Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
Welcome to our Math lesson on The History of Euler's Number, this is the second lesson of our suite of math lessons covering the topic of Natural Logarithm Function and Its Graph, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Although commonly associated with Leonhard Euler, the constant e was first discovered in 1683 by the mathematician Jacob Bernoulli. He was trying to determine how wealth would grow if interest were compounded more often than on an annual basis.
Imagine you are lending money to a bank that applies a 100% interest rate compounded once a year. From tutorial 5.4, we know that compound interest is calculated together with the principal P (i.e. by calculating the total amount A) through the compound interest formula:
where n is the number of times the interest is compounded in a year and r is the compound interest rate expressed as a decimal.
Give, the above conditions (n = 1 r = 100% = 1, and n = 1), after one year of deposit we have
If the interest rate decreases but the interest is compounded more often (for example, the interest rate becomes 50% and the interest is compounded twice a year, i.e. r = 50% = 0.5, n = 2, after one year of deposit, we have
If the interest is compounded 4 times a year at a 25% interest rate (n = 4, r = 0.25), we obtain
For 10 compounds a year at 10% interest (n = 10, r = 0.1), we obtain
For 100 compounds a year at 1% interest (n = 100, r = 0.01), we obtain
For 1000 compounds a year at 0.1% interest (n = 1000, r = 0.001), we obtain
As you see, this is a converging series and when the number of terms points to infinity, the value points towards a number that is slighter above 2.7.
Later on, Euler discovered that the series
converges to the same point as well. Therefore, he was the first to realize that this is a special number which needs more attention. Euler used the letter e to denote this new constant. Now when reading the constant 'e' we immediately refer to it as 'Euler's Number' as the letter e corresponds to the first letter of his surname. As said earlier, this is an irrational number written as
Usually, we consider only the four digits after the decimal place when dealing with Euler's Number. Hence, we write
Another feature that makes Euler's Number special is the fact that it is the only number for which the derivative and integral have the same value - a value that corresponds to the original number itself. (Derivatives and integrals are concepts that we are going to explain later in this course).
When Euler's number represents the base of an exponential expression, we often write exp(x) instead of ex ('exp' stands for 'exponential').
To summarize, Euler's number is the base of an exponential function whose rate of growth is always proportionate to its present (actual) 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.
Enjoy the "The History of Euler's Number" math lesson? People who liked the "Natural Logarithm Function and Its Graph lesson found the following resources useful:
Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
We hope you found this Math tutorial "Natural Logarithm Function and Its Graph" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.