Methods of computing square roots
There are a numbers of ways to calculate square roots of numbers, and even more ways to estimate them.
The mathematical operation of finding a root is the opposite operation of exponentiation, and therefore involves a similar but reverse thought process.
Firstly, one needs to know how precise the result is expected to be. This is because often square roots are irrational. For example, square root of a nice round whole number 28 is a fraction which in its decimal notation has infinite length, and therefore it is impossible to express it exactly:[math]\displaystyle{ \sqrt{28} \approx 5.291502622129181.... }[/math]
Moreover, for some real numbers the square root is a complex number. For example, square root of -4 is a complex number 2i :[math]\displaystyle{ \sqrt{-4} = 2 i }[/math]
In many cases there may be multiple valid answers. For example, square root of 4 is 2, but -2 is also a valid answer. One can verify that they are both valid answers by squaring each candidate answer and checking if you obtain 4 as the result of verification:[math]\displaystyle{ 2^2 = 2 \times 2 = 4 }[/math]
[math]\displaystyle{ (-2)^2=(-2) \times (-2) = 4 }[/math]
Please note that calculating a square root is a special case of the problem of calculating Nth root.
Calculating
Most calculators provide a function for calculation of a square root.
| General Steps | Example | |
|---|---|---|
| How to calculate a square root using a simple calculator. |
|
|
Estimating
If the result does not have to be very precise, the following estimation techniques could be helpful:
| Methodology | Example |
|---|---|
| Suppose you need to find square root of some number [math]\displaystyle{ N }[/math].
Find some number [math]\displaystyle{ A }[/math] such that [math]\displaystyle{ A^2 }[/math] (that is [math]\displaystyle{ A }[/math] squared, or [math]\displaystyle{ A }[/math] times [math]\displaystyle{ A }[/math]) is approximately equal to [math]\displaystyle{ N }[/math] (but how close? This needs to be expanded). Then we can think of [math]\displaystyle{ A }[/math] as being approximately a square root of [math]\displaystyle{ N }[/math]. |
Suppose we need to estimate the square root of 2.
We know that [math]\displaystyle{ 1^2 = 1 }[/math], and [math]\displaystyle{ 2^2 = 4 }[/math]. Therefore, one of the answers to [math]\displaystyle{ \sqrt{2} }[/math] is somewhere between 1 and 2. |
Methods Of Computing Square Roots Media
Semilog graphs comparing the speed of convergence of Heron's method to find the square root of 100 for different initial guesses. Negative guesses converge to the negative root, positive guesses to the positive root.
Graph charting the use of the Babylonian method for approximating a square root of 100 (±10) using starting values*x0 = 50,*x0 = 1,*and x0 = −5. Note that a positive starting value yields the positive root, and a negative starting value the negative root.
A chart showing the IEEE floating format approximating base-2 logarithm.
