Calculate Square root of any number ( don't teach anyone )

Added: 2026-05-16

[00:00:00]Want to calculate the square root of any number by hand to any decimal point?

[00:00:04]Let's start by finding the square root of 2025.

[00:00:08]First, prepare the number by grouping the digits into pairs, starting from the decimal point and moving left.

[00:00:15]So, we mark off 25 and then 20.

[00:00:19]We start with the group on the far left, which is 20.

[00:00:24]Ask yourself, what is the largest perfect square that does not exceed 20?

[00:00:29]That would be 4², which is 16.

[00:00:32]Write four as our first quotient digit.

[00:00:34]Subtract 16 from 20, leaving a remainder of four.

[00:00:39]Bring down the next pair, 25, giving us a new working number of 425.

[00:00:45]Here is the only tricky part of the process.

[00:00:48]Take your current answer, which is four, and double it to get eight.

[00:00:52]This eight becomes the base of our next divisor.

[00:00:55]We're going to append a mystery digit to it.

[00:00:58]Let's call it B.

[00:01:00]And then multiply that whole new number by B.

[00:01:03]We need to solve 80 + B all multiplied by B is less than or equal to 425.

[00:01:11]Using some quick estimation, 80 * 5 is 400. So, let's try making B equal to five. 85 * 5 is exactly 425.

[00:01:21]We subtract that, leaving a remainder of zero.

[00:01:24]The process concludes, and we have our answer.

[00:01:28]The square root of 2025 is exactly 45.

[00:01:33]Let's level up.

[00:01:35]Could you pass the MIT entrance exam from 1876?

[00:01:39]The question is, find the square root of 11.56.

[00:01:43]When dealing with decimals, pair the numbers moving outward from the decimal point.

[00:01:49]So, we have 11 on the left and 56 on the right.

[00:01:53]Start with 11.

[00:01:55]The largest square without exceeding is 3 squared, which is 9.

[00:02:00]Our first digit is 3.

[00:02:02]Subtract 9 from 11 to get 2.

[00:02:05]We have hit the decimal point, so place it in your answer.

[00:02:09]Now, bring down the 5 6 to make 256.

[00:02:12]Double our current answer of 3 to get 6.

[00:02:15]We need a digit B so that 60 + B all multiplied by B is less than or equal to 256.

[00:02:23]If you know your computer science numbers, you know 64 * 4 is exactly 256.

[00:02:29]Our next digit is 4. No remainder. The answer is 3.4. Congratulations. You are going to MIT in the 1800s. But why does this digit-by-digit method work? It is completely based on the algebraic expansion of a binomial square.