# Maths for Humans: Inverse Relations and Power Laws

Strengthen your algebra, geometry and thinking skills by learning about fascinating mathematical relations from daily life.

5,771 enrolled on this course

• Duration

4 weeks
• Weekly study

3 hours

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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.

Skip to 0 minutes and 8 seconds NORMAN WILDBERGER: We live in a highly complex and interconnected world. Making sense of it can be a challenge, but looking closer we can find simple connections that can be understood using high school mathematics.

Skip to 0 minutes and 23 seconds Hi, I’m Norman Wildberger and we’re here at the University of New South Wales. This course will show you how to use mathematics to explore relationships and answer questions about the real world. How much longer does it take to download a movie in HD? Does your city have enough gas stations? What’s the connection between how much you exercise and your life expectancy? These questions are about relationships between variable quantities in our world. Some relationships are very familiar from everyday life, such as how the price of a pork chop depends on its weight. Others express important physical or mathematical laws, like the fact that the acceleration of an object is directly proportional to the force on it.

Skip to 1 minute and 7 seconds Or that the area of a planar figure scales quadratically with its size. And some correspondences surprise us, such as how populations of cities are distributed in a given country. Historically, these ideas rest on 17th century discoveries

Skip to 1 minute and 21 seconds of Fermat and Descartes: that mathematical relations can be modelled with algebraic equations and visualised with a sheet of graph paper. These insights brought together Greek geometry and Arabic algebra and set the stage for calculus and the Newtonian revolution in physics. In this course, we’ll look at understanding linear, quadratic and inverse functions and their graphs, with applications to a wide variety of real life situations. You’ll strengthen your skills in algebra and geometry, connect with science and economics, and solve a wide variety of interesting, fun and sometimes challenging problems. Finishing this course will be valuable to senior high school and incoming college and university students wanting to review an essential pre-calculus topic.

Skip to 2 minutes and 11 seconds It’ll be useful to high school teachers and to anyone with an interest in how the remarkable power of mathematics helps us understand the world around us.

## What topics will you cover?

• The geometry of hyperbolas and the algebra of inverse relations
• Applications of inverse relations to gases, springs and circuits
• Zipf’s law, Benford’s law and logarithms
• Rational functions, Mobius transformations, and arithmetic with matrices
• Cubic relations, their geometry and connections with Fermat’s last theorem
• Power laws and their roles in economics and biology
• More general relations appearing in physics, graph theory and geometry

## Learning on this course

On every step of the course you can meet other learners, share your ideas and join in with active discussions in the comments.

## What will you achieve?

By the end of the course, you‘ll be able to...

• Model real world phenomena with mathematics
• Explore applications of power laws to biology, economics and physics
• Engage with the interaction between the algebra and geometry of conics and cubics
• Identify key properties of inverse relations and their applications

## Who is the course for?

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.

## Who will you learn with?

### Norman Wildberger

Norman is a research mathematician with a strong interest in education. He is the discoverer of Rational Trigonometry, and his YouTube site (user njwildberger) has more than 30K subscribers.

### Daniel Mansfield

Daniel likes to learn. Most of the time he does this vicariously through teaching Mathematics at UNSW Australia. His research interests are ergodic and dimension theory.

## UNSW Sydney

Established in 1949 with a unique focus on the scientific, technological and professional disciplines, UNSW is a leading Australian university committed to making a difference

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### Join a global classroom

• Experience the power of social learning, and get inspired by an international network of learners
• Share ideas with your peers and course educators on every step of the course