Mathematics · Calculus

Euler’s Number

Reference entry · last updated October 12, 2026

Euler’s number, denoted \(e\), is the base of the natural logarithm. Its value begins \(2.718281828459045\ldots\). The function \(e^x\) is called the natural exponential function.[1]

First principles and definitions

The natural logarithm can be defined by \(\ln x=\int_1^x dt/t\) for \(x>0\). Euler’s number is the unique number \(e>1\) for which this integral equals one:[1, 3]

\[\int_1^e \frac{dt}{t}=1,\qquad \ln e=1.\]

For real \(x\), the inverse relation gives \(\ln(e^x)=x\). For positive \(x\), \(e^{\ln x}=x\). The notation \(\exp(x)\) also means \(e^x\).[1, 3]

Limit and series definitions

The same number is obtained from a limit over positive integers \(n\):[2]

\[e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n.\]

A convergent series gives another definition. Here \(n!\) is the product of the integers from \(1\) through \(n\), with \(0!=1\):[1]

\[e=\sum_{n=0}^{\infty}\frac{1}{n!}=1+1+\frac{1}{2!}+\frac{1}{3!}+\cdots.\]

Continuous compounding

In an idealized compounding model, a principal \(P>0\) earns a fixed nominal annual rate \(r\ge 0\). There are no deposits, withdrawals, fees, or taxes. With \(n\) equal compounding periods per year and \(t\ge 0\) years, at a compounding date the amount is:[4]

\[A_n(t)=P\left(1+\frac{r}{n}\right)^{nt}.\]

As the compounding frequency tends to infinity, \(A_n(t)\) tends to \(Pe^{rt}\). Setting \(P=1\), \(r=1\), and \(t=1\) gives \(e\): a unit principal at a nominal rate of 100% for one year under continuous compounding.[2, 4]

Calculus and proportional change

The natural exponential equals its derivative. It is an antiderivative of itself:[3, 5]

\[\frac{d}{dx}e^x=e^x,\qquad \int e^x\,dx=e^x+C.\]

Among real exponential functions \(b^x\) with \(b>0\), the base \(e\) gives slope one at \(x=0\). For a differentiable real function \(g\), the chain rule gives \(\frac{d}{dx}e^{g(x)}=g'(x)e^{g(x)}\).[5]

With constant \(k\) and initial value \(y(0)=y_0\), the equation for change proportional to the current value has solution:[6]

\[\frac{dy}{dt}=ky,\qquad y(t)=y_0e^{kt}.\]

For \(y_0>0\), positive \(k\) gives growth, negative \(k\) gives decay, and \(k=0\) gives a constant value. The model assumes a fixed proportional rate.[6]

Euler’s number and Euler’s constant

Euler’s constant usually denotes the number \(\gamma\approx0.57721\). It is defined by the limit of \(1+1/2+\cdots+1/n-\ln n\) as \(n\to\infty\).[7]

See also

References

  1. NIST, Digital Library of Mathematical Functions, §4.2: Definitions, equations 4.2.11–14, 4.2.32, and 4.2.34.
  2. NIST, Digital Library of Mathematical Functions, §4.4: Special Values and Limits, equation 4.4.17.
  3. OpenStax, Calculus Volume 1, §6.7: Integrals, Exponential Functions, and Logarithms.
  4. OpenStax, Calculus Volume 1, §1.5: Exponential and Logarithmic Functions, “The Number e.”
  5. OpenStax, Calculus Volume 1, §3.9: Derivatives of Exponential and Logarithmic Functions, theorem 3.14.
  6. OpenStax, Calculus Volume 1, §6.8: Exponential Growth and Decay.
  7. NIST, Digital Library of Mathematical Functions, §5.2: Definitions, equation 5.2.3.