What Is Euler's Constant?

Euler's constant (sometimes called Euler's number or the Euler–Mascheroni constant.) is a mathematical expression for the limit of the sum of 1 + 1/2 + 1/3 + 1/4 ... + 1/n, minus the natural log of n as n approaches infinity.

Euler's constant is represented by either e or the lower case gamma (γ) and appears in calculus as a derivative of a logarithmic function. It is the difference between a harmonic series and the natural logarithm (log base e). There is no closed-form expression for the harmonic number, but gamma can provide an estimate of it.

Key Takeaways

  • Euler's constant is an important number that is found in many contexts and is the base for natural logs.
  • An irrational number denoted by e, Euler's constant is 2.71828..., where the digits go on forever in a never-ending series (similar it pi.)
  • Euler's constant is used in everything from explaining exponential growth to radioactive decay, to continuous compounding of interest rates.

Understanding Euler's Constant

Information on Euler's constant was first presented by the Swiss mathematician Leonhard Euler in the 18th century in his work "De Progressionibus Harmonicus Observations." Mathematicians have concluded that Euler's constant is an irrational and transcendental number like pi, in that it goes on repeating forever to the right of its decimal point.

There are several ways at arriving at e, one of which involves adding the sums of 1 + 1/2 + 1/3 + 1/4 + ... + 1/n. This is also expressed as (1 + 1/n)n. Interestingly e is also approximated by the same kind of series but taking the factorial (!) of the denominator, where 4! is equal to 4 x 3 x 2 x 1, etc. Thus, 1/0! + 1/1! + 1/2! + 1/3! + ... 1/n! = 1 + 1 + 1/4 + 1/6 + ... 1/n! = 2.71828...

Euler's constant is also known as the exponential growth constant since it is used as the base for natural logarithms (ln) and is used to compute exponential growth or exponential decay across a wide range of applications from population growth of living organisms to radioactive decay of heavy elements like uranium by nuclear scientists.

2.71828

The first digits of Euler's number are 2.71828..., although the number itself is a non-terminating series that goes on forever, like pi (3.1415...).

Euler's Constant in Finance: Compound Interest

Compound interest has been hailed as a "miracle" of finance, whereby interest is credited not only initial amounts invested or deposited, but also on previous interest received. Continuously compounding interest is achieved when interest is reinvested over an infinitely small unit of time—and while this is practically impossible in the real world, this concept is crucial for understanding the behavior of many different types of financial instruments from bonds to derivatives contracts.

Compound interest in this way is akin to exponential growth, and is expressed by the following formula:

FV = (PV)e(r x t)

where:

  • FV = future value
  • PV = present value of the balance or sum
  • e = Euler's constant
  • r = the interest rate that is being compounded
  • t = the amount of time in years

Therefore, if you had $1,000 paying 2% interest with continuous compounding, after 3 years you would have:

$1,000 x 2.71828(.02 x 3) = $1,061.84

Note that this amount is greater than if the compounding period were a discrete period, say on a monthly basis. In this case, the amount of interest would be computed differently: FV = PV(1+r/n)nt, where n is the number of compounding periods in a year (in this case 12):

$1000 (1 + .02/12)12x3 = $1,061.78

Here, the difference is only a matter of a few cents, but as our sums get larger, interest rates get higher, and the amount of time gets longer, continuous compounding using Euler's constant becomes more and more valuable relative to discrete compounding.