The Natural Exponential eˣ

This lesson introduces the number e and the natural exponential function y = eˣ. The constant e is arguably the most important number in A-Level Mathematics after π, and it arises naturally from calculus. Understanding eˣ is essential for differentiation, integration, and modelling.

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What Is the Number e?

The number e is an irrational constant approximately equal to:

e ≈ 2.71828...

It is defined as the limit:

e = lim(n→∞) (1 + 1/n)ⁿ

You can see how this works by computing (1 + 1/n)ⁿ for increasing n:

n	(1 + 1/n)ⁿ
1	2
10	2.5937...
100	2.7048...
1000	2.7169...
10000	2.7181...
1000000	2.71828...

As n increases, the value approaches e = 2.71828...

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Why Is e Special?

The function y = eˣ is the unique exponential function whose gradient equals its own value at every point:

If y = eˣ, then dy/dx = eˣ