How can I find the square root of 456 using the
trial divisor method?
-------------------------------------------------
Write 456 as 456.000000. Then group it into pairs
of digits, right and left of the decimal, Consider
the introductory 4 to be the pair of digits 04. Each
SINGLE digit of the answer will be placed over a PAIR
of digits.
.
---------------
|04 56. 00 00 00
The largest digit that does not exceed the square root
of the digit pair 04 is 2, so write 2 above the pair of
digits 04.
2
---------------
|04 56. 00 00 00
Square 2, getting 04. Write it below the pair of digits 04
Draw a line under it
2 .
---------------
|04 56. 00 00 00
04
---
Subtract getting 00 then bring down the next pair of digits
56. Draw a vertical line to the left of 00 56
2 .
---------------
|04 56. 00 00 00
04
---
|00 56
So far we have 2 in the answer. Multiply 2 by 20. (Double it
and annex a 0). This gives 40. Write the 40 to the left of
the vertical line:
2 .
---------------
|04 56. 00 00 00
4
---
40|00 56
40 is the TRIAL divisor. Divide 40 into "00 56" (interpreted as
just 56). It goes 1+ times, so write 1 as the next digit in the
answer.
2 1.
---------------
|04 56. 00 00 00
4
---
40|00 56
Now to get the ACTUAL divisor, add the 1 to the the trial divisor,
getting 41, and write this actual divisor under the 40 and place
another vertical line to the right of it.
2 1.
---------------
| 4 56. 00 00 00
4
---
40|00 56
41|
Multiply the 1 times the ACTUAL divisor, getting 41. Write it
as "00 41" and write it beside the vertical line at the bottom
underneath the 00 56. Draw a line underneath and subtract, getting
15. Then bring down the next pair of digits 00. Then draw a
vertical line to the left of the 15.
2 1.
---------------
|04 56. 00 00 00
4
---
40|00 56
41|00 41
-----
|15 00
The digits in the answer so far are 21. So multiply this by 20
(Double 21, getting 42, then annex a 0 to get 420). Write 420
left of the vertical line. 420 is the next TRIAL divisor.
2 1.
---------------
|04 56. 00 00 00
4
---
40|00 56
41| 41
-----
420| 15 00
Divide the TRIAL divisor 420 into "15 00" (interpreted as
1500). It goes 3+ times, so write 3 as the next digit in the
answer.
2 1. 3
---------------
|04 56. 00 00 00
4
---
40|00 56
41| 41
-----
420| 15 00
Now to get the next ACTUAL divisor, add the 3 to the the trial divisor,
getting 423, and write this actual divisor under the 420 and place
another vertical line to the right of it.
2 1. 3
---------------
|04 56. 00 00 00
4
---
40|00 56
41|00 41
-----
420| 15 00
423|
Now multiply the 3 times the ACTUAL divisor, 423, getting 1296, and
write it as "12 96" underneath the "15 00" and subtract, getting 231.
So write this as "02 31", and draw a vertical line left of it
2 1. 3
---------------
|04 56. 00 00 00
4
---
40|00 56
41|00 41
-----
420| 15 00
423| 12 69
-------
|02 31
The digits in the answer so far are 213. So multiply this by 20
(Double 213, getting 426, then annex a 0 to get 4260). Write 4260
left of the vertical line. 4260 is the next TRIAL divisor.
2 1. 3
---------------
|04 56. 00 00 00
4
---
40|00 56
41|00 41
-----
420| 15 00
423| 12 69
-------
4260|02 31 00
4265|
Continue this process as long as you like. Here is the
finished problem taken to the nearest thousandth:
2 1. 3 5 4
---------------
|04 56. 00 00 00
4
---
40|00 56
41|00 41
-----
420| 15 00
423| 12 69
-------
4260|02 31 00
4265|02 13 25
---------
42700|17 75 00
42704|17 08 16
--------
66 84
To get another decimal place annex another "00" pair
to bring down.
Why does it work? I won't show the whole thing. But
here's an idea to get you started if you are interested.
(10u + t)² = 100u² + 20tu + t
Notice that 20u is the coefficient of t in the second term.
That is related to why we multiply u by 20 to get the TRIAL
divisor 20tu. The t on the end is related to why we must
add the t to get the ACTUAL divisor.
Edwin