What is e on a Calculator? (Euler's Number Explained)
e — Euler's number — is the irrational constant approximately equal to 2.71828. It is the base of the natural logarithm and the foundation of every exponential growth formula. Every scientific calculator has either an e key or an e^x key for using it.
The numerical value of e
e ≈ 2.71828182845904… The decimal expansion goes on forever without repeating (e is irrational). For most calculator work, 2.71828 or 2.71828183 is enough precision. The full value sits in your calculator's internal precision — press e or ALPHA-e to insert it, and the calculator uses the full value behind the scenes.
Why e matters
e is the unique base where the derivative of bˣ equals bˣ itself. That property makes e the natural choice for any continuously growing or decaying quantity — compound interest paid continuously, radioactive decay, capacitor discharge, population growth, drug clearance from the body. The exponential function eˣ is the most common growth model in science and finance.
e in compound interest
If you invest P at interest rate r compounded continuously for t years, the final amount is P × e^(rt). For P = ₹10,000 at r = 0.08 (8% annual) over t = 5 years: A = 10,000 × e^(0.4) = 10,000 × 1.4918 ≈ ₹14,918. Compare to simple annual compounding: 10,000 × 1.08^5 ≈ ₹14,693. Continuous compounding gives a small extra return — about ₹225 in this example.
e and the natural logarithm
ln(x) is the logarithm base e. ln(e) = 1 because e¹ = e. ln(1) = 0 because e⁰ = 1. The natural log is the inverse of the exponential function — if y = eˣ, then x = ln(y). Use ln for any equation where the unknown sits inside an exponent. Most calculators have a dedicated ln key separate from log.
How to enter e on a calculator
On most scientific calculators, press the e key directly (sometimes labelled 'e' or 'EXP'). On an online calculator, press the e button. The calculator inserts e into the current expression — pressing 2 × e + 5 = computes 2 × 2.71828… + 5 ≈ 10.4366. The eˣ key combines insertion and exponentiation: pressing eˣ then 2 then = computes e² ≈ 7.389.
e in series expansion
e^x can be computed as a Taylor series: e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + … This converges for every real x. At x = 1 you get e itself: 1 + 1 + 1/2 + 1/6 + 1/24 + … ≈ 2.71828. The series gives a way to compute e to any precision you want — early mathematicians used it before electronic calculators existed.
Frequently asked questions
What does e equal?
e ≈ 2.71828182845904. It is an irrational, transcendental constant — the base of natural logarithms. Found by Jacob Bernoulli around 1683 while studying compound interest, and used heavily by Leonhard Euler in the 1700s (hence the name 'Euler's number').
How is e different from π?
Both are irrational constants used in calculus, but they come from different problems. π comes from geometry — the ratio of circumference to diameter. e comes from growth — the limit of (1 + 1/n)ⁿ as n grows. They appear together in Euler's identity e^(iπ) + 1 = 0, which links five of the most important constants in mathematics.
Where does e show up in real life?
Continuously compounded interest (banking, finance). Radioactive decay (physics). Capacitor charging and discharging (electronics). Population growth models (biology). Drug elimination half-lives (pharmacology). Carbon dating (archaeology). Any process where the rate of change is proportional to the current amount uses e.
What is e to the power of 1?
e¹ = e ≈ 2.71828. Any number to the power of 1 equals itself. The expression e^1 is sometimes used in formulas just to emphasise that the exponent is one — but the value is the same as plain e.
Why is ln(e) = 1?
ln is logarithm base e. The logarithm of a number is the power you raise the base to in order to get the number. e¹ = e, so ln(e) = 1. The same logic gives ln(1) = 0 (because e⁰ = 1) and ln(e²) = 2 (because e² is e raised to the power 2).