How to find a square root

Here is the original π video:

Watch the long version of the extra working out here:

My calculation gave the square root of 10,005 to be 100.02499687578103

Compared to the true value of 100.0249968757810059447921878763577780015950243686963146571...

CORRECTIONS
- None yet. Let me know if you spot anything!

Thanks to my Patreon supporters who enable me to spend a day doing a lot of maths by hand. Here is a random subset:

David McCarthy JR
Derek Chandler
Mauro Cioni
Kevin Petrychyn
James Tanner

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Music by Howard Carter
Filming and editing by Trunkman Productions
Design by Simon Wright

MATT PARKER: Stand-up Mathematician

[RasperHelpdesk]

Tripping over a root while discussing roots... priceless 0:26

[GuanoLad]

The dedication to do an opening bit in the woods just so you could trip on a root at the right time is an admirable commitment to comedy.

[Chazlar]

Start = root
End = square
You have a great sense of humour Mr Parker.

[sunglow]

0:27 goes to the middle of the forest just ro make a root pun. 0.0

[Odothuigon]

I had a Dynamics professor, Dr. Penrod, at Auburn Univ. who could work out roots and powers simultaneously (e.g. 2.68^2/3) and give an answer to about 3 or 4 decimal places. It was absolutely amazing.

[genessab]

Oh so it’s just what I watched 2 days ago but again. *watches again anyways*

[TyrtlekingCR]

The root joke was levels of comedic genius never before witnessed by the human race

[SuselLee]

Did you stumble (0:27 s) over the root on purpose? Maybe because you end the video on the city square. ;)

[ElagabalusRex]

When I find a square root, I sketch a big parabola and then estimate distances using an elaborate system of paperclips and hourglasses.

[MrCyanGaming]

0:27 - Looks like you found a different type of root :)

[ronniebrown]

00:26
“Find the square ROOT (trips over tree root, notices that he accidentally made a Parker pun, and smiles)

[mfwplayinggames]

in high school i once overestimated a really simple square root on a test, and couldn't figure it out. so to make sure i never got stuck on any root again, i learned an iterative method by heart which only uses simple operators (so i could do it on paper, albeit very slowly). naturally, we never got any square roots on tests after that

[volbla]

I predict that the frequency at which Matt's hair is going to change will not be random but rather contain some sort of coded message.

[alexandergallon]

He tripped over a root when talking about square roots

[ruichen]

0:27 when he was saying “root” and literally tripped over the root.

[YCCCm]

"Find the ROOT-" trips on actual tree root. Love it.

[eamonnsiocain]

I wouldn't mind if my hair made an hilarious reappearance.

[OriginalPiMan]

The method I learned to do square roots looks a lot like long division but it is done two digits at a time.

[jiaming]

Newton's method for √N:
Next term = xₙ/2 + N/2xₙ

[cigmorfil]

There is a sort of long division method to find square roots that works in binary very easily:
1. write the number in digit pairs left and right of the <insert base> point, adding a zero before the first digit if odd number of digits before point, appending a zero after the ladt digit idmf an odd number of digits after the point. (extra pairs of zeros can be added after the nunber after rhe point.)
2. Do a long division taking the pairs of digits of the nunber (dividend) as a unit and creating a new divisor at each step:
2.1. double the answer so far and multiply by 10 (the base).
2.2.. Now find a digit to put in the ones place to create a divisor that when multiplied by this digit is the largest result that does not exceed the "working" digits (of the division)
2.3. Put this digit over the pair of digits that form the lat pair of digits of the "working" number.
2.4. Multiply the divisor by this digit and subtract (like long division)
2.5. Bring down next pair of digits to continue the division.
3. Continue the "division" until all pairs of digits have been used and result of last subtraction is zero, or until enough precision has been reached.

In binary the double is a shift left 1 bit, and multiply by 10 (the base - in decimal this is also 2) shift left anothe bit; only possible values for the units bit are 1 and 0, so try 1 - put a 1 in the units place and if divisor is less than "working" number can subtract and a 1 goes in the answer; otherwise a 0 goes in answer and next 2 bits of dividend are appended to the end of the working number. - highly efficient for a digital computer.

For 10005 the division is:
10005 -> 01 00 05 . 00 00 00 00
Put a decimal point above the point in the "dividend"
first pair is 01; answer do far is 0 -> 0_ so need units digit 1; 1 in answer above 01, 01×1=01 subtract to get 0 and bring down next pair 00 to make the "working" number 0 00
As this is zero can put a 0 above the 00 and bring down the next pair 05 to make the working number 0 00 05
Double answer and multiply by 10 gives 20_ which is greater than 5 so put 0 above 05. Bring down next pair 00 to make working number [0 00] 05 00
Double etc answer -> 200_ again 0 in answer; bring down next pair for 05 00 00
Double etc -> 2000_ this time can put a 2 in the units place -> 20002, multiply by that 2 to get 40004 and subtract to get 99 96, bring down 00
Double etc -> 20 00 4_ this time a 4 to give 20 00 44 × 4 = 80 01 76 to subtract = 19 94 24; bring down 00
Double etc -> 2 00 04 8_ units = 9 to give 2 00 04 89 x 9 = 18 00 44 01 to subtract = 1 93 79 99
and repeat as required.
(The "division" is easier to see (and do) when written out as a long division with the new divisor written down each time along side each "working" number, extending the vertical line of the divisiom "bus stop shelter" down the page.)