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How to Calculate the Square Root of Any Number, Digit by Digit Method

MindYourDecisions

669,907 views

MIT and Harvard entrance exams specifically tested if students knew how to calculate square roots by hand. In this video I will teach you the square root algorithm to calculate the square root of any number, digit by digit, by hand without a calculator.

0:00 Intro
3:47 2025
7:48 831744
10:50 13689
13:18 11.56
15:15 1234
19:12 531.5
23:47 inventor
25:15 why it works
28:58 place value

Harvard 1869 problem
https://graphics8.nytimes.com/package...
MIT exam 1876
https://web.archive.org/web/201904040...
Naval Mathematics, 1966
https://www.google.com/books/edition/...
Goals for the Correlation of Elementary Science and Mathematics, 1969
https://archive.org/details/goalsforc...
Aryabhata's methods
https://arxiv.org/pdf/math/0608793
Translation of Sanskrit
https://www.ganitamkalakriyagolam.com...
Route to Learn
How to find square roots by long division ...
Āryabhaṭa's Algorithmic Square Rooting
https://itihaasa.substack.com/p/aryab...
The square root algorithm
https://www.cantorsparadise.com/the-s...
Byjus
https://byjus.com/maths/square-root/
Find Square Root by Hand without Calculator
Find Square Root by Hand without Calculator
Square root algorithm
https://www.basic-mathematics.com/squ...
Calculating Square Root by Hand (Early Grades)
Calculating Square Root by Hand (Early Gra...
Ancient formula calculate ANY square root
How To Calculate Any Square Root. An Ancie...
Abacus picture
https://en.wikipedia.org/wiki/Abacus#...
Margarita philosophica nova
https://archive.org/details/margarita...
Laplace quote
https://en.wikipedia.org/wiki/History...
Laplace picture
https://en.wikipedia.org/wiki/File:La...

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20 Comments

@DrunkenUFOPilot 10 months ago Pinned

Thanks!

@crosshairs007 11 months ago

Why the hell did they never teach me this? This is EXACTLY the sort of thing that is the MOST interesting. Who cares if you can type in the right numbers and symbols on a calculator, I want to be able to figure it out by myself with no calculator.

@belaayya5094 10 months ago

I wish my dad was still alive so I could call him and tell him I can calculate a square root by hand now! He spent hours at the table trying to teach me math, and it often ended in tears for me, and him feeling helpless. But now I realize -- in the 80s, they never taught us long division. That came from my dad. They told us to memorize the times tables, but I struggled. Daddy taught me how to add to get the answer. And alllllll that came together now for me to understand how to calculate a square root. I sure hope he's up there and beaming with pride. When school failed me, he taught me the only way he knew how, and it culminated in something magical tonight. Thank you so much for this! Now...can you teach me algebra?

@kurtisburtis 11 months ago

Mom got me homeschooling textbooks when I was a kid, which is where I first learned this method.

@ryanbrown982 11 months ago

This sounds a hell of a lot like "balanced reading" where you teach kids to guess what a word might be based on pictures and context. Rather than, ya know, teaching kids to read.

@kevinjohnston8637 11 months ago

"Let's stop teaching the square root algorithm, let's teach guess/divide/average instead. And also, let's stop teaching divide". Brilliant.

@MmmM-mf3zd 11 months ago

Hidden gem of a topic I’d been searching for years and years: how to calculate roots by hand.

@probag8414 11 months ago

This is why I love this channel. In Math classes, the teacher would always give us the 2025 example. Clean. No curveballs.
The rest would be on the test and we'd all drown.

@joetyndall3382 8 months ago

I did this as a kid, got into MIT, used it until calculators came into existence during fall 1971, got a calculator for Christmas and promptly stopped using this algorithm cause the calculator was faster.

@kvetter 11 months ago

In binary this algorithm simplifies nicely. The tricky step of finding "b" comes down to only two choices, zero or one, and thus simplifies into a comparison. Everything else is shifting and subtracting.

@coffinrodriguez3150 10 months ago (edited)

13:02 spotted a 117 in the wild. Finish the fight, Chief!

@aguyontheinternet8436 11 months ago

Calling long division “meaningless rote” is crazy to me. For beginner students, long division is the only method of dividing polynomials that HAS meaning. It’s the only one students can understand, a method they could easily come up with themselves if they really needed to. But something like synthetic division is inherently more obtuse, because you need to have a thorough understanding of long division to get why it works.

@pabloverdi7543 8 months ago

I went to public schools in 60s/70s and was taught New Math. My Mom taught college accounting. When she saw my homework, she taught me long division, multiplication, basic algebra, fractions, etc. served as a solid base for a career in Computer science. Thanks Mom.

@JB-od1pi 11 months ago

I turned 11.56 into a fraction, which was 289/25, which is a quotient of perfect squares!

@Yeetmaster982 10 months ago

I tried searching online for like an hour trying to find a method of calculating by hand, and I could never find one. and this thing just pops up in my recommended page.

@KevinByrne-o5j 11 months ago

THANK YOU for presenting this algorithm for finding square roots.
I was taught this algorithm many years ago, but I hadn't used it for so long that I'd forgotten it.
I searched the Internet, but I couldn't find it. Now it suddenly appeared among my YouTube recommended videos.
Again, thank you.

@CristobalWatsonHernandez 6 months ago

The tuition in NZ for square numbers was "use a calculator." I really wish my old home country had better educational standards. Never skip the basics!

@verkuilb 11 months ago (edited)

My mother is about to turn 80. She was taught this method. When I wasn’t taught it in high school, she was horrified, and taught it to me.

She was a nurse prior to giving it up to have and raise a family—years later, as she approached empty nest status, she decided to return to school to get more training to be a lab technician. For that, she needed to retake college entrance exams, which didn’t allow a calculator. One of the problems involved a large square root, which she answered correctly using this method. She was accused of cheating by the test administrator, because “solving that problem requires a calculator”. She went into the office of the school dean, and demonstrated how she answered the question. She passed, and the test administrator got demoted.

@Gusttz20i 11 months ago

Now a division-like algorithm to logarithm

@RawwkinGrimmie64 11 months ago

Growing up, I was always fascinated by squares and square roots. I HATED that we were always told to just round non-squares to the nearest Integer, and now seeing how simple it could have been to actually learn the calculation just blows my mind!
Thank you for this video. You have made my inner child very happy!