Can you calculate Square Roots in your Head?

How to use the amazing Babylonian Method to get quick Estimates of Square Roots

Truthfully, I can only do a tiny bit of calculations in my short-term memory before I lose fo… Oh look, there is football on the telly! Wait, what I was I saying? Oh yes, I can only do a tiny itty bitty mental maths which can be useful if your calculator was eaten by your dog or lost it on the bus.

Table of Contents

• The Square Root of 2
• The Square Root of 10
• The Babylonian Method for extracting Square Roots
• In Conclusion

The Square Root of 2

Here is the recipe. Yes, you have heard correctly. Recipe!

• (i) What is the number that you would like to extract the square root of?
• Let’s pick the denary number 2, for example.
• (ii) What are the two nearest square numbers to your number?
• Those are 1 and 4.
• (iii) What is the square root for the smallest of those two numbers?
• That would be 1.
• (iv) Divide your number by that smallest root.
• Okay, divide 2 by 1, which gives us 2.
• (v) Add the smallest root to your result.
• Sure, 2 + 1 which is 3.
• (vi) Last but not least, halve the last sum.
• Okay, 3 halves is the same as 1 and a half.
• (vii) Are you happy with that estimate? If not, repeat the whole process from the third step with the new estimate.

Nope, not happy. The first estimate of 1.5 is over the actual square root of 2 = 1.41421356… by one tenth. We can attempt to do this in our heads but might cause some brain cells overheating! Let’s try the second iteration.🥴

• (iii) In this step replacing the smallest root with your previous estimate.
• First estimate was 1 ½ or simply 3/2.
• (iv) Divide your number by the previous estimate.
• Okay, divide 2 by (3/2), which gives us 4/3.
• (v) Add the previous estimate to your result.
• Sure, 4/3 + 3/2 which is 8/6 + 9/6 = 17/6 (my poor brain is already getting cooked 😶‍🌫️).
• (vi) Last but not least, halve the last sum.
• Halving 17 sixths give us 17/12 or 1 ⁵/₁₂.
• (vii) Are you happy with that estimate? If not, repeat the whole process from the third step with the new estimate.

Yes, absolutely.

At this stage, I grab a scrap piece of paper to work out 5/12:

Therefore, 17/12 = 1.41666666… which is an overestimate by more than two thousandths. That’s not bad going for doing most of the arithmetic in your head.

The Square Root of 10

Let us try a more challenging example.

• (i) What is the number that you would like to extract the square root of?
• This time let’s pick the denary number 10.
• (ii) What are the two nearest square numbers to your number?
• Those are 9 and 16.
• (iii) What is the square root for the smallest of those two numbers?
• That would be 3.
• (iv) Divide your number by that smallest root.
• Okay, divide 10 by 3, which gives us 10 thirds or 3 and one third.
• (v) Add the smallest root to your result.
• Sure, 10/3 + 3 is the same as 10/3 + 9/3 which gives 19 thirds.
• (vi) Last but not least, halve the last sum.
• Okay, 19 thirds halved that is (19/3)/2 which gives 19/6 or 3 and a sixth.
• (vii) Are you happy with that estimate? If not, repeat the whole process from the third step with the new estimate.

Honestly, I can only really do one iteration in my head before it short-circuits and goes up in smoke. This whole recipe does take some serious brain processing power to calculate, but it is sort of doable.

I would say that 3 ⅙ = 3.16666666… is close enough to the actual square root of 10 = 3.16227766… for mental calculations.

If you want to take it one step further, I would advise doing these calculations on paper. Unless you have superhuman brain powers…

Thus, after two iterations you get to a pretty decent estimate of 3 ³⁷/₂₂₈ = 3.16228... for the square root of 10.

The Babylonian Method for extracting Square Roots

We have evidence from clay tablets such as the YBC 7289 that from around 1800 to 1600 BC that the Babylonians already knew how to extract roots in their far superior sexagesimal number system i.e., in base 60.

But why did I call their method a ‘recipe’?! That is because arithmetic was done that way for thousdands of years! Standardised algebraic notation as we know it nowadays was only introduced by René Descartes in 1637 AD.

For ease of use let’s rewrite the recipe in modern algerbraic notation. That is, to find the square root x of a positive number a i.e., x = √a, we do the following:

• To start with, we need a good guess of x₀.
• First iteration, we calculate x₁ = ((a / x₀) + x₀) / 2.
• Second iteration, calculate x₂ = ((a / x₁) + x₁) / 2.
• Repeat until you reach a satisfactory estimate.

This method is actually called Heron’s method from the first century AD. As the original Babylonian method involved a bit more geometry and which boiled down to using this estimate xₙ₊₁ = a / (2xₙ) + xₙ, Heron streamlined it into an easier to understand algorithm as presented in this article.

Regardless, I still call this xₙ₊₁ = ((a / xₙ) + xₙ) / 2 the Babylonian method after the rightful originators.

In Conclusion

If you are looking for an explanation of how and why this method works so efficiently, rest assured that I will write an article on it in the near future with python code 🐍 included as well. 🤓

As it happens, this is also the method used by most computers to extract square roots.

In practice, the method for finding x = √a is failry straight forward if you follow the recipe or the formula for every iteration n:

It all depends on how comfortable you are with mental arithmetic. I tend to resort to writing my workings on scrap paper, if my dog hasn’t eaten those too… 🐶

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