The number that multiplies by itself to reach yours
Find the square root of any positive number, with enough decimal places for geometry, construction and physics work.
A square root answers a geometric question: given an area, how long is the side of the square? The area 144 gives a side of 12.
Most roots are irrational — they run forever without repeating. √2 starts 1.41421356 and never settles, which is exactly what unsettled the ancient Greeks about it.
The square root to several decimal places Instant identification of perfect squares Precision suitable for diagonals, distances and areas Returns the positive root. Strictly, 9 has roots of both 3 and −3. Negative inputs have no real square root; they belong to the imaginary numbers. What is the square root of 144? 12, because 12 × 12 = 144. It is a perfect square, so the answer is a whole number.
Why can't I take the square root of a negative number? Because any real number times itself is positive. Mathematics gets around it by inventing i, where i² = −1, but that leaves the real number line.
How would I estimate √50 without a calculator? It sits between √49 = 7 and √64 = 8, much closer to 7. The actual value is 7.07.
What is a perfect square? A number whose root is a whole number: 1, 4, 9, 16, 25, 36 and so on. Everything between them has an irrational root.
Worked example The diagonal of a square room
Input: √32
Output: 5.657
Note: A room 4 m by 4 m has a diagonal of √(16+16) = √32, about 5.66 m — worth knowing before carrying a sofa through it.
