"What are the practical uses of e? How can e be used for practical mathematics? This is for a presentation on (among other numbers) e, aimed at people between the ages of 10 and 15. To clarify what I want: Not wanted: eiπ+1=0 is cool, but (as far as I know) it can't be used for practical applications outside a classroom. What I do want: e I think is used in calculations regarding compound interest. I'd like a simple explanation of how it is used (or links to simple explanations), and more examples like this."

Kelton Molina

Kelton Molina

Answered question

2022-09-20

What are the practical uses of e?
How can e be used for practical mathematics? This is for a presentation on (among other numbers) e, aimed at people between the ages of 10 and 15.
To clarify what I want:
Not wanted: e i π + 1 = 0 is cool, but (as far as I know) it can't be used for practical applications outside a classroom.
What I do want: e I think is used in calculations regarding compound interest. I'd like a simple explanation of how it is used (or links to simple explanations), and more examples like this.

Answer & Explanation

pedradauy

pedradauy

Beginner2022-09-21Added 8 answers

e is not useful. It just happens to be the value at x=1 of the exponential function
e x = n = 0 x n n !
which is incredibly useful because it is its own derivative. This fundamental property helps us solve differential equations, which are a very powerful language for understanding the universe.
The boring example is modeling something like the growth of bacteria. (As Raymond Manzoni's comment says, a more interesting example involving exponential growth is carbon dating.) A much more interesting example (to me, anyway) occurs if you allow x to be something more general than a real number: allow it to be complex and you get Euler's formula
e i x = cos x + i sin x
which tells you that the exponential can represent rotations in the plane. It also tells you that cosx and sinx can be used to describe simple harmonic motion, a basic and fundamental example of a differential equation describing real-world phenomena (such as circuits) that we know how to solve exactly thanks to the exponential function.
To be more specific, the current I(t) in an LC circuit satisfies the differential equation
d 2 I d t 2 + 1 L C I = 0
where L is the inductance and C is the capacitance. The complex exponential can be used to deduce that the solutions to this differential equation look like
I ( t ) = A cos ω t + B sin ω t
where ω 2 = 1 L C . This is a very precise description: the current is periodic and we can explicitly compute its period. Add in a resistor and you get a similar differential equation whose solutions can now be damped, and again we can explicitly compute how quickly the current is damped.

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