Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — proving that the square root of 2 is irrational
Prove that 2 is irrational.
- 1Suppose, for contradiction, that 2 is rational.Every proof of irrationality at this level starts by assuming the opposite.
- 2Then 2=qp with p,q integers having no common factor.The words 'no common factor' are what will be contradicted.
- 3Squaring: 2=q2p2, so p2=2q2.
- 4p2 is even, so p is even.An odd number squared is odd, so an even square forces an even root.
- 5Write p=2m: 4m2=2q2, so q2=2m2.
- 6q2 is even, so q is even.
- 7Both p and q are even — a common factor of 2, contradicting the assumption.So 2 cannot be rational. ■
Example 2 — converting a recurring decimal
Express 0.1˙8˙ as a fraction in its lowest terms.
- 1Let x=0.181818…Name the number first.
- 2The repeating block has two digits, so multiply by 100.One full period is shifted past the point.
- 3100x=18.181818…
- 4Subtract: 99x=18The infinite tails cancel exactly — that is the whole trick.
- 5x=9918=112Divide top and bottom by 9. Any recurring decimal is rational.
Example 3 — a rational plus an irrational
Prove that the sum of a rational number and an irrational number is always irrational.
- 1Let r be rational and i irrational, and suppose for contradiction that r+i=s is rational.
- 2Rearrange: i=s−rBoth s and r are rational.
- 3Q is closed under subtraction, so s−r is rational.This is the step that does the work.
- 4That makes i rational, contradicting the assumption.
- 5Therefore r+i is irrational. ■Contrast with two irrationals, where no such rule holds.