A student claims that 11000\displaystyle \frac{1}{1000}10001 is the smallest positive rational number.
Their teacher gives this argument.
Step 1: Assume there is a smallest positive rational number, and call it rrr.
Step 2: Let s=r2\displaystyle s = \frac{r}{2}s=2r.
Step 3: Then s s\,s is rational and 0<s<r0 < s < r0<s<r.
Explain why Step 3 gives a contradiction.
Use a similar method to prove by contradiction that there is no largest negative rational number.
303 exam-style questions on OCR (MEI) A Level Maths 1.2 Algebra, covering 1.2.1 Algebraic vocabulary and notation, 1.2.2 Solve linear equations, 1.2.3 Change the subject of a formula, 1.2.4 Solve quadratic equations, 1.2.5 Discriminant of a quadratic, 1.2.6 Linear simultaneous equations, 1.2.7 One linear one quadratic simultaneous equations, 1.2.8 Points of intersection and solutions, 1.2.9 Linear inequalities, 1.2.10 Quadratic inequalities, 1.2.11 Expressing solutions of inequalities, 1.2.12 Use and manipulate surds, 1.2.13 Rationalise the denominator, 1.2.14 Laws of indices, 1.2.15 Negative, fractional and zero indices, 1.2.16 Proportional relationships, 1.2.17 Partial fractions (A-level only), and 1.2.18 Simplify rational expressions (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.