Euler's Constant

What Is Euler's Number?

Euler's number is a mathematical expression for the base of the natural logarithm. It is usually represented by the letter e and is commonly used in problems relating to exponential growth or decay.

Another way to interpret Euler's number is 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 ex increases at a rate of ex for every possible x.

Key Takeaways

  • Euler's number is an important constant that is found in many contexts and is the base for natural logarithms.
  • An irrational number denoted by e, Euler's number is 2.71828..., where the digits go on forever in a series that never ends or repeats (similar to pi).
  • Euler's number is used in everything from explaining exponential growth to radioactive decay.
  • In finance, Euler's number is used to calculate how wealth can grow due to compound interest.

Understanding Euler's Number

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

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. Bernoulli found that 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.

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

e = 1 + 1 1 + 1 2 + 1 1 × 2 × 3 + 1 1 × 2 × 3 × 4 + . . . + 1 n ! 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 from population growth of living organisms to radioactive decay of heavy elements like uranium by nuclear scientists. It also has applications in trigonometry, probability, and other areas of applied mathematics.

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 Number 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 t where: FV = Future value PV = Present value of balance or sum e = Euler’s constant r = Interest rate being compounded t = Time in years \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 constant} \\&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 constantr=Interest rate being compoundedt=Time in years

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

$ 1 , 000 × 2.7182 8 ( . 02 × 3 ) = $ 1 , 061.84 \$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 ( 1 + . 02 12 ) 12 × 3 = $ 1 , 061.78 \$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.

Euler's number (e) should not be confused with Euler's constant, denoted by the lower case gamma (γ). Also known as the Euler-Mascheroni constant, the latter is related to harmonic series and has a value of approximately 0.5772....

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.

Why Is Euler's Number Important?

Euler's number frequently appears in problems related to growth or decay, where the rate of change is determined by the present value of the number being measured. One example is in biology, where bacterial populations are expected to double at reliable intervals. Another case is radiometric dating, where the number of radioactive atoms is expected to decline over the fixed half-life of the element being measured.

How Is Euler's Number Used in Finance?

Euler's number appears in problems related to compound interest. Whenever an investment offers a fixed interest rate over a period of time, the future value of that investment can easily be calculated in terms of e.

What Is Euler's Number Exactly?

To put it simply, Euler's number is the base of an exponential function whose rate of growth is always proportionate to its present value. The exponential function ex always grows at a rate of ex, a feature that is not true of other bases and one that vastly simplifies the algebra surrounding exponents and logarithms. This number is irrational, with a value of approximately 2.71828....

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

Article Sources

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  1. Popular Mechanics. "What's the Big Deal with Euler's Number?" Accessed Dec. 5, 2021.

  2. School of Mathematics and Statistics, University of St. Andrews. "The Number e." Accessed Dec. 5, 2021.

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