Why do some numbers seem magical?
Start with a simple rule: add the previous two numbers. Soon Fibonacci numbers lead to rectangles, spirals, the golden ratio, irrational numbers and infinite fractions. Then explore other numerical curiosities where simple rules produce astonishing patterns.
The six lessons
The Fibonacci sequence
Discover the famous sequence built by adding the previous two numbers, and investigate the patterns hidden inside it.
Fibonacci rectangles and spirals
Turn Fibonacci numbers into growing rectangles and see how a simple number pattern creates a spiral.
The golden ratio
Explore the number φ, its connection with Fibonacci numbers, and why the same ratio keeps appearing.
Numbers that go on forever
Meet decimal expansions that never end and investigate the difference between repeating and non-repeating patterns.
Infinite fractions
Build fractions inside fractions and discover how an infinite process can lead to a surprisingly precise number.
Other magical numbers
Explore more remarkable numbers and patterns, and ask what makes a number feel mathematically special.
Magical Numbers – Complete Course
Download the complete six-lesson course in one bundle, including all six teaching handouts and the complete exercises booklet.
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Magical Numbers – Exercises
Download the complete six-lesson exercise booklet with warm-ups, core practice, reasoning, investigations, challenges and answers.
Get the exercises →How to use this course
There’s no need to rush. Try the calculations and puzzles before opening answers, and use the learning-tip icons only when you want a nudge.
The main route through each lesson is designed for learners from about age 10. Sections marked Advanced extension or Advanced challenge are optional: skip them now and come back later if you like.
The aim is not to memorise a collection of famous numbers. Keep asking: Why does this pattern happen?, What stays the same? and What happens if I change the rule?
Stopping with an interesting question still in your head is a perfectly good place to finish.