How this instrument works
Euler's number e is defined as the limit of (1 + 1⁄n)ⁿ as n grows without bound, a quantity that settles at 2.718281828459045… and never repeats or terminates. Raising it to a power x, written eˣ, gives the natural exponential function — the one exponential curve whose slope at every point equals its own height. No other base shares that outright; 2ˣ and 10ˣ climb too, but their slopes are scaled copies of themselves, off by a constant factor of ln 2 or ln 10.
The shape follows from the series eˣ = 1 + x + x²⁄2! + x³⁄3! + ⋯, an infinite sum that converges for every real x and is how the underlying arithmetic actually gets done. Near x = 0 only the first couple of terms matter, which is why eˣ ≈ 1 + x for small x — a shortcut engineers lean on constantly when a full exponential would be overkill for the precision needed.
Push x negative and the curve never reaches zero; it only halves the remaining distance, the classic asymptote of exponential decay. e⁻¹ is about 0.36788, and e raised to a large negative number becomes a vanishingly small positive figure, never zero — the same shape that describes a cooling cup of coffee or a charge draining from a capacitor, where 'approaches but never touches' is the entire point of the curve.
- Type your exponent into the x field — any real number works, positive, negative, or zero.
- Read the result in eˣ; it updates the instant you change x.
- Set x to 0 to confirm the universal identity eˣ = 1 when the exponent vanishes.
- Try x = -1 to watch the reciprocal of e appear as roughly 0.36788.
- Nudge x using the 0.1 step to see eˣ curve smoothly rather than jump between whole numbers.
Worked example — x = 1 returns e itself
Set x to 1 and the calculator returns value = e¹ = 2.718281828459045 — Euler's number itself, accurate to fifteen decimal places. There is no rounding trick here: the constant is the definition of the base, so raising it to the power 1 simply hands back e unchanged, the same figure mathematicians have carried since Euler formalized the notation in the eighteenth century.
Nudge the exponent down to x = 0 and the reported value snaps to exactly 1, confirming the identity that holds for every nonzero base. Push it to x = -1 instead and the result flips to the reciprocal, 0.36787944117144233 — proof that a negative exponent never produces a negative number, only a fraction of the original base, here about 36.8 percent of e's own value.
Questions
What is Euler's number e?
e is the limit of (1 + 1⁄n)ⁿ as n grows without bound, an irrational constant beginning 2.718281828459045… First tied to compound growth by Jacob Bernoulli and later named for Leonhard Euler, it is the unique base whose exponential curve eˣ has a slope equal to its own height at every point — no other number shares that property.
Why is eˣ its own derivative?
Differentiating the series eˣ = 1 + x + x²⁄2! + x³⁄3! + ⋯ term by term returns the identical series, because each term's power drops by one while its factorial cancels the coefficient exactly. That self-matching is unique to base e; differentiating 2ˣ or 10ˣ produces the same curve scaled by ln 2 or ln 10, a leftover factor that only disappears when the base is e.
How does eˣ differ from 2ˣ or 10ˣ?
All three pass through the point (0, 1) and climb as x grows, but their slopes differ from their heights by a constant factor: the derivative of bˣ is bˣ times ln b. Only when b = e does ln b equal 1, so eˣ is the one member of the family whose growth rate exactly matches its current value at every x.
What does e raised to a negative power give?
A negative exponent still means a reciprocal: e⁻ˣ = 1 ⁄ eˣ. At x = -1 that works out to about 0.36788, the inverse of e's own value — never negative, and never zero, only shrinking toward it as x falls further.
What is e to the power 0?
Exactly 1, by the same rule that makes any nonzero base to the power 0 equal 1. It is also the point where the eˣ curve crosses the vertical axis, the anchor from which the curve rises for positive x and falls, without ever touching zero, for negative x.
Where does the eˣ series come from, and does it always converge?
It comes from matching every derivative of eˣ at x = 0, which forces the coefficients 1, x, x²⁄2!, x³⁄3!, and so on — a Taylor series centred at zero. It converges for every real x, however large, which is why this sheet returns a value instantly rather than only for x close to zero.