What you’ll learn
Compare irrational and rational numbers.
Key idea
For nonnegative values, comparing squares preserves order; an approximation is not an exact equality.
For nonnegative values square order is preserved; irrational does not mean larger.
Worked example
Classify examples and non-examples for the concept 'Compare irrational and rational numbers'.
Compare √2 with 1.6 by exact squares. Choose the relation with evidence.
- Both √2 and 1.6 are positive; compare their squares 2 and 64/25.
- Since 2<64/25, √2<1.6; neither radical notation nor irrationality alone sets order.
Answer: √2<1.6 — 2<1.6²=64/25, and both quantities are positive.
Quick Check
Classify examples and non-examples for the concept 'Compare irrational and rational numbers'.
Compare √2 with 4.1 by exact squares. Choose the relation with evidence.