Where does e-driven growth appear in nature and finance?

Opens the investigation by juxtaposing two scenes: a bank statement showing continuous-compounding growth and a Petri dish where a colony expands hour by hour. Both contexts use e, prompting the driving question.

• Compound interest formulas and biological growth formulas both contain e
• The same number cannot be an accident in two unrelated domains
• We will look for a single underlying rule

A single commitment question: why does e appear in both money growth and population growth? The learner picks the explanation that feels most plausible before any evidence is shown.

• Commit to one hypothesis before seeing evidence
• Surface the common assumption that e is a financial quirk

A simulator where the learner increases the number of compounding periods per year from 1 to 12 to 365 to 'continuous'. The growth multiplier converges toward the value produced by e.

• More frequent compounding yields slightly more growth
• Continuous compounding produces a fixed limit, not infinity
• The limiting base is exactly e

A visualization of a bacterial colony where the per-cell division rate can be tuned. The total count traces a curve that visually matches the continuous-compounding curve from the previous scene.

• Each cell divides at a rate proportional to how many cells exist
• The count-vs-time curve is the same shape as the money curve
• Biology and finance produce visually identical graphs

Presents the unifying equation dN/dt = k·N in plain language. Explains that whenever the instantaneous change is a fixed fraction of the current size, the solution is N(t) = N0·e^(kt). Money, cells, and decaying atoms all obey this rule for different signs of k.

• dN/dt = k·N is the single mechanism
• Its solution is N(t) = N0·e^(kt)
• Sign of k selects growth (k>0) or decay (k<0)

Shows a measured decay curve for a short-lived isotope alongside the formula N(t) = N0·e^(-λt). Reinforces that the same e-based exponential describes loss as well as gain, completing the picture across physics, biology, and finance.

• Decay is negative exponential growth
• Half-life emerges from the same e-based equation
• Carbon-14 dating rests on this curve

Shows two cases where e-based growth fails: a population approaching its food-limited carrying capacity, and a savings account eroded by inflation. Introduces the logistic curve as the honest boundary of the lesson's claim.

• Real populations saturate and become logistic, not purely exponential
• Inflation can flip finance from growth to decline
• The rule holds only while the proportionality between rate and amount remains constant

The learner is given a brand-new scenario — viral spread in a closed network — and adjusts the per-contact infection probability to predict the curve. The widget reveals whether their prediction matches an e-shaped growth or a logistic one.

• Recognize the rate-equals-amount pattern in a fresh context
• Distinguish pure exponential from saturating growth
• Justify the choice with the shared rule

Closes the investigation by directly answering the driving question: e is not borrowed from finance into biology, nor the other way around — it is the mathematical signature of self-proportional change, and money, microbes, and atoms all write the same equation.

• e is the natural base of continuous self-proportional change
• Finance, biology, and physics share the rule, not just the number
• The lesson's promise: one mechanism, one curve, one constant