E (mathematical constant)

Some numbers are very special.

This number helps us see how things grow. It can show how money grows in a bank.

A man named Jacob Bernoulli found it. He was looking at interest on money. This number is about 2.718. It never ends and has no pattern. It is used in many math puzzles. It is a very important tool for us.

Some numbers are more special than others. One such number is called e. It is about 2.71828. This number is a constant. A constant is a value that stays the same.

Jacob Bernoulli found this number in 1683. He was studying how money grows in a bank. This is called compound interest. If you earn interest, you get more money. Then, you earn interest on that new money too.

Bernoulli saw that the growth has a limit. Even if you add interest every single moment, the money does not grow forever. It reaches a specific point. That point is e.

Math experts also use e for many things. It helps us study growth and decay. Decay is when things get smaller over time.

We also see e in games of chance. For example, imagine a gambler playing a slot machine. If the chance of winning is one in e, the math uses this number. e is also used to find the best way to plan. It is a vital tool for many math problems.

Some numbers are more special than others. The number e is one of these very important values. It is a mathematical constant, which means its value stays the same. This number is approximately 2.71828. It is used as the base for natural logarithms and exponential functions. You might even hear it called Euler's number.

Scientists and mathematicians see this number everywhere in the world. It is considered just as vital as the numbers 0 and 1.

This number describes how things grow or change over time. Imagine you have money in a bank that earns interest. If the bank adds interest every month, your money grows. If they add it every day, it grows even faster. Jacob Bernoulli studied this idea of compound interest in 1683. He found that even if you add interest every single moment, the growth reaches a limit. That specific limit is the number e.

This shows how a small change can lead to a steady pattern.

Many famous people helped us understand this number. John Napier first used similar ideas in 1618. Later, the Swiss mathematician Jacob Bernoulli discovered the constant through his work on interest. Gottfried Leibniz was the first to use the letter e for this number. Around 1727, Leonhard Euler began using the letter e as well. Euler was a very famous mathematician from Switzerland. He showed that e is the sum of an infinite series of numbers.

His work made the symbol e a standard tool for everyone.

There are many interesting facts about how e works. The number e is irrational, so it cannot be written as a simple fraction. It is also transcendental, which is a special way to describe its math properties. In a graph, the slope of the function at one is exactly 1.

This unique trait makes it very helpful for calculus. It is also used to calculate the area under certain curves. If you look at the number to 30 decimal places, it stays very consistent.

You can see e working in many places in your life. It helps describe exponential growth, like when a population gets larger very quickly. It also describes decay, which is when something gets smaller over time. We even see it in games of chance and probability. For example, it can help predict what might happen in a slot machine game.

It even helps solve the "hat check problem" about random mixing. Even if you do not see it, e is helping to explain the patterns of our world.

The mathematical constant *e* is a fundamental value that serves as the base for natural logarithms and exponential functions. It is approximately equal to 2.71828. Along with 0, 1, $\pi$, and $i$, it is considered one of the most important numbers in all of mathematics. The number is irrational, meaning it cannot be expressed as a ratio of two integers. It is also transcendental, which means it is not the root of any non-zero polynomial with rational coefficients.

One way to understand *e* is through the mechanism of continuous compound interest. Imagine an investment that grows at an annual interest rate of 100%. If the interest is added once a year, the total grows to 2. If it is added every six months, the amount grows to 2.25. If interest is added every month, the value approaches a specific limit. Jacob Bernoulli studied this in 1683 and found that as the number of compounding intervals increases toward infinity, the growth approaches *e*.

There are several distinct mathematical ways to define this constant. It is the limit of an expression used to calculate compound interest as the number of intervals becomes infinite. It can also be expressed as the sum of an infinite series: $1/0! + 1/1! + 1/2! + 1/3!$ and so on. Furthermore, *e* is the unique positive number that makes the slope of the function $y = e^x$ exactly 1 at the point where $x = 0$.

The history of *e* involves many famous mathematicians. The first references appeared in 1618 in a work on logarithms by John Napier, though the constant itself was not explicitly named. In 1661, Christiaan Huygens studied logarithms geometrically but did not recognize the constant as a unique entity. Jacob Bernoulli introduced the constant in 1683 while solving problems regarding continuous interest. Later, Gottfried Leibniz used the letter *e* in correspondence around 1690.

Leonhard Euler played a massive role in making *e* a standard mathematical tool. He began using the symbol *e* around 1727 and published it in his work *Mechanica* in 1736. Euler proved that *e* is the sum of an infinite series involving factorials. He also connected *e* to complex numbers through Euler's identity. This identity is a famous formula that links *e*, $i$, $\pi$, 1, and 0 in a single equation.

Beyond finance, *e* is vital in the study of probability and randomness. In Bernoulli trials, such as playing a slot machine with a one-in-*e* chance of winning, the probability of losing every single time approaches approximately 36.79% as the number of tries increases.

Another example is the "hat check problem," or derangements. This problem asks for the probability that no guest at a party receives their own hat when hats are returned at random. As the number of guests grows, this probability approaches $1/e$.

Finally, *e* is essential for describing growth and decay in the physical world. Exponential growth occurs when the rate of change of a quantity is proportional to the quantity itself. This describes many natural processes where things increase at an ever-accelerating rate. Conversely, exponential decay describes quantities that decrease over time. *e* also appears in the standard normal distribution, which is a fundamental concept in statistics used to model many natural phenomena.