Roots Detective

Find √2 and √5. Then choose 9: why do the two limits become equal?

Explore the controls and explain what changed in your own words.

A diagonal of exact length

A square with side 1 has diagonal √2, since d² = 1² + 1² = 2. The expression √2 indicates the exact value. In decimal, we can write √2 ≈ 1,414. The symbol ≈ means approximately equal.

Surround the root with known numbers

Lower limit Comparison Upper limit
1 1² < 2 < 2² 2
1,4 1,4² < 2 < 1,5² 1,5
1,41 1,41² < 2 < 1,42² 1,42

The smaller the range, the more accurate the location. Draw a straight line from 1,4 to 1,5 to view the second range and then zoom in from 1,41 to 1,42.

Challenge: root or integer?

Sort √16, √2 and 1,414. Which one is a rational approximation of an irrational one?

Check the analysis

√16 = 4 is integer and rational. √2 is irrational. 1,414 = 1414/1000 is rational and approximates √2. A root is not irrational just because it is written with a radical.