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### Longhand Square Roots

```
Date: 03/30/98 at 11:48:43
From: Tim
Subject: Square root

How do you find the square root of any number? Is there an easy
formula?
```

```
Date: 03/30/98 at 15:33:57
From: Doctor Rob
Subject: Re: Square root

There are a couple of ways.  See

http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html

for one.  Here is another:

To find a square root by the "longhand" method, we proceed as follows.
I intersperse numbered steps with an example. We will find the square
root of 113 to three decimal places.

1. Draw a square root symbol, or radical, with the number whose root
off digits in both directions in groups of two. Put a decimal point
above the radical, and directly above the other decimal point.

.
/-------------
\/ 1 13.00 00 00

2. Start with the first group of 1 or 2 digits.  Find the largest
square of a single-digit integer less than or equal to it. Write
the single digit above the radical, and its square under the first
group.  Draw a line under that square, and subtract it from the
first group.

1  .
/-------------
\/ 1 13.00 00 00
1
----
0

3. Bring down the next group below the last line drawn. This forms the
current remainder. Draw a vertical line to the left of the
resulting number, and to the left of that line put twenty times the
number above the radical, a plus sign, a blank space (to be filled
in during step 4), an equals sign, and some blank space for the

1  .
/-------------
\/ 1 13.00 00 00
1
----
20+_=?? | 0 13

4. Pick the biggest digit D that you could put into the underscore
place, so that when you do the indicated addition and then
multiply the sum by D, the product is less than the current
remainder. (If you guess too large a D, the remainder will be
negative. If you guess too small a D, the remainder will be
greater than the number to the left of the vertical line.) Put it
above the radical above the last group of digits brought down,
and also put it in the blank space you left in step 3. Compute
the number given by the expression, and put it after the equals
sign. Multiply D times that number, and put that below the current
remainder, draw a horizontal line below that, and subtract, to give
a new current remainder.

1  0.
/-------------
\/ 1 13.00 00 00
1
----
20+0=20 | 0 13
0
-----
13

5. If the current answer, above the radical, has the desired accuracy,
stop.  Otherwise, go back to step 3.

Step 3:
1  0.
/-------------
\/ 1 13.00 00 00
1
----
20+0=20 | 0 13
0
-----
200+_=??? | 13 00

Step 4:
1  0. 6
/-------------
\/ 1 13.00 00 00
1
----
20+0=20 | 0 13
0
-----
200+6=206 | 13 00
12 36
---------
64

Step 3:
1  0. 6
/-------------
\/ 1 13.00 00 00
1
----
20+0=20 | 0 13
0
-----
200+6=206 | 13 00
12 36
---------
2120+_=???? | 64 00

Step 4:
1  0. 6  3
/-------------
\/ 1 13.00 00 00
1
----
20+0=20 | 0 13
0
-----
200+6=206 | 13 00
12 36
--------
2120+3=2123 | 64 00
63 69
--------
31

Step 3:
1  0. 6  3
/-------------
\/ 1 13.00 00 00
1
----
20+0=20 | 0 13
0
-----
200+6=206 | 13 00
12 36
--------
2120+3=2123 | 64 00
63 69
--------
21260+_=????? | 31 00

Step 4:
1  0. 6  3  0
/-------------
\/ 1 13.00 00 00
1
----
20+0=20 | 0 13
0
-----
200+6=206 | 13 00
12 36
---------
2120+3=2123 | 64 00
63 69
--------
21260+0=21260 | 31 00
0
-----
31 00

Step 5:  Stop.

Thus the square root of 113 to three decimal places is 10.630.
Checking, 10.630^2 = 112.9969, and 10.631^2 = 113.0182, so the

The underlying principle is (10*a+b)^2 = 100*a^2 + b*(20*a+b),
or even better,

(100*N+n) - (10*a+b)^2 = (100*[N-a^2]+n) - b*(20*a+b).

Here is a more detailed explanation of what is going on:

N is the integer made up of first groups of digits, and a is the
integer part of the square root of N already calculated. N - a^2 is
the current integer remainder. When you bring down the next two-digit
group n, and append it to the integer remainder, you are multiplying
the integer remainder by 100 and adding n to give 100*[N-a^2] + n.

decimal place, and the "+b" and the "b*" are the new digit you are

The old square root was a, and the new one is 10*a + b,
gotten by appending the digit b to the old square root.  Then
(100*N+n) - (10*a+b)^2 is the new integer remainder.
b is chosen so that this remainder is positive, but as small as
possible.  That ensures that at each step, b will be just a single
digit. Now replace N by 100*N+n, and a by 10*a+b, and repeat.

-Doctor Rob,  The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Square & Cube Roots
Middle School Square Roots

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