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This content is taken from the UNSW Sydney's online course, Maths for Humans: Linear, Quadratic & Inverse Relations. Join the course to learn more.
3.1

## UNSW Sydney

Skip to 0 minutes and 13 seconds This week, we’re going to be looking at inverse relationships, which is all about the geometry of the hyperbola, going also back to Apollonius and the ancient Greeks. The hyperbola has special properties that make it perhaps the coolest conic section. With a very simple algebraic equation, y equals 1 over x, it’s particularly amenable to the Cartesian approach. In fact, it really contains a lot of interesting calculus type notions underneath the story. So we’re going to talking about applications to physics, including the ideal gas law– very important result from chemistry– and also Ohm’s law from electricity. These are very important applications of inverse relations.

Skip to 0 minutes and 55 seconds Then we’re going to shift to a much more human aspect, looking at distributions of words and also populations of cities in the modern world. We’re going to look at a remarkable result called Zipf’s law, which is also all about inverse relationships. And then, we’ll have a look at Benford’s Law, which is all about the distribution of digits in numbers in documents. Finally, we’re going to talk about what happens when we scale and shift hyperbolas around. We get more general kinds of rational functions called fractional linear transformations going back to the great German geometer Mobius. So the geometry of the hyperbola and inverse relations sort of a precursor to some calculus ideas, but of independent interest, lots of interesting examples.

# Inverse relations, hyperbolas and Zipf's Law

Here we introduce week 3: Inverse relations, hyperbolas and Zipf’s law, giving an overview to what is ahead.

The main object of study this week is the basic inverse relation given by

$\Large y=\frac{a}{x}$

for some fixed constant $$\normalsize a$$. This is also called an inverse proportionality, and is based on the geometry of the hyperbola – perhaps the most interesting of the conic sections.

Inverse relations are very important in physics, for example Boyle’s Law in the theory of gases, and Ohm’s Law in the theory of electricity. Zipf’s law is a fascinating and surprising phenomenon that occurs in linguistics, population studies and quite a few other aspects of modern economies. Benford’s Law is a mysterious but surprisingly useful numerical pattern hidden in many types of data. And the logarithm function will also make a surprise appearance. All of these ultimately are connected to the simple inverse relation $$\normalsize y=\frac{a}{x}$$.