Euler’s number

The number e is a very mysterious constant. Unlike “pi,” it’s not at all obvious where it came from or what it actually means. And it’s far from clear what “magical” properties it possesses, given that this number is one of the most popular.

What is the number e

The number e is Euler’s constant. So far, everything seems quite simple; it’s just another constant. But here’s the question: where does it come from?

For example, the number pi is the ratio of a circle’s circumference to its diameter. Clear and simple.

But the number e is the base of the natural logarithm… Its origin is a bit more mysterious; this constant didn’t arise from geometry, where understanding its essence is as simple as seeing it. Here, things are a bit more complex but far more interesting…

The natural logarithm ln x is a logarithm with the number e as its base. In mathematical terms:

ln x = log_e x

But mathematicians love to simplify things, so they most often use the notation ln x. Why write extra symbols if it’s already clear? While the notation is straightforward, the number itself is not yet clear. To understand what the number e is, we need to recall the history of its discovery.

Where did the number e come from

It’s quite strange, but the number e was first discovered by Jacob Bernoulli, not Euler. Bernoulli was solving a problem about compound interest. Here’s the gist:

You deposit 1 dollar in a bank at 100% annual interest (these numbers are chosen for simplicity; multiplying by one is always easy, no need to know the multiplication table).

If the interest is credited at the end of the year, the bank client earns 2 dollars. That’s clear. But what if interest is credited more frequently? For example, every quarter or every month? Or even more often than once a month?

If interest is credited twice a year and the accrued interest is added to the deposit, it’s obvious that the client’s income will increase.

Simple math:

After 6 months, the bank credits 100% annual interest for half a year. That’s 100/2=50%. So, 1*0.5=0.5, and this 0.5 is added to the initial deposit amount.
After a year, the bank credits 100% annual interest for half a year again, but now not on 1 dollar, but on 1.5. That’s 1.5*0.5=0.75. Add this to the initial amount: 1+0.5+0.75=2.25. That’s 0.25 more than with the option without compounding and with annual payouts. The magic of compound interest is actually very simple.

Since compound interest is beneficial (but only if we’re talking about a deposit, not a loan), it’s better to receive payouts as often as possible. Of course, adding them to the deposit amount.

Once a year. Deposit 1 dollar, earn 2 at 100% annual interest.
Twice a year — 2.5, that’s already 125%, not 100%!
Four times a year — 2.44!
Every month — 2.61, that’s 161.3%!

The more frequently interest is compounded on the deposit, the better for us. So, let’s ask the bank to pay out interest daily, or even better, hourly!

To avoid calculating daily or even hourly payouts, we need to derive the compound interest formula. Here it is:

$1*(1+1/n)ⁿ

Interesting facts:

The number e is irrational. It’s an infinite decimal approximately equal to 2.71. The number of digits after the decimal point is infinite.
The number e is a transcendental number; it cannot be the root of an algebraic equation.
The sequence of digits in the number e is random. Thus, in its infinite “tail,” you can find any combination of digits.
The informal holiday of Euler’s number is celebrated on February 7. Because in the American date format, it’s 2/7.
In the binary system, the number e is written as 10.10110… Also an infinite fraction. In any number system.
The derivative of the exponential function is equal to itself. This is a unique property of such a function.

f(x)’=de^x/dx=e^x

This means that the rate of change of such a function at any point is always equal to the function’s value. In simple terms, the extent to which we change the function’s value is exactly the extent to which its rate of change changes.

For example, the further we try to move, the stronger the resistance of the environment, or the more often we demand interest payouts, the slower the final profitability increases.

On the other hand, the integral of the exponential function is also equal to the function itself plus a constant.

∫e^x dx = e^x + C

In simple terms, the function e^x remains unchanged under both integration and differentiation.

Where the number e is used

The number e can primarily be found where exponential growth processes are involved. That is, when using the function f(x)=e^x. The number e raised to the power of x is the exponential!

The most vivid example is the growth of a population, such as bacteria. Their numbers will grow exponentially until they reach a limit, for example, when space for further growth runs out or the nutrient medium is depleted (there’s nothing left to eat).

The number e can also be found in radioactive decay formulas. Chemical reactions also follow an exponential law, where the number e is used. For example, the relationship between activation energy and the rate of a chemical reaction.

The charging and discharging of a capacitor are also described by an exponential law using Euler’s number.

Additionally, Euler’s number is used in the normal distribution function. This function shows the distribution of a random variable, such as the height or weight of cats, the intelligence quotient of people, or how far an average grain of sand might roll from the spot where a bucket of sand was poured.

So, the number e is used frequently. It appears in mathematics, physics, biology, and especially in economics. That’s why it’s so popular.