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Maths for Humans: Inverse Relations and Power Laws

University of New South Wales via FutureLearn

This course may be unavailable.


What’s the connection between the size of a city and the number of gas stations in it? How would you describe this relationship? How would you try to visualise and understand it?

Study real-world connections using algebra and geometry

In this free online course, we’ll look at a wide spectrum of interesting, and often surprising, mathematical relationships in our everyday world. We’ll be studying inverse relations, which complement direct relationships; power laws that occur in economics, physics and biology; as well as curious applications of the logarithm function. We’ll visualise these real-world interconnections using graph paper and pencil, and also modern graphing software such as GeoGebra and Desmos.

The functions and graphs in this course help us to make predictions, evaluate actions and test theories about population distributions, the frequencies of common words in texts, scaling laws in biology and much more. We’ll even see some applications to detecting fraud!

Explore inverse and power relationships

In Week 1 you’ll meet hyperbolas and inverse relations, which are complementary to the direct proportionality that we studied in Maths for Humans: Linear and Quadratic Relations. You’ll learn to represent such relationships algebraically, and understand them geometrically in the Cartesian plane in a direct visual way.

In subsequent weeks, we’ll look at Zipf’s law and Benford’s law, check out applications in engineering and physics, investigate power relations and higher degree polynomials, and explore allometry in biology. You will see the history, look at lots of examples, and get useful insights into: lifespans, whether exercise really is good for you, the distribution of wealth, and the crazy claw of the Fiddler crab. And lots of other juicy applications of advanced high school mathematics.

Gain valuable skills for further study

Understanding basic mathematical relationships is vital to many fields of study: biology, engineering, business, economics, political science and design. By the end of this course, you’ll have hands-on experience with a wide range of explicit examples, be familiar with a core area of pre-calculus mathematics, and be ready to go on to more advanced study of calculus or linear algebra.

Whether you’re encountering these topics for the first time or brushing up on your high school mathematics, we hope you’ll actively join our community on this journey through some fascinating and practical topics that have contributed much to our understanding of the world around us.

This course is aimed at:

  1. senior high school students wanting to strengthen and enrich their understanding of a core pre-calculus topic;
  2. new undergraduates wishing to review and consolidate their background in algebra and geometry in preparation for STEM studies;
  3. high school maths teachers;
  4. anyone with an interest in mathematics and a curiosity about how mathematics is intertwined with the real world.

Taught by

Daniel Mansfield and Norman Wildberger


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