AI Tutor
If you build a square garden with an area of 5 ft25\text{ ft}^2, each side must measure exactly 5 ft\sqrt{5}\text{ ft}. But what mark on your tape measure do you cut?

Trapped Between Squares

An irrational number is a value that cannot be written as a fraction of two integers, meaning its decimal expands forever without repeating. Around 500 BCE, the Greek mathematician Hippasus discovered that the diagonal of a unit square was irrational, shattering the belief that all quantities could be expressed as simple ratios.
Because the square root function increases steadily, if a<x<ba < x < b, then a<x<b\sqrt{a} < \sqrt{x} < \sqrt{b}. To estimate any square root, first find the two nearest perfect squares — integers created by multiplying a whole number by itself.
We know 5\sqrt{5} lies strictly between 22 and 33, but how do we pin down its first decimal place?

Zooming In to Tenths

Since 55 is much closer to 44 than to 99, we test tenths near 22: (2.2)2(2.2)^2 and (2.3)2(2.3)^2.
Because 4.84<5<5.294.84 < 5 < 5.29, 5\sqrt{5} is between 2.22.2 and 2.32.3. Since 5−4.84=0.165 - 4.84 = 0.16 while 5.29−5=0.295.29 - 5 = 0.29, 55 is closer to 4.844.84, making 5≈2.2\sqrt{5} \approx 2.2 to the nearest tenth.
A common mistake is guessing the halfway decimal 2.52.5 because 55 feels halfway between integers. But squaring is non-linear: (2.5)2=6.25(2.5)^2 = 6.25, which overshoots 55 by a huge margin.
What if an engineering project demands even tighter precision, like the nearest hundredth?

Refining to Hundredths