How to estimate square roots quickly in your head
RIP calculators 🪦
👋 I remember when I was in elementary school, I was taught different techniques for mental math, and estimating square roots in my head in seconds was one of them. I didn’t realize until much later that not many folks knew how to do this and really understood why it worked. Even simple multiplication of double digits is taught in many classrooms in ways that blindly force unnecessary rote memorization on students and their interest in math is quickly lost. Beyond math, unlearning and re-learning how we do things by shifting perspectives is a critical skill in life to cultivate that will help us become a better version of ourselves in the long run.
Now let’s not get ahead of ourselves too far, and start with estimating square roots. How do we do this quickly in our head without using a calculator?
How does it work?
Let’s try to estimate the square root of 23 for instance.
Steps to find a fairly accurate estimate of the square root of 23:
Ask yourself what is the square number (or perfect square) less than and closest to 23? We know that 4 squared is 16, which is less than and the closest to 23. While 5 squared or 25 is closer to 23 than 16, 25 > 23. Remember 4 and 16—we’ll use them in step 2.
\(4^2 = 4 \times 4 = 16 < 23 < 25 = 5 \times 5 = 5^2\)The square root of 23 should be greater than 4 but less than 5. In decimal, it should be 4 point something. Now subtract 16 from 23 and multiply 4 by 2. We’ll see in the next section why we do these.
\(23 - 16 = 7\)\(4 \times 2 = 8\)Write 7 in the numerator and 8 in the denominator of the fraction that we are going to append to 4.
\(\sqrt{23} = 4.79583152331... \approx 4\frac{7}{8} = 4.875\)Our approximation 4.875 is close to 4.7958! You can literally do this in your head without using a calculator for the square root of any positive integer.
Let’s do another one and estimate the square root of 453. The perfect square less than and closest to 453 is 21 because 20 squared is 400 and 21 squared is 441. In future posts, I’ll also show you how to quickly calculate the squares because the key to estimating square roots is knowing the perfect squares on the fly.
Subtract 441 from 453 to get 12 and multiply 21 by 2 to get 42. 12 is the numerator and 42 the denominator of the fraction we append to 21.
We see that the approximation 21.28 is correct up to two decimal places.
Why does it work so well?
Let’s say we want to estimate the square root of a positive integer N (say 23 from the previous example). We can break down N into two components: a (4) squared (16) and b (23 - 16 = 7). a squared is the perfect square less than and closest to N, and b is the remaining difference. Essentially, we are proposing that is approximately equal to a + b / 2a.
We can rewrite the right-most expression as the following:
So we have the approximation:
If we want both sides to be nearly equal to each other, we want the last term on the right-side inside the square root to be very small:
The term b over 2a is actually always less than 1 and squaring it will make it even smaller. We can prove this statement by contradiction. Assuming b / 2a is greater than or equal to 1, we have that b is greater than or equal to 2a:
Plug in 2a for b everywhere. But we see the square root of N cannot be greater than or equal to a + 1 because we assumed a squared to be the perfect square less than N. Otherwise N would have been the perfect square of a + 1.
Hence, by contradiction, b / 2a is always less than 1. Q.E.D.
When will I ever use it?
If you’re prepping for ACT, SAT, GMAT or whichever standardized test you need to take, you can quickly narrow down the multiple choices or even double check your answers. Every minute you spend calculating matters in these tests! Trust me, with enough practice, you’ll be faster than your calculator.
Besides being practically useful in real-life and being able to impress your friends, there is another benefit to practicing estimating square roots mentally in your head. Too many folks these days can’t do simple arithmetics without relying on a calculator. Sometimes we learn to do it in a wrong way at an early age that prevents us from quickly doing it in our head. Doing mental exercises like this will keep your mind sharp and help you boost your confidence.
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About this newsletter
I’m Seong Hyun Hwang, and I’m a data scientist, machine learning engineer, and software engineer. Casual Inference is a newsletter about data science, machine learning, tech startups, management and career, with occasional excursions into other fun topics that come to my mind. All opinions in this newsletter are my own.