1) The Pythagorean Theorem : This theorem is foundational to our understanding of geometry. It describes the relationship between the sides of a right triangle on a flat plane: square the lengths of the short sides, a and b, add those together, and you get the square of the length of the long side, c.
This relationship, in some ways, actually distinguishes our normal, flat, Euclidean geometry from curved, non-Euclidean geometry. For example, a right triangle drawn on the surface of a sphere need not follow the Pythagorean theorem.
2) Logarithms : Logarithms are the inverses, or opposites, of exponential functions. A logarithm for a particular base tells you what power you need to raise that base to to get a number. For example, the base 10 logarithm of 1 is log(1) = 0, since 1 = 10 0 ; log(10) = 1, since 10 = 10 1 ; and log(100) = 2, since 100 = 10 2 .
The equation in the graphic, log(ab) = log(a) + log(b), shows one of the most useful applications of logarithms: they turn multiplication into addition.
Until the development of the digital computer, this was the most common way to quickly multiply together large numbers, greatly speeding up calculations in physics, astronomy, and engineering.
3) Calculus : The formula given here is the definition of the derivative in calculus. The derivative measures the rate at which a quantity is changing. For example, we can think of velocity, or speed, as being the derivative of position - if you are walking at 3 miles per hour, then every hour, you have changed your position by 3 miles.
Naturally, much of science is interested in understanding how things change, and the derivative and the integral - the other foundation of calculus - sit at the heart of how mathematicians and scientists understand change.
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