Calculating the root of a number by hand

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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.
  • #1
NODARman
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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.
$$
x^{\prime}=\frac{1}{2}\left(x+\frac{n}{x}\right)
$$
 
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  • #2
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.?
In your example are you asking about the square root of 1.2; i.e. ##\sqrt{1.2}##?
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:
 
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  • #3
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.
 
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  • #4
Mark44 said:
It's not clear to me what you are asking here.
Which root -- square root, cube root, etc.?
In your example are you asking about the square root of 1.2; i.e. ##\sqrt{1.2}##?

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.
 
  • #5
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.
 
  • #6
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.
 
  • #7

FAQ: Calculating the root of a number by hand

1. What is the method to 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 "guess and check" method. In the long division method, you group the digits in pairs from right to left, find the largest number whose square is less than or equal to the first group, and then use that to estimate the square root. You then bring down the next pair and repeat the process until you reach the desired precision.

2. How do I estimate the square root of a non-perfect square?

To estimate the square root of a non-perfect square, you can find the two perfect squares that the number lies between. For example, for the square root of 10, you know that it lies between 3 (since 3² = 9) and 4 (since 4² = 16). You can then average these two numbers or use a method like linear interpolation to get a more precise estimate.

3. Can I use the Babylonian method (or Heron's method) to find square roots by hand?

Yes, the Babylonian method (also known as Heron's method) is an effective way to find square roots by hand. Start with an initial guess (x₀) for the square root. Then, use the formula x₁ = (x₀ + n/x₀) / 2, where n is the number for which you want the square root. Repeat this process until the value stabilizes to your desired precision.

4. What is the difference between calculating square roots and cube roots by hand?

The main difference is in the methods used. Square roots can often be calculated using the long division method or the Babylonian method. In contrast, for cube roots, you might use a method similar to estimating square roots but with a focus on finding the cube of numbers. You can also use trial and error, testing small integers until you find the closest cube.

5. Are there any shortcuts for calculating roots of specific numbers?

Yes, there are some shortcuts for calculating roots of specific numbers. For example, the square roots of perfect squares (like 1, 4, 9, 16, etc.) can be quickly recalled. For other numbers, recognizing patterns, such as knowing that the square root of 2 is approximately 1.414, can help in estimation. Additionally, for cube roots, knowing the cubes of small integers can aid in quickly finding the value.

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