The Value of e: Definition & Example

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  • 0:03 Euler's Number
  • 0:39 Irrantional
  • 1:23 Calculating It
  • 2:47 Uses
  • 4:54 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will learn that the mathematical constant e has a name. You will also learn how it is used in mathematics along with what characteristics it has.

Euler's Number

The number e is sometimes called 'Euler's number' because Leonhard Euler, the Swiss mathematician, used this particular letter to represent this mathematical constant. I think it's easy to remember the name of this number. The letter 'e' is also the first letter in Leonhard Euler's last name!

As a mathematical constant, the number e remains the same value no matter what math you do to it. It will always be approximately 2.71828. I say approximately because the number e, like the number pi, is an irrational number.


Being an irrational number, the number e is a real number that goes on forever and cannot be written as the fraction of two numbers. Just like how you can't write a simple fraction to represent pi, you can't write a fraction to represent e. You have to use the symbols to accurately represent these numbers.

But we can approximate irrational numbers. Just like we approximate the number pi to 3.14, we approximate e to 2.71828. We do, however, keep in mind that the number does go on forever, and by using these approximations, our answers will also be approximations and won't be 100% accurate.

Calculating It

There are two ways to calculate e, but both involve infinity.

The first is when we calculate (1 + 1 / n)^n when n is infinity. Sure we can't calculate infinity; we make calculations that get progressively higher. Can we ever reach infinity? No, we can't, but we can get closer and closer to it if we keep making higher and higher calculations. As we do this, we see that we get closer and closer to the number e. For example, (1 + 1 / 100,000)^100,000 = 2.71827, which is correct to four decimal places. If we keep making higher calculations, our answer will be closer and closer to the value of e.

The second way of calculating e is to keep adding onto this series 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5!. . . The exclamation mark is the symbol for factorial, which means we take the number and multiply it by all the numbers it takes to count to that number starting with 1. So 5! means we multiply 1, 2, 3, 4 and 5 together. 4! means we multiply 1, 2, 3 and 4 together. The more we add, the closer we get to the value of e. Try it for yourself.


The number e has a very important use in math as the base of the natural logarithm. You might have seen equations such as y = ln (x). What this means is that if we take the number e to the y power, we will get x as our answer. So, ln (5) will equal 1.609, because if we take e to the power of 1.609 we will get 5 as our answer. On calculators, you will see a button for ln, which is this natural logarithm function.

Another famous use of the number e is used by bankers when they make continuous compounding calculations. What is this? Well, when you have a savings account, your interest, the money you earn, is usually calculated once a month or once a year. This is called periodic compounding. When you increase the frequency of these calculations to the point where it is continuously being calculated and you are paid continuously, it is called continuous compounding.

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