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What you will learn in Number Systems & Arithmetic
1. Naturals, Integers and the Number Line
The sign rules stop being rules once negative numbers are directions on a line and absolute value is a distance along it.
2. Fractions, Decimals and Recurring Decimals
Why 1/7 recurs and 1/8 does not — a question about the denominator's prime factors, answered by watching the remainders.
3. Rational Numbers and Density
Between any two rationals lies another — shown by constructing one, not by asserting it.
4. Irrational Numbers and Surds
A length you can draw and can never write as a fraction, with the classical proof that √2 is one of them.
5. The Real Line and Completeness
The rationals are dense and still full of holes. Filling them is what the real line is for.
6. Index Laws, Roots and Rationalisation
Zero, negative and fractional indices derived by keeping a pattern intact, rather than declared and memorised.
7. Standard Form and Orders of Magnitude
Comparing a proton to a galaxy without computing anything, by counting the zeros instead of writing them.
8. Ratio, Proportion and Rates
Direct, inverse and compound proportion — and the reason two ratios cannot simply be added.
9. Percentage, Profit, Loss and Interest
Percentage as a multiplier, which is what makes reverse percentage a division rather than a second subtraction.
10. Primes, Divisibility, HCF and LCM
One factorisation, read two ways: the lower powers give the HCF and the higher powers give the LCM.
11. Estimation, Significant Figures and Error
How many of a calculator's digits you are entitled to keep, and why subtracting two close measurements destroys most of them.