Worked Example 1
Trap the square root of 2 between two decimals with two places.
Start with whole numbers1² = 1 and 2² = 4, so the root is between 1 and 2
Narrow to one place1.4² = 1.96 and 1.5² = 2.25, so it is between 1.4 and 1.5
Narrow again1.41 < √2 < 1.42
What never happensNo decimal you ever write down squares to exactly 2.
Worked Example 2
Prove that the square root of 2 cannot be written as a fraction.
Assume the opposite√2 = a/b, in lowest terms
Square both sidesa² = 2b², so a² is even, so a is even
Write a as 2k4k² = 2b², so b² = 2k², so b is even too
The contradictionBoth even, so they share a factor of 2. Impossible. √2 is irrational.
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