Easy as 1, 2, e

Sometimes I want to press ‘3’ on the keyboard and I misjudge where my finger goes and I hit ‘e’ instead. Anyone else get that? No? Just me?

However, today I did mean to press e.

Say you put £1.00 in an extremely generous but realistically non-existent bank account that credited 100% interest per year. At the end of the year, you would have £2.00.

If instead, the bank credited 50% interest every half a year, after the first six months you would have £1.00 × 1.5 = £1.50. After the second six months you would have £1.50 × 1.5 = £2.25, slightly more.

If the bank credited 25% interest every quarter of a year, you would end up with £1.00 × 1.254 = £2.44 (to the nearest penny).

What happens if this interest is credited more frequently?

  • Every month: £2.6130…
  • Every week (assuming 52 weeks in a year): £2.6926…
  • Every day (assuming 365 days in a year): £2.7146…
  • Every hour (assuming 8760 hours in a year): £2.7181…
  • Every second (assuming 31536000 seconds in a year): £2.7183…

The values can be calculated using the expression:

Limit - Exponential

n is the number times per year the interest is credited. You can see that this sequence is converging to some value of around 2.7. As n becomes larger and larger, the value of the above expression approaches the number e. Yes, you heard correctly. The number e.

Here is the value of e to fifty decimal places for your pleasure:

2.71828182845904523536028747135266249775724709369995

But of course, nobody wants to write that all out so it has been assigned the letter e. e is one of the most important constants in maths and in physics, and pops up in many equations and fields of study.

e is irrational, meaning it cannot be expressed as a simple fraction (therefore it has an infinite number of decimal places). e is also transcendental, meaning that it cannot be expressed as the solution of an algebraic equation.

One of the very special properties of e I will describe here. If you plot a graph of y = ex, you get one that looks like this (credits to Desmos, the excellent free graph plotter):

y = e^x

You can find the gradient of the curve at any point by drawing a tangent (a straight line that only just touches that point). The blue line is the tangent of the curve at x = 1:

y = e^x and y = ex

The slope of the tangent shows how steep the curve is at that point. What is special though, is that the gradient of y = ex at any point is equal to its y-value. In the case above, the gradient at x = 1 is e, and the y-value of the point where they touch is also e.

Therefore, if you plot the gradients at every point as y-values (this is the graph of the derivative), you get the exact same graph. This is the reason why e is fundamental to many mathematical and physical theories.

Mind = blown, right?

Don’t forget to share our page! It is indeed as easy as 1, 2, e.

Yanhao

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