mathematics has many important constants that give structure to the discipline, such as pi and i, the imaginary number equal to the square root of -1. But a constant that is equally important, though perhaps less well known, is Euler’s constant, e.
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turns up all the time in mathematics and physics, most commonly as a base in logarithmic and exponential functions. is used to calculate compound interest, the rate of radioactive decay, and the amount of time it takes to discharge a capacitor. as stefanie reichert says in natural physics, “we can’t escape euler’s number”.
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but where does euler’s constant come from? And what exactly is it?
what is euler’s constant?
Euler’s constant, which you’ll also see some math experts refer to as Euler’s number, is an irrational number, meaning you can’t reduce it to a simple fraction. like pi, the decimals of e go on forever without repeating. if you want to get technical, this is what e looks like to the hundredth decimal point:
If you’ve ever taken a basic calculus course, you’ve probably come across Euler’s constant, since it’s the base of natural logarithms. looks like this: eln x= x.
As you graph the equation y=ex, you will find that the slope of that curve at any given point is also ex, and the area under the curve from negative infinity to x is also ex. euler’s constant is the only number in all of mathematics that can be substituted into the equation y=nx for which this pattern is true.
who discovered euler’s constant?
The history of e is a bit complicated and includes the contributions of three mathematicians: John Napier, Jacob Bernoulli, and Leonard Euler. For the long version, check out this article in Cantor’s Paradise, a midsize math-focused publication. for the short version, read on.
In the 17th century, Napier, a Scottish mathematician, physicist, and astronomer, began searching for an easier way to multiply very large numbers. specifically, he wanted to find a shortcut for exponents. Although Napier did not discover the number e, he did devise a list of logarithms that he unwittingly calculated with the constant. he published his work, mirifici logarithmorum canonis descriptio, in 1614.
It would be another 70 years before this list of logarithms would be associated with exponents. In 1683, Swiss mathematician Jacob Bernoulli discovered the constant E while solving a financial problem involving compound interest. he saw that through increasingly compound intervals, his sequence approached a limit (the force of interest). bernoulli noted this limit, since n keeps growing, like e.
Finally, in 1731, Swiss mathematician Leonhard Euler gave the number e his name after proving that it is irrational by expanding it into a convergent infinite series of factorials.
use euler’s constant to calculate compound interest
Because e is related to exponential relationships, the number is useful in situations that show constant growth.
A common example, which Bernoulli explored, involves compound interest: the interest you pay on a loan when you include both the initial principal (the amount of the loan) and the interest accumulated over previous periods in the calculation. That’s why you can make a minimum payment on your credit card every month, but never pay it off in full.
Suppose you deposit some money in the bank and the bank compounds that money annually at a rate of 100 percent. after a year, you would have double the amount you invested.
Now suppose the bank compounds the interest every 6 months, but only offers half the interest rate, or 50 percent. in this case, you would end up with 2.25 times your initial investment after one year.
Let’s move on. Suppose the bank offers 8.3 percent compound monthly interest (1/12 of 100 percent), or 1.9 percent weekly compound interest (1/52 of 100 percent). in that case, you would earn 2.61 and 2.69 times your investment.
let’s write an equation for this. if we let n equal the number of times the interest is compounded, then the interest rate is the reciprocal, or 1/n. the equation for how much money you would earn in a year is (1+1/n)n. For example, if your interest compounded five times a year, you would earn (1+⅕)5 = (1+0.2)5 = (1.2)5 = 2.49 times your initial investment.
To calculate compound interest, use the equation: a = p(1 + r/n)^n, where a = the ending amount, p = the beginning principal balance, r is the interest rate, n is the number of times interest is applied in a given time period, and t is the number of time periods elapsed.
so what happens if n becomes very large? say, large infinity? This is the question Bernoulli was trying to answer, but it took Euler 50 years to get there and figure it out. It turns out that the answer is the irrational number e, which is approximately 2.71828….
what else can you do with euler’s constant?
Euler’s constant is not only useful in finance. some other common use cases include:
☐ Probability Theory: If you play a game of roulette and bet on a single number, the probability that you lose each game, over a span of 37 games, is approximately 1/ mi.
☐ calculation of the half-life of radioactive chemical substances.
☐ equations for waves (such as light, sound, and quantum waves) in physics.