What is e in math? The number 2.718 behind natural growth

In math, e is a constant, about 2.71828, that appears whenever something grows or shrinks in proportion to its own size, continuously. It is what one dollar becomes after a year at 100% interest if the interest is added not yearly or daily but at every instant. Like pi, e is irrational, its digits never ending or repeating, and it is the base of the natural logarithm and of the exponential function that calculus finds simplest.

In short

• Compounding 100% growth more and more often gives 2, then 2.61, then 2.7146, and closes in on e = 2.71828…; e is that limit.
• The curve y = e^x has a slope equal to its own height at every point, which makes e the natural base for growth and decay.
• e turns up in radioactive decay, the bell curve and puzzles of pure chance, and joins pi, i, 1 and 0 in Euler's identity.

Where does e come from?

The cleanest way to meet e is through money, as an illustration rather than a real offer. Put 1 dollar in an account that pays 100% interest a year. Paid once at the end of the year, you have 2 dollars. Now split the year: pay 50% every six months, and the second payment earns interest on the first, giving 1.5 × 1.5 = 2.25 dollars. That is compound interest, and the more often it compounds, the more you end with. The question Jacob Bernoulli asked in 1683 was whether that keeps growing without limit.

It doesn't. With n payments a year, each of 100% ÷ n, the year-end amount is (1 + 1/n)^n:

Compounded	n	Amount after one year
Yearly	1	2.00000
Twice a year	2	2.25000
Quarterly	4	2.44141
Monthly	12	2.61304
Weekly	52	2.69260
Daily	365	2.71457
Hourly	8,760	2.71813
Every minute	525,600	2.71828
Continuously	no limit	2.718281828…

Going from yearly to monthly adds 61 cents; going from daily to every minute adds less than a cent. The amounts creep toward a ceiling, and that ceiling, the value of (1 + 1/n)^n as n grows without end, is e. Bernoulli showed it lies between 2 and 3, the MacTutor History of Mathematics archive records, and if that is taken as e's definition, it was the first time a number was defined by a limit.

How is e written and calculated?

The letter comes from Leonhard Euler, who began using it in 1727 or 1728 in an unpublished paper on cannon fire, and in a letter to Christian Goldbach in 1731. MacTutor calls the story that he chose it for his own name ridiculous; e may simply have been the next vowel free after a, which he was already using. In his Introductio in analysin infinitorum of 1748, Euler gave e to 18 decimal places: 2.718281828459045235.

He didn't say how he got them, but a second recipe would have done it: an infinite sum that reaches e much faster than compounding does: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + 1/720 and so on, where each denominator is the previous one times the next whole number (1 × 2, then 1 × 2 × 3, then 1 × 2 × 3 × 4). Adding the first seven terms gives 2.71806; the first eight give 2.71825. About 20 terms are enough for Euler's 18 decimals.

Euler also proved that e is irrational, so no fraction equals it, and in 1873 Charles Hermite proved it is transcendental: like pi, it is not the solution of any polynomial equation with whole-number coefficients. That puts it among the stranger kinds of real numbers, though one of the most useful.

Why is e^x its own derivative?

Every exponential curve, 2^x or 3^x or 10^x, has a slope that is proportional to its height: the higher it gets, the steeper it climbs, which is the definition of exponential growth. The only question is the constant of proportion. For 2^x, the slope at any point is about 0.693 times the height. For 3^x it is about 1.099 times the height. Somewhere between 2 and 3 there must be a base for which that constant is exactly 1, and that base is e.

So the curve y = e^x has a slope equal to its own value everywhere: at x = 0 it stands at height 1 and climbs at slope 1; at x = 2 it stands at about 7.39 and climbs at 7.39. In the language of calculus, the derivative of e^x is e^x, the only function (apart from multiples of it) that is unchanged by taking its rate of change. That is why calculus, physics and engineering write almost every growth and decay with base e, and why logarithms to base e are called natural logarithms, written ln. The constants 0.693 and 1.099 above are just ln(2) and ln(3), the natural logarithms of 2 and 3, and 0.693 is where the rule of 70 for doubling times comes from.

Where does e show up?

Anything whose rate of change depends on how much of it there is follows e. Some examples:

• Decay. A radioactive sample loses a fixed fraction of its atoms per second, so the amount left after time t is N₀ × e^(−λt), where N₀ is the starting amount and λ, the decay constant, sets the pace. A cooling cup of coffee and a drug clearing from the blood follow the same shape, approximately.
• Continuous growth. Money compounding continuously at a rate r for t years grows by a factor of e^(rt). At 5% for 10 years that is e^0.5, about 1.649.
• The bell curve. The formula for the normal distribution, the shape of heights and measurement errors, is built on e^(−x²/2), along with a factor involving pi.
• Pure chance. If n people check their hats and get them back at random, the chance that nobody gets their own hat approaches 1/e, about 36.8%, as n grows. It settles fast: with 5 people it is 36.7%, with 10 people 36.788%. The same 37% appears in the "secretary problem", where the best strategy for choosing the top candidate from a stream is to watch the first 37% without picking and then take the next one better than all of those.

One formula ties all of this together: Euler's identity, e^(iπ) + 1 = 0, which links e to pi, to the imaginary unit i (the square root of −1), and to 1 and 0. It falls out of a formula Euler found connecting exponentials to circles: raising e to an imaginary power makes it travel around a circle rather than grow.

What is e easily confused with?

Three different things share the name or the letter. Euler's number, e = 2.71828…, is the subject of this note. Euler's constant, also called the Euler–Mascheroni constant and written γ, is a different number, about 0.5772, from the theory of sums of fractions. And the E on a calculator display, as in 6.02E23, isn't e at all: it is shorthand for "times 10 to the power", here 6.02 × 10²³.

The other misconception is that e was chosen, the way a unit is chosen. It wasn't. Unlike the number 10, which is a convenient base because humans have ten fingers, e is forced on anyone who asks how a quantity changes when its growth depends on itself. That is why it keeps appearing in fields that have nothing to do with money.

Check yourself

• Without looking back, explain why compounding more and more often gives an amount that approaches a limit instead of growing forever.
• What property of the curve y = e^x makes e the "natural" base for growth?
• Name two places outside finance where e appears, and say what is changing in proportion to itself in each.

Sources

• The number e — MacTutor History of Mathematics, University of St Andrews
• Exponential and logarithmic functions — OpenStax, Calculus Volume 1, Rice University, 2016
• Derivatives of exponential and logarithmic functions — OpenStax, Calculus Volume 1, Rice University, 2016
• e (mathematical constant) — Wikipedia