Euler’s number, written e, is an irrational constant approximately equal to 2.71828. It is the number that ( 1 + 1n )n gets closer and closer to as n grows. It is the base of natural exponential growth, ex, and of the natural logarithm, lnx.
In Spanish
número de Euler
Also called
e, the number e, natural base, base of the natural logarithm
Grade level
Algebra 2
Algebra 2PrecalculusCalculusCCSS HSF-LE.A.4SATACT
Euler's Number in a picture
If 1 dollar earns 100 percent interest compounded n times a year, it grows to ( 1 + 1n )n dollars. More compounding helps less and less: the amounts level off at e.
Formal definition and notation
Euler’s number is the limit of ( 1 + 1n )n as n → ∞. It also equals the endless sum 1 + 11! + 12! + 13! + ⋯, so e ≈ 2.718281828. It is irrational (its decimal never ends or repeats). The function ex is the natural exponential function, and its inverse is the natural logarithm, lnx = logex.
e ≈ 2.718281828. Like π, it is irrational, so any decimal is an approximation.
Compounding more and more often makes ( 1 + 1n )n creep up toward e: monthly gives about 2.613, daily about 2.715.
ex is always positive, passes through (0, 1) and grows faster and faster. In calculus it is the function whose slope at every point equals its own value.
lne = 1 and ln 1 = 0, because e1 = e and e0 = 1.
Do not confuse e with the “E” on a calculator screen, which means times a power of 10 in scientific notation.
Formulas with e
Formulas with e
e ≈ 2.71828 A = Pert
A
the final amount
P
the starting amount (principal)
r
the annual rate as a decimal (5 percent is 0.05)
t
the time in years
Use the ex key on a calculator. Keep e exact (write e0.5) until the last step, then round.
Euler's Number examples, step by step
Example 1: Evaluate a power of e
Warm-up
Find e2 to two decimal places.
e2 = e × e, with e ≈ 2.71828.
On a calculator, press ex and enter 2: about 7.389.
Round to two decimal places.
Answer:e2 ≈ 7.39.
Example 2: Continuous compound interest
Core
You deposit 1,000 dollars in an account that pays 5 percent interest compounded continuously. How much is in the account after 10 years?
A calculator display such as 3.2E5 means 3.2 × 105. It has nothing to do with Euler’s number.
Quick check: 3 questions
Question 1 of 3
Which is the best approximation of e?
Show answer and explanation
Answer: B.e ≈ 2.71828. 3.14159 is π, 1.61803 is the golden ratio, and 2.5 is too rough.
Question 2 of 3
What is ln (e4)?
Show answer and explanation
Answer: D. The natural logarithm undoes ex, so ln (e4) = 4. The other choices treat ln as multiplication or forget that it reverses the exponent.
Question 3 of 3
5,000 dollars is invested at 3 percent compounded continuously. To the nearest dollar, what is it worth after 8 years?
Show answer and explanation
Answer: C.A = 5000e0.03 × 8 = 5000e0.24 ≈ 6356. 6,200 is simple interest, 6,334 is yearly compounding, 5000(1.03)8, and 5,150 is only one year of interest.
Euler's Number: frequently asked questions
Why is e called the natural base?
Because it shows up on its own whenever something grows at a rate proportional to its size: continuous interest, population growth, radioactive decay and cooling. In calculus, ex is the only exponential function whose rate of change equals itself, which makes formulas simplest in base e. Logarithms with base e are therefore called natural logarithms.
What is e equal to?
e ≈ 2.718281828459, and its digits go on forever without repeating, because e is irrational. Two exact ways to describe it are the limit of ( 1 + 1n )n as n grows without bound and the endless sum 1 + 1 + 12 + 16 + 124 + ⋯, whose denominators are factorials.
How do you type e on a calculator?
Most scientific and graphing calculators have an ex key, often the second function of the ln key. To get e itself, enter e1. Do not use the EE or E key, which enters a power of 10 for scientific notation. In spreadsheets, the function EXP(1) returns e.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-LE.A.4
Cite this entry
Effortless Math Team. “Euler's Number.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/eulers-number/
Flashcard
TermEuler's Numbere ≈ ?
DefinitionAn irrational constant, about 2.71828, the limit of ( 1 + 1n )n. It is the base of ex and of lnx.
Irrational NumberAn irrational number cannot be written as a fraction of two integers. Its decimal form does not end and does not repeat in a pattern. Examples are √2 = 1.41421… and π = 3.14159…. The square root of any whole number that is not a perfect square is irrational.Number System · Grade 8, Algebra 1Exponential FunctionAn exponential function is a function in which the input is the exponent: f(x) = a · bx, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.Functions · Algebra 1, Algebra 2, PrecalculusNatural LogarithmThe natural logarithm of a positive number x, written lnx, is the power you must raise e ≈ 2.718 to in order to get x. So lnx = y means ey = x. For example, ln 1 = 0, lne = 1, and ln 20 ≈ 3.0 because e3 ≈ 20.1.Functions · Algebra 2, Precalculus, CalculusLogarithmA logarithm is the exponent you need to raise a base to in order to get a given number. logbx = y means by = x, so log2 32 = 5 because 25 = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.Functions · Algebra 2, PrecalculusCompound InterestCompound interest is interest computed on the principal plus all the interest already earned, so the balance grows exponentially. The formula is A = P( 1 + rn )nt, where P is the principal, r the annual rate, n the compoundings per year and t the years. It beats simple interest after the first period.Functions · Grade 7, Algebra 1, Algebra 2SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. 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Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7CircumferenceCircumference is the distance around a circle, found by multiplying the diameter by pi, or the radius by 2 pi: C = π d = 2π r. It is the circle’s perimeter, measured in units of length. A circle with diameter 10 has a circumference of about 31.4, because π ≈ 3.14.Geometry · Grade 6, Grade 7, Grade 8DiameterThe diameter of a circle is the distance straight across it through the center. It is twice the radius, d = 2r, and it is the longest chord any circle has. The word also names the line segment itself: a segment that passes through the center and touches both sides of the circle.Geometry · Grade 6, Grade 7, Grade 8FactorialThe factorial of a whole number n, written n!, is the product of all the whole numbers from n down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. It counts the ways to arrange n different objects in a row. By agreement, 0! = 1.Probability · Algebra 2, Precalculus, StatisticsProperties of LogarithmsThe properties of logarithms are rules for rewriting logs: the log of a product is a sum, logb(MN) = logbM + logbN; the log of a quotient is a difference; and the log of a power is a multiple, logb(Mp) = plogbM. They come from the rules of exponents.Functions · Algebra 2, PrecalculusLimitA limit is the value a function gets closer and closer to as the input approaches a certain number. We write limx → af(x) = L. The function need not even be defined at a. For f(x) = x2 − 1x − 1, f(1) is undefined, but the limit as x → 1 is 2.Calculus · Precalculus, CalculusPercent ChangePercent change measures how much a value grew or shrank relative to where it started: the difference divided by the original amount, written as a percent. From 50 to 60 is a 20% increase; from 80 to 60 is a 25% decrease. The original value is always the base, which is the step people get wrong.Ratios and Proportions · Grade 7, Grade 8, Algebra 1