.

Real-World Application of e

1. Finance: Continuous Compounding and Global Investments

 

                                                    Diagram 6.0: The Growth of Compounding in Investment     

 

Euler’s number plays a foundational role in continuous compounding, a financial principle where interest is added to the principal at every possible instant. The formula is:

 

Where:

• A (t)     = The amount of money after t years.           
• P          = The initial principal (e.g., $1,000).
• r            = Annual interest rate (0.04).
• t           = Time in years.

Example:

If investing $1,000 at a 4% annual interest rate for 5 years:

This result is higher than standard monthly compounding, showing how e leads to smoother and more efficient growth. Continuous compounding is used worldwide in investment portfolios, savings accounts, and economic forecasting.

2. Epidemiology: Modeling Disease Spread

 

Diagram 7.0: Virus Growth Vector

2. Epidemiology: Modeling Disease Spread

Euler’s number also helps model how infections spread during pandemics, especially during early, uncontrolled stages.

Scenario A: Gradual Exponential Growth

Assume a city starts with 100 cases and the virus spreads at a 10% daily rate. The infection growth can be modelled as:

Where:

Thus, the infection count would grow to approximately 165.

 

⚠️ Clarification

If the scenario were instead that each infected person transmits the virus to two others per day, it’s not a 10% growth model. Instead, we use doubling:

Diagram 8.0: The impact of protective measure to flatten the growth of virus

This significant difference shows how crucial it is to choose the correct exponential model—e fits proportional growth, while base 2 models is binary spread. Health organizations use such models to project the spread and plan interventions like vaccinations, social distancing, and quarantines.

3. Artificial Intelligence: Smarter Learning with Sigmoid Functions

Diagram 9.0: Exponential Growth of Smart Technology Era

 

In machine learning and neural networks, Euler’s number appears in activation functions, particularly the sigmoid function:

Diagram 10.0: Logistic (Sigmoid) Function Representation of the Technology Adoption Lifecycle

 

This function transforms input values into smooth probabilities between 0 and 1, ideal for classification tasks like:

• Email spam filtering
• Image recognition
• Medical diagnosis

 

Diagram 11.0: Comparison of Sigmoid Curve vs Exponential Function

 

 

The graph in Diagram 11.0 compares two growth models in technology trends:

The sigmoid's S-shaped curve allows models to learn non-linear relationships, making e essential in building responsive and adaptive AI systems. Additionally, e appears in optimization techniques such as gradient descent and loss functions (like cross-entropy), supporting intelligent decision-making in machines.

4. Nature’s Signature Constant

Euler’s number doesn’t just reside in science and finance—it’s present in nature’s architecture too:

• The spiral patterns of seashells
• Logarithmic spirals in galaxies
• Population models in ecology
• Radioactive decay in physics

These phenomena all showcase continuous and proportional change, which is elegantly described by e. As such, e rightfully earns the title of the natural base, as it models the very pulse of change in the universe.

Diagram 12.0: The natural phenomena

Euler’s number e is far more than a mathematical curiosity where it is actually the mathematical key to understanding natural growth, decay, learning, and adaptation. From compounding interest and pandemics to artificial intelligence and biological patterns, e describes the processes that evolve and grow over time.

Its unique mathematical properties, especially in calculus and exponential modelling, solidify its role as a foundational pillar in mathematics, science, and technology. Indeed, wherever there is continuous growth or transformation, Euler’s number is silently at work in defining how the world changes, moment by moment.

Prepared by;

Puan Nurul Amira binti Aslan

Mathematics Unit

Date of Input: 15/04/2025 | Updated: 16/04/2025 | hasniah