e Calculator — What is eˣ and How to Calculate e to the Power of x
Euler's number e ≈ 2.718281828459045… is one of the most important constants in mathematics. Like π, it is irrational and transcendental — its decimal expansion is infinite and non-repeating. Our e calculator lets you compute eˣ (e raised to any power) instantly, with a step-by-step breakdown and practical applications.
What is e on a Calculator?
On a scientific calculator, the button eˣ computes the natural exponential function: it raises Euler's number e to the power of x. The result grows very rapidly for positive x and approaches zero for large negative x. Key properties:
- e⁰ = 1 (any number to the power zero equals 1)
- e¹ ≈ 2.71828
- e² ≈ 7.38906
- e⁻¹ ≈ 0.36788
- The slope of eˣ at any point equals its value at that point: d/dx(eˣ) = eˣ
- e is the base of the natural logarithm: ln(eˣ) = x
How to Calculate eˣ — Step by Step
The standard method uses the Taylor series expansion:
eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + x⁵/5! + …
For example, to compute e¹⁰:
- Using the approximation e ≈ 2.718281828:
- e¹⁰ = 2.718281828¹⁰ ≈ 22,026.47
Our calculator handles this automatically — just enter x and get the result with up to 10 decimal places.
eˣ Calculator — Examples
| x | eˣ (approx.) | Context |
|---|---|---|
| 0 | 1 | Base case |
| 1 | 2.71828… | Euler's number itself |
| 2 | 7.38906… | Population doubling |
| 3 | 20.0855… | Exponential growth |
| 5 | 148.413… | Continuous compounding |
| 10 | 22,026.47… | Rapid growth |
| −1 | 0.36788… | Decay / half-life |
| −2 | 0.13534… | Rapid decay |
| iπ | −1 (Euler's identity) | eiπ + 1 = 0 |
Applications of eˣ
- Continuous Compound Interest: A = P × eˣ, where x = r × t (rate × time). For example, $1,000 invested for 5 years at 10% continuous compounding: A = 1000 × e^(0.1×5) = $1,648.72
- Population Growth: N(t) = N₀ × eˣ, where x = r × t
- Radioactive Decay: N(t) = N₀ × e^(−λt)
- Physics: Biot number, Reynolds number, f-number formulas
- Probability: Normal distribution, Poisson distribution
Euler's Number — Fun Facts
- e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + … (sum of reciprocals of factorials)
- e = lim(n→∞) (1 + 1/n)ⁿ — the compound interest limit
- Euler's identity: e^(iπ) + 1 = 0 — considered the most beautiful equation in mathematics
- e appears in the normal (Gaussian) distribution: f(x) = (1/√(2π)) × e^(−x²/2)
Metric vs. US / Imperial Measurement Systems
The value of eˣ is a dimensionless pure number. However, our calculator can interpret the result in real-world units to help you appreciate its scale:
- Metric System (SI): Interprets eˣ as a length (nm, cm, m, km), mass (g, kg, t), and volume (mL, L, m³)
- US / Imperial System: Interprets eˣ as a length (in, ft, mi), mass (oz, lb, short tons), and volume (fl oz, gal)
Compound Interest with Multiple Currencies
Using the formula A = P × eˣ, our calculator shows the result in your chosen currency. Simply enter the principal (P) and select from 12 world currencies: USD, EUR, GBP, JPY, CNY, CAD, AUD, CHF, INR, BRL, MXN, and RUB.
FAQs
- What does e mean on a calculator?
- On a scientific calculator, "e" refers to Euler's number ≈ 2.71828. The eˣ button computes the natural exponential function.
- What is e to the power of 0?
- e⁰ = 1, because any nonzero number raised to the power of 0 equals 1.
- What is e to the power of 1?
- e¹ = e ≈ 2.718281828459045
- What is e to the power of 2?
- e² ≈ 7.38905609893065
- Can x be negative?
- Yes. e^(−x) = 1/eˣ. Negative exponents produce values between 0 and 1.
- What is the maximum value of x?
- For standard floating-point arithmetic, x ≤ 709. Beyond that, eˣ exceeds the maximum representable float (overflow).