Methods of computing square roots

The square root of a number ${\displaystyle n}$ is whatever number equals ${\displaystyle n}$ when multiplied by itself (squared). There are many ways to calculate square roots, from estimations to precise calculations.

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 square roots are usually irrational. For example, the square root of 28 is a number with infinite digits, and therefore it is impossible to express it exactly:$${\displaystyle {\sqrt {28}}\approx 5.291502622129181....}$$

Moreover, calculations to calculate square roots usually only give one answer, so it is important to remember that all square roots except for ${\displaystyle {\sqrt {0}}}$ have two answers. Each answer is the negative of the other answer. For example, the square roots of 4 are 2 and -2. You can confirm this by squaring 2 and -2, and seeing that they both result in 4:$${\displaystyle 2^{2}=2\times 2=4}$$

$${\displaystyle (-2)^{2}=(-2)\times (-2)=4}$$

Additionally, the square roots of a negative number are two complex numbers. For example, the square root of -4 is 2i and -2i.

$${\displaystyle {\sqrt {-4}}=2i}$$

$${\displaystyle 2i={\sqrt {-4}}}$$

$${\displaystyle 2i^{2}=-4}$$

$${\displaystyle 2{\sqrt {-1}}^{2}=-4}$$

$${\displaystyle 2{\sqrt {-1}}*2{\sqrt {-1}}=-4}$$

$${\displaystyle 4*-1=-4}$$

$${\displaystyle -4=-4}$$

Please note that calculating a square root is a special case of the problem of calculating N^th root.

Most calculators provide a function for calculation of a square root.

If the result does not have to be very precise, the following estimation techniques could be helpful:

Methodology

Example

Suppose you need to estimate the square root of some number ${\displaystyle N}$. Find some whole number ${\displaystyle A}$ such that ${\displaystyle A^{2}}$ is less than ${\displaystyle N}$, while ${\displaystyle (A+1)^{2}}$ is greater than ${\displaystyle N}$. Now we know that the square root of ${\displaystyle N}$ is somewhere between ${\displaystyle A}$ and ${\displaystyle A+1}$, so we can use ${\displaystyle A+0.5}$ as an approximate square root of ${\displaystyle N}$.

Suppose we need to estimate the square root of 2. ${\displaystyle 1^{2}=1}$, which is less than 2, and ${\displaystyle 2^{2}=4}$, which is more than 2. We now know ${\displaystyle {\sqrt {2}}}$ is somewhere between 1 and 2. Therefore, ${\displaystyle {\sqrt {2}}}$ is approximately 1.5.

You can get more accurate results by doing this technique again on the result. For example:

• ${\displaystyle 1.5^{2}}$ is more than 2, so ${\displaystyle {\sqrt {2}}}$ is somewhere between 1 and 1.5. The halfway point of 1 and 1.5 is 1.25, so ${\displaystyle {\sqrt {2}}}$ is approximately 1.25.
• ${\displaystyle 1.25^{2}}$ is less than 2, so ${\displaystyle {\sqrt {2}}}$ is somewhere between 1.25 and 1.5. The halfway point of 1.25 and 1.5 is 1.375, so ${\displaystyle {\sqrt {2}}}$ is approximately 1.375.
• ${\displaystyle 1.375^{2}}$ is less than 2, so ${\displaystyle {\sqrt {2}}}$ is somewhere between 1.375 and 1.5. The halfway point of 1.375 and 1.5 is 1.4375, so ${\displaystyle {\sqrt {2}}}$ is approximately 1.4375.
• ${\displaystyle 1.4375^{2}}$ is more than 2, so ${\displaystyle {\sqrt {2}}}$ is somewhere between 1.375 and 1.4375. The halfway point of 1.375 and 1.4375 is 1.40625, so ${\displaystyle {\sqrt {2}}}$ is approximately 1.40625.

With each step, the range of possible values that ${\displaystyle {\sqrt {2}}}$ can be is cut in half.

The Babylonian method is a much faster way to estimate the square root of a number.

Methodology	Example
Suppose you need to estimate the square root of some number ${\displaystyle N}$. Start by guessing the result, as accurately as possible. The better this guess is, the faster the method is. We'll call this guess ${\displaystyle G}$. Calculate ${\displaystyle (G+N/G)/2}$. This will be a more accurate guess. Set this calculation as the new value of ${\displaystyle G}$. Go back to step 2.	Suppose we need to estimate the square root of 2. We'll start with a guess of 1.5. The new guess will be ${\displaystyle (1.5+2/1.5)/2}$ = 1.4166666... Let's repeat for an even better guess. ${\displaystyle (1.4166666+2/1.4166666)/2}$ = 1.41421569...

• "Square Root".

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