Level 7: Building number systems

Numbers · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

Every family of numbers you know was invented to answer a question the older family could not. This level looks back at that ladder (counting numbers, integers, fractions, real numbers) and then takes the last step of school mathematics: a number \(i\) whose square is \(-1\). You will calculate with complex numbers, draw them as points on a plane, use them to solve every quadratic equation, and finish with a careful look at distance on the real line: intervals, absolute value and the triangle inequality.

Lesson	You will be able to
Why each family of numbers was invented	Explain which equation forced each new family, what "closed under an operation" means, and why \(x^2 = -1\) needs something new
The number i	Use \(i^2 = -1\), find any power of \(i\) from the cycle of four, write square roots of negative numbers, and tell the cubic-equation story honestly
Complex numbers a + bi	Add, subtract, multiply and divide complex numbers, and use the conjugate and its rules
The complex plane	Plot complex numbers, add them as arrows, find the modulus and the distance between two numbers, and see multiplying by \(i\) as a quarter turn
Every quadratic has roots	Solve any quadratic with real coefficients, explain why the roots come in conjugate pairs, and read the discriminant in the complex world
Intervals and absolute value	Write sets of numbers in interval notation, solve absolute value equations and inequalities as distance problems, and prove the triangle inequality

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