Placing an Irrational Number

√2 lies between 1 and 2, because 1² = 1 and 2² = 4, and 2 sits between them.

Narrowing it down

1.4² = 1.96 and 1.5² = 2.25. So √2 is between 1.4 and 1.5, and closer to 1.4.

And again

1.41² = 1.9881 and 1.42² = 2.0164, so √2 is between them. The process never finishes.

What irrational means

The decimal never terminates and never repeats. Any written value is an approximation.

Trapping it between two numbers

Since 1² = 1 and 2² = 4, and 2 lies between them, √2 lies between 1 and 2. Squaring 1.4 gives 1.96 and 1.5 gives 2.25, so √2 is between 1.4 and 1.5. Narrowing the trap gives as much precision as you want.

The method is systematic

Guess, square, compare, adjust. Each round adds a decimal place. This is genuinely how square roots were computed before calculators, and doing a couple of rounds by hand makes irrational numbers feel real.

Irrational numbers have places on the line

√2 sits at a definite point on the number line — it is the diagonal of a unit square, which you can construct exactly with a ruler. Being unable to write it as a fraction does not make it any less a number.

Ordering roots

To compare √10 and 3.2, square both: 10 against 10.24. So 3.2 is larger. Squaring turns a comparison of roots into a comparison of ordinary numbers, and it works whenever both are positive.