What a square root is, and why there are two of them
A square root of n is a number whose square is n. Since 7 × 7 = 49, seven is a square root of 49 — and so is −7, because a negative times a negative is positive. Every positive number therefore has exactly two real square roots, equal in size and opposite in sign.
The radical symbol √ is defined to mean the principal root, the non-negative one. So √49 = 7, full stop, and if you want both you write ±√49. This is a convention rather than a theorem, and it exists so that √ is a proper function — one input, one output. It is also why the quadratic formula carries an explicit ±: the formula needs both roots, and the radical only supplies one.
Geometrically, the square root is the side length of a square with area n. A square of area 50 square metres has sides of 7.07 m. That reading explains both the name and the fact that negative inputs have no real answer: no square has a negative area, and no real length squares to −16.
Negative radicands are not meaningless, though. Mathematicians define the imaginary unit i by i² = −1, and then √−16 = 4i. Complex numbers built this way are not a curiosity: they are the standard working language of alternating-current circuit analysis, signal processing and quantum mechanics.
How square roots are actually computed
No formula produces √2 in a finite number of arithmetic operations, because √2 is irrational — its decimal expansion never terminates and never repeats. So every square root of a non-perfect square is computed by successive approximation. The method that matters is Newton's iteration, known in this form to Babylonian mathematicians and sometimes still called Heron's method:
xk+1 = (xk + n / xk) ÷ 2
The idea behind it is simple enough to hold in your head. If your estimate x is too small, then n/x must be too large, and the true root lies between them. Averaging the two gives a better estimate. Repeat.
What makes it valuable is the speed. The error is squared at each step, so the number of correct digits roughly doubles: starting from a rough guess, four or five iterations give you full double-precision accuracy. Watch it converge on √50 from a deliberately poor start of 25: 25 → 13.5 → 8.6019 → 7.2073 → 7.0724 → 7.071068, with the next step correct to every digit a calculator displays. Five steps from an estimate three and a half times too big.
Two other methods are worth knowing. Bracketing by perfect squares gives a fast mental estimate: 50 lies between 49 and 64, so √50 lies between 7 and 8, much nearer 7. Long division for square roots, taught before calculators, produces one digit at a time and is exact at every stage, but it is slow and nobody uses it now except as an exercise.
Whether a whole number is a perfect square is a separate question with an exact answer: compute the root, round it, square the rounded value, and see whether you get back what you started with. That test is exact in integer arithmetic and is what this calculator uses, rather than inspecting decimal digits.
Worked example: √50 by hand
Find √50 to six decimal places without a calculator.
- Bracket it. 7² = 49 and 8² = 64. Since 50 is only just above 49, the root is a little above 7. First estimate: 7.1.
- One Newton step. 50 ÷ 7.1 = 7.042254. Average with 7.1: (7.1 + 7.042254) ÷ 2 = 7.071127. Already correct to four decimals.
- A second step. 50 ÷ 7.071127 = 7.071009. Average: (7.071127 + 7.071009) ÷ 2 = 7.0710678. That is correct to every digit shown.
- Check. 7.0710678² = 50.0000000 to the precision carried. The check is the definition, and it costs one multiplication.
- Both roots. The solutions of x² = 50 are +7.0710678 and −7.0710678. The radical denotes only the first.
A useful cross-check. √50 = √(25 × 2) = 5√2, and √2 = 1.41421356, so 5 × 1.41421356 = 7.0710678. Pulling perfect-square factors out from under the radical is the standard way to simplify a surd, and it turns an unfamiliar root into a multiple of one you have memorised.
And an estimate you can do in your head. Between consecutive squares, the root moves almost linearly. From 49 to 64 is 15 units of n for 1 unit of root, and 50 is one unit past 49, so the root is about 7 + 1/15 = 7.0667. That is 0.062% below the true value, which is more than enough for a sanity check on a calculator answer.
Reading the result
Check the bracket first. The root must lie between the two perfect squares shown, and much closer to whichever square the number is nearer. If your answer falls outside that range, something has been mistyped.
Then note the size relationship, which reverses at 1. For a number greater than 1, the square root is smaller than the number: √50 = 7.07. For a number between 0 and 1, the root is larger than the number: √0.25 = 0.5. At exactly 1 and exactly 0 the root equals the number. Rooting always moves a positive value toward 1, which is the same behaviour as raising it to any power between 0 and 1.
Treat a non-terminating decimal as an approximation, and keep the exact form when you can. √2 is exact; 1.41421356 is not, and squaring the rounded value gives 1.99999999. In geometry, trigonometry and algebra, carrying the radical through to the end and only evaluating at the last step avoids accumulating that error. This is why answers are conventionally left as 5√2 rather than 7.0710678.
Finally, remember that the calculator's eight decimal places are display precision, not measurement precision. If the number under the radical came from a measurement good to three significant figures, the root deserves three significant figures too — the significant figures calculator applies the rules properly.
Perfect squares and common irrational roots
| n | n² | Non-square | √n |
|---|---|---|---|
| 1 | 1 | 2 | 1.414214 |
| 2 | 4 | 3 | 1.732051 |
| 3 | 9 | 5 | 2.236068 |
| 4 | 16 | 6 | 2.449490 |
| 5 | 25 | 7 | 2.645751 |
| 6 | 36 | 8 | 2.828427 |
| 7 | 49 | 10 | 3.162278 |
| 8 | 64 | 11 | 3.316625 |
| 9 | 81 | 12 | 3.464102 |
| 10 | 100 | 13 | 3.605551 |
| 11 | 121 | 14 | 3.741657 |
| 12 | 144 | 15 | 3.872983 |
| 13 | 169 | 17 | 4.123106 |
| 14 | 196 | 18 | 4.242641 |
| 15 | 225 | 19 | 4.358899 |
| 16 | 256 | 20 | 4.472136 |
| 17 | 289 | 21 | 4.582576 |
| 18 | 324 | 22 | 4.690416 |
| 19 | 361 | 23 | 4.795832 |
| 20 | 400 | 24 | 4.898979 |
The right-hand pair lists every whole number from 2 to 24 that is not a perfect square, and every one of those roots is irrational. Knowing √2, √3 and √5 to four figures lets you estimate most others by factoring: √18 = 3√2 and √20 = 2√5.
Square-root pitfalls
- Writing √49 = ±7. The radical means the principal root only, so √49 = 7. The equation x² = 49 has two solutions; the symbol does not.
- Assuming √(a + b) = √a + √b. √(9 + 16) = 5, while √9 + √16 = 7. Roots distribute over products and quotients, never over sums.
- Cancelling a square with a root without care. √(x²) is |x|, not x. For x = −5 the answer is 5, which matters whenever the variable can be negative.
- Rounding the root and then squaring it. 1.414² = 1.999396, not 2. Carry the exact form as far as you can and evaluate once.
- Expecting a negative input to produce a negative root. It produces an imaginary one. √−16 is 4i, not −4, because (−4)² is +16.
- Reading eight displayed decimals as eight known decimals. The precision of the answer is limited by the precision of the input, not by the width of the display.
Where square roots come from and what to use next
A square root is a power with exponent one half, so everything here is a special case of the exponent calculator: √n = n0.5, and the same tool handles cube roots as n1/3 and any other root as n1/k. Roots can also be computed through logarithms — halve the logarithm and take the antilog — which the logarithm calculator does, and which is how slide rules extracted roots.
The most common source of square roots in school algebra is the quadratic formula, where the root of the discriminant decides whether a quadratic has two real solutions, one, or none. The quadratic equation calculator works that through, including the case where the discriminant is negative and the roots are complex — the same imaginary result this calculator reports for a negative input.
Simplifying a surd means pulling perfect-square factors out from under the radical, which requires knowing the factorisation: √72 = √(36 × 2) = 6√2. The prime factorization calculator gives you the factors with their exponents, and any prime appearing an even number of times comes out whole.
Roots of very large or very small numbers are easier to handle in scientific notation, because taking a square root halves the exponent: √(9 × 1016) = 3 × 108. Make the exponent even before you halve it, and the arithmetic stays trivial.