√5 is irrational — 2.2361 to four decimal places, 2.24 to two
√5 ≈ 2.2361. 5 is not a perfect square, so the root is an irrational number and 2.2361 is a rounded value. Below is where that answer comes from, how to check it in one step, and how to estimate any square root without a calculator.
√5 ≈ 2.2361. Rounded to two decimal places that is 2.24.
5 is not a perfect square, so its root is not a whole number. The nearest whole numbers either side are 2 and 3: 2 × 2 = 4 and 3 × 3 = 9, and 5 sits between those two, so √5 sits between 2 and 3.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 2.2361 is a rounded value, not the exact one — the exact answer can only be written as √5 itself.
Square your answer and see how close it lands. 2.24 × 2.24 = 5.0176, which is within a whisker of 5 — close enough to confirm the root without a calculator agreeing digit for digit.
Because rounding happens at the end, squaring a rounded root will never give back exactly 5. That gap is the rounding, not an error in the root.
Find the two square numbers it falls between and you have already bracketed the answer. 4 and 9 are 2² and 3², and 5 lies between them, so the root is between 2 and 3.
To sharpen the estimate, see how far along the gap the number sits. From 4 to 9 is a jump of 5, and 5 is 1 of the way in, so the root lands roughly 2.2. That linear guess is usually right to within a tenth, which is plenty for a sanity check.
Knowing the square numbers up to 25² by heart is what makes this instant. They are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576 and 625.
Square roots turn up any time an area, a distance or a spread has to be turned back into a length. The diagonal of a rectangle, the standard deviation of a set of numbers, the distance between two points on a grid — all of them finish with a square root.
That is why estimating one quickly is worth more than memorising a decimal. Being able to say "√5 is a bit over 2" in a second is what keeps a longer calculation moving.
The square root of 5 is approximately 2.2361, or 2.24 to two decimal places. It is an irrational number, so the decimal does not end.
No. 5 falls between 4 (2²) and 9 (3²), so its square root is not a whole number.
2 × 2 = 4. Squaring is the inverse of taking a square root, so it is also the way to check a root you have just worked out.
Bracket it between the two nearest square numbers — 4 and 9 — then judge how far along the gap 5 sits. That gets you to roughly 2.2, which is close enough for almost any mental check.
The square numbers up to 25² each have a page here, and the square numbers index lists them all in order.