Mathematical Numbers: What Is 'e'?
An Interesting Interest Problem
Suppose you put £1 into a savings account at your bank which gives an incredible 100% interest rate paid at the end of the year. 100% of £1 is £1, so at the end of the year you have £1 + £1 = £2 in your bank account. You've basically doubled your money.
Now Let's Make It More Interesting
Now suppose instead of getting 100% at the end of the year, your interest is halved to 50%, but paid twice per year. Furthermore suppose that you get compound interest i.e. you earn interest on any earlier interest received as well as interest on the original lump sum.
Using this method of interest, after 6 months you get your first interest payment of 50% of £1 = 50p. At the end of the year you get 50% of £1.50 = 75p, so you end the year with £1.50 + 75p = £2.25, 25p more than if you had 100% interest in a one-off payment.
Using this method of interest, after 6 months you get your first interest payment of 50% of £1 = 50p. At the end of the year you get 50% of £1.50 = 75p, so you end the year with £1.50 + 75p = £2.25, 25p more than if you had 100% interest in a one-off payment.
Splitting The Interest Into Four
Now let's try the same thing but this time split the interest into four so you get 25% interest every three months. After three months we have £1.25; after six months it is £1.5625; after nine months it is £1.953125 and finally at the end of the year it is £2.441406. We get even more this way than we did by splitting the interest into two payments.
Splitting The Interest Further
Based on what we have so far, it looks like if we keep splitting our 100% into smaller and smaller chunks paid out with compound interest more frequently, then the amount that we end up with after one year will keep on increasing forever. Is this the case however?
In the table below, you can see how much money you will have at the end of the year when the interest is split up into progressively smaller chunks, with the bottom row showing what you would get if you earned 100/(365×24×60×60)% every second.
In the table below, you can see how much money you will have at the end of the year when the interest is split up into progressively smaller chunks, with the bottom row showing what you would get if you earned 100/(365×24×60×60)% every second.
How Much is in the Savings Account at the End of the Year?
How often is the interest paid? |
Amount at the end of the year (£) |
Yearly |
2 |
Half-yearly |
2.25 |
Quarterly |
2.441406 |
Monthly |
2.61303529 |
Weekly |
2.692596954 |
Daily |
2.714567482 |
Hourly |
2.718126692 |
Every minute |
2.71827925 |
Every second |
2.718281615 |
The Limiting Value
You can see from the table that the numbers are tending towards an upper limit of 2.7182... . This limit is an irrational (never ending or repeating decimal) number which we call 'e' and is equal to 2.71828182845904523536... .
Perhaps a more recognisable way of calculating e is:
e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + ... where ! is factorial, meaning multiply all of the positive integers up to and including the number e.g. 4! = 4×3×2×1 = 24.
The more steps of this equation you type into your calculator, the closer your answer will be to e.
Perhaps a more recognisable way of calculating e is:
e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + ... where ! is factorial, meaning multiply all of the positive integers up to and including the number e.g. 4! = 4×3×2×1 = 24.
The more steps of this equation you type into your calculator, the closer your answer will be to e.
Why is 'e' Important?
e is an extremely important number within the world of mathematics. One major use of e is when dealing with growth such as economic growth or population growth. This is particularly useful at the moment when modelling the spread of coronavirus and the increase in cases across a population.
It can also be seen in the bell curve of the normal distribution and even in the curve of the cable on a suspension bridge.
It can also be seen in the bell curve of the normal distribution and even in the curve of the cable on a suspension bridge.
Portrait of Leonhard Euler by Jakob Emanuel Handmann (1753)
Euler's Indentity
One of the most incredible appearances of e is in Euler's Identity, named after the prolific Swiss mathematician Leonhard Euler (1707 - 1783). This identity brings together five of the most important numbers in mathematics (π, e, 1, 0 and i = √-1) in a beautifully simple way.
e^iπ + 1 = 0
Euler's Identity has been compared to a Shakespeare sonnet and described by the renowned physicist Richard Feynmann as the 'most remarkable formula in mathematics'.
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