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What Is e in Math?
e is a fixed number, approximately 2.718281828459045, known as Euler’s number. It is the value that continuous growth converges to, which is why it appears in compound interest, population change and radioactive decay alike. This guide covers what e means, where the number comes from, how it differs from π, how to get it out of a calculator, and why the natural logarithm is built on it.
7 min read
What does e mean in math?
e is a specific number, roughly 2.71828, in the same way that π is a specific number roughly 3.14159. It is not a variable and it does not stand for anything you get to choose. Its full value begins:
e = 2.718281828459045…
The decimals never repeat and never end. e is irrational, so no fraction equals it exactly, and it is also transcendental, which means no polynomial equation with whole-number coefficients has it as a root.
It is called Euler’s number, after Leonhard Euler, who gave it the letter in the 1720s. The digits 1828 appearing twice in a row is a coincidence and a useful one: most people who remember e at all remember it as “two point seven, eighteen twenty-eight, eighteen twenty-eight”.
What makes it worth a name is not the digits. It is that e is the growth rate of anything that grows in proportion to how much of it there already is, which is why it turns up in compound interest, in populations, in radioactive decay and in the natural logarithm.
Where e comes from: compounding, taken to the limit
e is not a measurement. It is the answer to a question about compound interest, and you can watch it appear.
Put $1 into an account paying 100% a year. Pay the interest once at the end and you finish with $2. Pay half of it at six months and half at twelve, and the first half earns interest for the second half of the year, so you finish with slightly more. The more often the interest is added, the more it earns on itself. The formula for n payments a year is (1 + 1/n)ⁿ:
| Compounded | n | (1 + 1/n)ⁿ |
|---|---|---|
| Annually | 1 | 2.000000000 |
| Quarterly | 4 | 2.441406250 |
| Monthly | 12 | 2.613035290 |
| Daily | 365 | 2.714567482 |
| Every millionth of a year | 1,000,000 | 2.718280469 |
The numbers keep rising but they are converging, not running away. Their limit is e. That is the definition: e is what (1 + 1/n)ⁿ approaches as n grows without bound. Compounding continuously rather than daily is worth about a thirtieth of a cent on the dollar, which is a reasonable answer to the question of whether it is worth chasing.
There is a second route to the same number that converges far faster. Add up the reciprocals of the factorials:
e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + …
Which is 1 + 1 + 0.5 + 0.1666… + 0.041666…. Five terms already give 2.7166666667. Ten terms give 2.7182818011, correct to seven decimal places. The millionth compounding step above was still only correct to five.
e vs pi
Both are irrational, both are transcendental, both are written as a single letter, and both turn up where you did not put them. They answer different questions.
| e ≈ 2.71828 | π ≈ 3.14159 | |
|---|---|---|
| Answers | How fast does continuous growth compound? | How does a circle’s circumference relate to its diameter? |
| Defined by | The limit of (1 + 1/n)ⁿ | Circumference divided by diameter |
| Shows up in | Interest, decay, probability, logarithms | Circles, waves, rotation, geometry |
| Its own function | eˣ is its own derivative | Sine and cosine repeat every 2π |
They meet in Euler’s identity, eⁱⁿ + 1 = 0, which ties both constants to the imaginary unit and to 0 and 1. It is quoted more often than it is used, but it is not a trick: it falls out of writing eˣ as a series and noticing that the terms sort themselves into sine and cosine.
The practical difference is which one you reach for. A question about circles wants π. A question about something growing or shrinking at a rate proportional to its own size wants e.
How to find e on a calculator
Almost no calculator has a key labelled e. It has a key that raises e to a power, and you use it to get e itself.
- Find the key marked eˣ. On many calculators it is the shifted function of the ln key, because they are inverses.
- Enter e¹, that is e to the power 1. The display shows 2.718281828.
- If the keyboard only offers ln, the same fact runs backwards: ln(e) = 1, so whatever number has a natural log of exactly 1 is e.
- In a spreadsheet, =EXP(1) returns it. There is no =E() function.
Two keys are commonly mistaken for it. The EXP or EE key on a scientific calculator usually means times ten to the power, so 3 EXP 4 is 30,000 and not 3 × e⁴. And the e that appears in a display like 2.5e6 is that same times-ten shorthand, unrelated to Euler’s number: it is standard form abbreviated for a display with no superscripts. That collision is worth knowing about before it costs you an answer.
The log calculator takes natural logs directly, which is the operation e exists to invert.
Worked examples
e earns its place in three settings that look unrelated and are the same calculation.
- Continuous compounding. $1,000 at 5% compounded continuously for 3 years is 1000 × e⁰·⁵ × ³, that is 1000 × e⁰·¹⁵ ≈ $1,161.83. Compounded monthly instead it is $1,161.47, so continuous compounding is worth 36 cents over three years.
- Decay. The same formula with a negative exponent. A quantity falling at 5% a year for 3 years keeps e⁻⁰·¹⁵ ≈ 0.861 of itself, so about 86.1% remains.
- Half-life. Written as e⁻ᵏᵗ, the time to fall to half is ln(2)/k ≈ 0.6931/k. That single expression is why half-life does not depend on how much you started with.
All three are the same statement: something changing at a rate proportional to its own size. e is the number that describes that behaviour, which is why one constant covers savings accounts and radioactive isotopes without modification.
e and the natural logarithm
The natural logarithm, written ln, is the question “what power do I raise e to in order to get this number?” So ln(x) and eˣ undo each other exactly.
- ln(e) = 1, because e to the power 1 is e.
- ln(1) = 0, because e to the power 0 is 1.
- e⁽ⁿ⁽ˣ⁾⁾ = x for any positive x.
- ln is not log. A bare log usually means base 10. On a calculator they are separate keys and they give different answers: log(100) = 2 but ln(100) ≈ 4.605. The change of base formula converts between them.
The reason this logarithm is called natural rather than arbitrary is that eˣ is the only exponential function that is its own rate of change. Its slope at every point equals its height at that point. Every other base picks up a stray constant factor, so base e is the one where the calculus comes out clean.
The log calculator handles natural logs and base-10 logs side by side, which makes the difference between them easy to see on your own numbers.
Common mistakes
- Treating e as a variable. It is a fixed number. In eˣ the x varies and the e never does.
- Confusing e with the EXP or EE key. On most calculators that key means ×10ⁿ, so 5 EE 3 is 5,000.
- Using log where ln is meant. Base 10 and base e are different functions and the answers differ by a factor of about 2.303.
- Rounding e to 2.7 mid-calculation. That is wrong in the third significant figure and compounds through an exponent.
- Reading 1828 twice as a pattern. The digits do not repeat; the coincidence stops after 2.718281828 and the next digit is 4.
Quick summary
- e ≈ 2.718281828459045, irrational and transcendental.
- It is the limit of (1 + 1/n)ⁿ, the value continuous compounding converges to.
- It is also 1/0! + 1/1! + 1/2! + …, which converges much faster.
- eˣ is its own derivative, which is what makes base e the natural one.
- ln is the logarithm in base e, and ln(e) = 1.
The log calculator takes natural logs, and the compound interest calculator runs the compounding schedule that e is the limit of.