What is e, Euler's Number?

Introduce the driving question and surface the surprise that a single number rules compound interest, radioactive decay, and bell curves.

• e is approximately 2.71828...
• It appears in growth, decay, probability, and calculus
• Why this number, and where does it come from?

A simulation widget lets learners change the compounding frequency n (1, 2, 4, 12, 52, 365, 10000) for a $1 investment at 100% annual interest and observe the resulting value of (1 + 1/n)^n.

• Start at n = 1: value is 2
• Increase n: the value creeps upward
• Predict: does it stop, blow up, or settle somewhere?

Before seeing the limit revealed, ask the learner to commit to one prediction about the value (1 + 1/n)^n converges to as n → ∞.

• Make one independent prediction

Visualization widget plots the value of (1 + 1/n)^n for n from 1 to one million on a logarithmic scale, with a horizontal line drawn at e ≈ 2.71828 to make convergence visible.

• The curve flattens as n grows
• Values approach but never quite reach e
• e is a limit, not a final value of the sequence

Present the series definition e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ... and show partial sums converging to e, reinforcing that e has multiple equivalent definitions.

• Add the terms one by one
• Partial sums approach 2.71828...
• Different roads, same destination

Explain that for any base b, the derivative of b^x is (ln b) · b^x. Only when b = e does the ln b factor equal 1, so d/dx(e^x) = e^x. This is why e is the natural base for modeling continuous change.

• Derivative of a^x contains a factor of ln(a)
• Set ln(a) = 1 to get a = e
• Only e^x is its own rate of change

A simulation widget lets learners model continuous bacterial growth using P(t) = P₀ · e^(kt) and compare it to non-natural bases such as 2^t or 10^t, seeing that only the e-based formula has rate of change exactly equal to its current size.

• Plug in different growth rates k
• Observe rate of change matches population for e-based model
• Recognize e as the natural unit of continuous growth

Clarify the limits of the intuition: e is not the only way to write exponentials (any base b works as b^x), and in discrete situations (interest paid annually, generations counted in whole steps) e^x is an approximation rather than an exact law.

• Discrete vs continuous modeling matters
• e^x is the continuous limit of (1 + x/n)^n
• Other bases are valid; e is just the natural one

Resolve the driving question directly: e is the limit of (1 + 1/n)^n as n → ∞, approximately 2.71828, and it is the unique base for which the exponential function equals its own derivative, making it the natural language of continuous change.

• e ≈ 2.71828 from the compounding limit
• e is the only base where d/dx(e^x) = e^x
• e unifies growth, decay, probability, and calculus