# Calculating the root of a number by hand

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• NODARman
In summary, the conversation discusses various methods for calculating roots by hand, such as the Cosine formula and the Babylonian method. It also mentions a technique for calculating square roots taught in the 8th grade. The conversation also mentions the Babylonian clay tablets which show how to compute the square root of 2 using sexagesimal representation. The conversation concludes by mentioning that there are many other algorithms for computing square roots.
NODARman
TL;DR Summary
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Hi, is it possible, is there any formula that can help me to take root from (for example) 1,2 without a calculator (by hand)?
For example, there is a cos(x) formula that can be calculated on the paper:
$$\cos x=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}$$

There is the Babylonian method for roots, but it's not as accurate as the cos(x) formula.
$$x^{\prime}=\frac{1}{2}\left(x+\frac{n}{x}\right)$$

NODARman said:
Hi, is it possible, is there any formula that can help me to take root from (for example) 1,2 without a calculator (by hand)?
It's not clear to me what you are asking here.
Which root -- square root, cube root, etc.?
NODARman said:
For example, there is a cos(x) formula that can be calculated on the paper:
$$\cos x=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}$$
This formula has nothing to do with roots. It is the Maclaurin series representation for the cosine function. There's a corresponding series for the sine function and many other functions.

NODARman said:
There is the Babylonian method for roots, but it's not as accurate as the cos(x) formula.
$$x^{\prime}=\frac{1}{2}\left(x+\frac{n}{x}\right)$$
I don't believe this derives from the Babylonians, since they knew nothing about derivatives. This formula derives from a technique called the Newton (or Newton-Raphson) method.

If you're interested in calculating square roots by hand, I was taught a technique back when I was in the 8th grade, a long time ago. The technique is somewhat akin to long division. As far as I know, it's no longer taught. Here's a link to a youtube video:

Last edited:
FactChecker, PeroK, DrClaude and 3 others
Mark44 said:
I don't believe this derives from the Babylonians, since they knew nothing about derivatives.
You don't need to know anything about derivatives to derive this method, you simply need to realise that if ## x ## is an understimate of ## \sqrt n ## then ## \frac n x ## is an overestimate (and vice versa) and therefore the midpoint ## x^{\prime}=\frac{1}{2}\left(x+\frac{n}{x}\right) ## is a better estimate.

The term "Babylonian method" is commonly used, although I am not aware of a confirmed source. See e.g. https://demonstrations.wolfram.com/BabylonianAlgorithmForComputingSquareRoots/.

Edit: https://www.sciencedirect.com/science/article/pii/S0315086098922091 seems to provide a source.

DrClaude, PeroK and NODARman
Mark44 said:
It's not clear to me what you are asking here.
Which root -- square root, cube root, etc.?

This formula has nothing to do with roots. It is the Maclaurin series representation for the cosine function. There's a corresponding series for the sine function and many other functions.I don't believe this derives from the Babylonians, since they knew nothing about derivatives. This formula derives from a technique called the Newton (or Newton-Raphson) method.

If you're interested in calculating square roots by hand, I was taught a technique back when I was in the 8th grade, a long time ago. The technique is somewhat akin to long division. As far as I know, it's no longer taught. Here's a link to a youtube video:

x' is not a derivative of x, it means the new result of x.

NODARman said:
x' is not a derivative of x, it means the new result of x.
Without any explanatory context, a "primed" variable would ordinarily be interpreted to mean the derivative of that variable.

These algorithms fall into something we call algorithms in mathematics. A numerical analysis book, has many of these types of solutions, not just for roots.

To give you a better answer. What are you trying to find the root of? a square root, cubic, function? A particular example would help.

## 1. How do I calculate the square root of a number by hand?

To calculate the square root of a number by hand, you can use the long division method or the Babylonian method. Both methods involve repeatedly dividing and averaging until you reach a close enough approximation of the square root.

## 2. What is the difference between finding the square root and finding the cube root?

Finding the square root of a number means finding a number that, when multiplied by itself, gives the original number. Finding the cube root of a number means finding a number that, when multiplied by itself twice, gives the original number. In other words, finding the cube root is like finding the square root twice.

## 3. Can I use a calculator to find the square root of a number?

Yes, most calculators have a square root function that can quickly and accurately find the square root of a number. However, it is still important to understand how to calculate it by hand for educational purposes.

## 4. Is there a shortcut or formula for finding the square root of a perfect square?

Yes, for perfect squares (numbers whose square roots are whole numbers), there is a shortcut called the "square root method" or "square root formula." This involves breaking down the number into its prime factors and taking the square root of each factor.

## 5. How accurate is calculating the square root of a number by hand?

Calculating the square root of a number by hand can be accurate up to a certain number of decimal places, depending on the method used and the precision of your calculations. However, it may not be as accurate as using a calculator or computer, which can calculate to a much higher degree of precision.

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