Euler's Number (e) Explained, and How It Is Used in Finance

What Is Euler's Number (e)?

The term Euler's number (e) refers to a mathematical expression for the base of the natural logarithm. This is represented by a non-repeating number that never ends. The first few digits of Euler's number are 2.71828. The number is usually represented by the letter e and is commonly used in problems relating to exponential growth or decay. You can also interpret Euler's number as the base for an exponential function whose value is always equal to its derivative. In other words, e is the only possible number such that e^x increases at a rate of e^x for every possible x.

Understanding Euler's Number (e)

As noted above, Euler's number is used to express the base of the natural logarithm. E is a series of numbers that begin with 2.71828. Just like pi, it is non-terminating, which means it goes on and on. It is also an irrational number, which means it can't be expressed as a fraction. You can use it to calculate the decay or growth of a particular factor over time, such as compound interest.

Imagine lending money at a 100% interest rate, compounded every year. After one year, your money would double. But what if the interest rate were cut in half, and compounded twice as often? At 50% every six months, your money would grow by 225% in one year.

As the interval gets smaller, the total returns get slightly higher. If interest is calculated n times per year, at a rate of 100%/n, the total accreted wealth at the end of the first year would be slightly greater than 2.7 times the initial investment if n is sufficiently large.

History of Euler's Number (e)

Although commonly associated with and named after the Swiss mathematician Leonhard Euler, it was first discovered in 1683 by mathematician Jacob Bernoulli. He was trying to determine how wealth would grow if interest were compounded more often, instead of on an annual basis.

The most pivotal work surrounding the number was not performed until several decades later, by Leonhard Euler. In his book Introductio in Analysin Infinitorum (1748), Euler proved that it was an irrational number, whose digits would never repeat. He also proved that the number can be represented as an infinite sum of inverse factorials:

$e = 1 + \frac{ 1 }{ 1 } + \frac { 1 }{ 2 } + \frac { 1 }{ 1 \times 2 \times 3 } + \frac {1 }{ 1 \times 2 \times 3 \times 4 } + ... + \frac { 1 }{ n! }$e=1+11​+21​+1×2×31​+1×2×3×41​+...+n!1​

Euler used the letter e for exponents, but the letter is now widely associated with his name. It is commonly used in a wide range of applications, including population growth of living organisms and the radioactive decay of heavy elements like uranium by nuclear scientists. It can also be used in trigonometry, probability, and other areas of applied mathematics.

Euler's Number (e) 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. 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:

$\begin{aligned}&\text{FV}=\text{PV}e^{rt}\\&\textbf{where:}\\&\text{FV} = \text{Future value}\\&\text{PV} =\text{Present value of balance or sum}\\&e = \text{Euler's formula}\\&r = \text{Interest rate being compounded}\\&t = \text{Time in years}\end{aligned}$​FV=PVertwhere:FV=Future valuePV=Present value of balance or sume=Euler’s formular=Interest rate being compoundedt=Time in years​

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

$\$1,000 \times 2.71828 ^ { ( .02 \times 3 ) } = \$1,061.84$$1,000×2.71828(.02×3)=$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):

$\$1,000 \Big ( 1 + \frac { .02 }{ 12 } \Big ) ^ { 12 \times 3 } = \$1,061.78$$1,000(1+12.02​)12×3=$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.

The Bottom Line

Euler's number is one of the most important constants in mathematics. It frequently appears in problems dealing with exponential growth or decay, where the rate of growth is proportionate to the existing population. In finance, e is also used in calculations of compound interest, where wealth grows at a set rate over time.

Correction–Dec. 5, 2021: An earlier version of this article incorrectly conflated Euler's number with Euler's constant.