The
following was a question from the HSTUAC home school discussion list.
"I
am wondering why so many lessons when teaching Square roots tell students to
use a calculator? I have found it very
difficult to find simple instruction on how to find the square root of a
number. Also does someone have a simple
instruction on this......other than memorizing them all."
My
answer:
The
reason you can't find simple instructions for finding square roots is the same
reason you can't find an intelligent cat.
They don't exist.
If
you're talking about square roots for numbers that are perfect squares, it's
probably best to memorize them, to a point.
I don't think there is any point trying to remember that the square root
of 34012224 is 5832. However, up through
20^2 = 400, I think that is reasonable.
For things like the square root of 376 I'm perfectly willing to let my
calculator tell me it is approximately 19.39071943.
Here's a
method for extracting square roots.
Suppose you want the square root of 5216239. Pair the numbers moving left from the decimal
point (that would be there if I wasn't using a whole number).
-----------------------------
05 21
62 39 . 00 00
Find the
largest square less than 05. It's 4 which is 2^2.
2 will be the first digit of the answer.
Subtract 4 from the first pair.
2
-------------------------------
05 21
62 39 . 00 00
-4
----------
1 21 bring
down the next pair.
Now,
take your answer so far (2) and multiply it by 20, obtaining 40. Find the largest number x, such that (40 + x)
x is less than or equal to 121. (40 + 3)
3 = 129 which is too big so (40 + 2) 2 = 84 is it. The 2 is the next part of our answer and
subtract off 84.
2
2
-------------------------------
05 21
62 39 . 00 00
-4
----------
1 21
- 84
---------
37 62 bring down the next pair.
Now,
take your answer so far (22) and multiply it by 20, obtaining 440. Find the largest number x, such that (440 +
x) x is less than or equal to 3762. (440
+ 8) 8 = 3584 is it. The 8 is the next
part of our answer and subtract off 3584.
2
2 8
-------------------------------
05 21
62 39 . 00 00
-4
----------
1 21
-
84
---------
37 62
- 35 84
------------
1 78 39 bring down the next pair.
Now,
take your answer so far (228) and multiply it by 20, obtaining 4560. Find the largest number x, such that (4560 +
x) x is less than or equal to 17839.
(4560 + 3) 3 = 13689 is it. The
3 is the next part of our answer and subtract off 13689. This is the answer to the one's place. You can keep going as long as you want until
you tire out.
2
2
8 3
-------------------------------
05 21
62 39 . 00 00
-4
----------
1 21
- 84
---------
37 62
- 35 84
------------
1 78 39
-1 36 89
-------------
41 50 00 bring down
the next pair.
Now,
take your answer so far (2283) and multiply it by 20, obtaining 45660. Find the largest number x, such that (45660 +
x) x is less than or equal to 415000.
(45660 + 9) 9 = 411021 is it.
The 9 is the next part of our answer and subtract off 411021.
2
2
8 3 . 9
-------------------------------
05 21
62 39 . 00 00
-4
----------
1 21
- 84
---------
37 62
- 35 84
------------
1 78 39
-1 36 89
-------------
41 50 00
-41 10 21
----------------
39 79 00 bring down
the next pair.
Now,
take your answer so far (22839) and multiply it by 20, obtaining 456780. Find the largest number x, such that (456780
+ x) x is less than or equal to 397900.
(456780 + 0) 0 = 0 is it. The 0
is the next part of our answer and subtract off 0.
2
2
8 3 . 9 0
------------------------------------
05 21
62 39 . 00 00
00
-4
----------
1 21
- 84
---------
37 62
- 35 84
------------
1 78 39
-1 36 89
-------------
41 50 00
-41 10 21
----------------
39 79 00
- 0
---------------
39 79 00 00 bring down
the next pair.
Now,
take your answer so far (228390) and multiply it by 20, obtaining 4567800. Find the largest number x, such that (4567800
+ x) x is less than or equal to 39790000.
(4567800 + 8) 8 = 36542464 is it.
The 8 is the next part of our answer and subtract off 36542464.
etc.
So, we
have 2283.908 as an approximation for the square root of 5216239.
My
calculator says it's 2283.908711.
Maple,
the computer algebra system gives the first 2000 digits as
2283.9087109602257480505791566381621441789731009946311896933054247890899614452615333964513908175474726687548061753978212878025250894088817350974975374411327736425058726272007084476364753670977184920935454478696637101636078156689364759002823618316310492089207625366251563552068663606898998004834734738721331166966391885797300502598213444405045205111435920972235024084824178555230472748612725376184229076814537063942269426459222548338780966327947340613138173676840000087287251293776159489237606886197627676370903095136495855001511223117683764248291963663829681288469993102238154538360428500118679586844156080299004200668607803278192272073868261371250797193348454567569281123916782299018686560754921517177675920872156035749918344334376062694996111773707939299043276828119443119130500489385268960409445639677838666159082574549545234881551835562824103179799735647960861742150234486175471788096270907257784519697280805327392386330112305070233389128565829831058804601708051202912240090361767259411452989181607482150936621989484535026126327381853349031617926087905783460151596272123398738285796854842002066229652353701034733587823297450682873285212275776804476742781085492033737272524207967363998548791766583509247704622034650759310559238249463692130299874211941402990381519708375164130075786209664274168990384986225686820275813957638265367967919546993685101857516338315394634152372064263773669400799926105322739872911691012523989567844349390378036364754917558586674623129536301777488056990283965683660481698971945232909165664852082548448987716239939685588050802824672549098446120774868825181064634626839648287977567710140181306631704257309195798568591362041515261877162835022031434225891006843551609034468271472004304489232376000958384672610713153108667744015163059244810529518992202119753898722041439599394366823775204516008887394874343469929650616360180788112066070730444558363844844168175129793053983117836187572008448268569934077978425340252111662561286840739102429053905178286044636470429153177768047549
Except
for the counting numbers whose square root are
themselves counting numbers, the square roots will have infinitely long decimal
expressions.
The
method can be used for numbers that are perfect squares.
For
example, let's find the square root of 104976.
3
2 4
-------------
10 49 76
-9 3 (20) = 60
------
1 49 62 (2) = 124
-1 24
-------
25 76 32 (20) = 640 644 (4) = 2576
-25 76
----------
0
I'm
willing to live with using the calculator.