What is Euler’s number and Pi ?
In: Mathematics
If you were to take something cylindrical, like a vial of Carmex (the lip balm), and make a mark at one point on the edge, and roll out one complete roll; and then measured the diameter of the Carmex container, you’d notice that the distance of one complete roll (that is, the circumference) equaled *pi* times the diameter.
And the cool thing is it happens with every circle.
Circumference equals diameter times *pi*.
Every. Time.
Written another way, *pi* = Circumference divided by Diameter.
Every time. Every circle.
Circumference divided by Diameter *always* equals 3.14159…
* * * * *
“*e”* is a little more difficult to conceptualize, but it is the result of adding smaller and smaller fractions:
1/1 + 1/(1×2) + 1/(1x2x3) + 1/(1x2x3x4) + 1/(1x2x3x4x5) … = 2.71828
Another way of expressing *e* is *y=1/x*
*pi* and *e* are curious constants that happen to be very useful in maths and physics.
Pi is the ratio of the circumference (distance around) to twice the radius (distance from the center to the edge) of a circle. It’s about 3.14159 but the written decimal goes on forever. Because of a pretty deep connection between circles and trigonometry, and between trigonometry and Euler’s number, pi shows up in a *ton* of places in physics and math.
Euler’s number, e, is about 2.718 (but also goes in forever). It has a number of unique properties but it’s not as easy to intuitively “see” as pi. It shows up in calculus a ton because the slope of certain functions involving e is the the same as the value of the function. Which sounds really abstract but it shows up in all kinds of natural processes where the increase of something depends on how much of it you already have, like compound interest or rabbits reproducing.
As soon as you get even vaguely near differential equations e and pi start popping up all over the place.
Edit: clarified radius vs diameter
PI has already been explained by the other comments, it is just the ratio between a circle’s diameter and circumference. So a circle with diameter 1 has a circumference of pi (around 3.141).
Euler’s number e is a little bit harder to explain. In short, Euler’s number is a way of capturing how things grow or shrink when they change at a continuous rate, like populations, investments, or even certain natural processes. I think the easiest explanation is with interest at a bank.
Imagine you have $1 in a bank that offers 100% interest per year. If the bank pays the interest once a year, you’d get $1 of interest after one year, ending up with $2 in total.
If the bank instead pays out 50% interest every 6 months, this is still 100% interest per year, but you end up with slightly more than $2 in the end, because in the last 6 months the interest from the first 6 months is already in your account, helping to generate more interest during the remaining time of the year. The smaller the intervals of the payout are, the stronger this effect and the more money you end up with after one year.
Now, if the bank pays the interest at the smallest possible interval, which is continuous payment (imagine it’s adding a tiny amount of interest an infinite number of times per second, but still only 100% per year in total), the final amount you’d have after one year is exactly the euler number, which is roughly $2.718.
So e is the factor by which an amount grows when it increases continuously at a rate of 100% for one unit of time—starting from 1, it grows to about 2.718 (instead of 2.0 which is for a single non-continous step of growth).
Looking forward to a good answer here. Tried to explain the concept of Euler’s number to my grade 12 math students in like less than 2 minutes. Used the concept of the fibonacci series. Didn’t work.
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