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One of my closest math tutors once told me he had a student who asked the question: "Was math discovered or invented?"

At this website calculus is not scary, but it is a bit oversimplified. Calculus is concerned with finding the slope of a line tangent to a function (derivative, think of acceleration), and the area under a function (integral, think of distance traveled). These two concepts make up the fundamental theorem of calculus, which shows that the two concepts are related. It is amazing that derivatives and integrals are inverse operations, like dividing and multiplying. We know intuitively that the faster you speed up, the more ground you cover during that period. Calculus defines and explores the relationship between these concepts.

Consider a letter that is also a number: e = 2.718281828459045235...
When graphed a special way [f(x) = e^x], it looks like this:



The number e is important because it is tied to compounding growth. If a bank compounded interest daily instead of monthly, then hourly, then every minute, then every second, then every millisecond, and so on, mathematicians have found the number e pops out. It represents the maximum performance level of any compounding activity. This applies to interest in the bank, reproducing bacteria, or the growing number of Napster shares in 2000.

Where calculus fits in: The function e^x is it’s own derivate and integral.

For multiplication and division, 1 is the identity value: 1 * 1 = 1 and 1/1 = 1. In calculus, e^x is that identity. Key point: The number e is somehow like the number one, in that it is an identity for the inverse operations of calculus.



At all points along the curve f(x) = e^x, the slope is equal to e^x, the area under the curve is equal to e^x, and the y value (height) is equal to e^x. Using the car example, ‘speed’, ‘acceleration’, and ‘distance’ are equal at all times.

Illustrated graphically:


How is it, that a property of nature, and calculus can be related?

This one is also interesting:

e^pi*sqrt(-1) = -1

Try typing it into your calculator. When we see these seemingly unrelated complex numbers combine in this way it is hard not to wonder what the heck is going on.

pi, and e are letters and not numbers. Regardless of the numeric value their real values are inherently the same. The numbers 2.71... and 3.14... were assigned arbitrarily, tied to the numbering system we use (base 10), but in hex, binary, or anything else they would still have the same meaning.