Download Finding a Square Root Using an Algorithm

Finding a Square Root Using an Algorithm
The use of this method, similar to long division, diminished with the advent of calculators. This pencil-and-paper
method works well if you cannot use a calculator, especially if your number is large and you need your answer to
several decimal places. The guess-and-check method then becomes time-consuming.
We will demonstrate with an example: Find the square root of 432 to the tenths place.
Step 1: Set up the problem.
Group the digits in pairs from right to left. The leftmost
digit will either be in a pair or by itself, depending on the
number of digits. (Odd number of digits = one number;
even number of digits = a pair). Each pair of numbers will
yield one digit in the square root. Because you want the
answer to one decimal place, add two pairs of zeroes to
the right of the decimal point, so you can round correctly.
The blanks are to help you keep track of where you are in
working out the problem; they should be ignored in the
final solution.
__________
√4 32.00 00
Step 2: Find the first digit in the square root.
Work from left to right, as you do in long division. Find a
number whose square is less than or equal to the first
number or pair. Square it, subtract from the first
number/pair, and bring down the next pair of numbers. In
this example, choose 2 because 2 squared = 4. Square 2,
obtain 4, and write that underneath the 4. Subtract and
bring down the next pair of digits.
2_________
√4 32.00 00
-4
0 32
Step 3: Double and leave a space.
Now, consider the number above the square root sign.
Double it, and write it down off to the left with an empty
space for another digit. The work in the example is shown
in red. 2 doubled is 4, and we write it to the left of 032
with an empty space for another digit.
2_________
√4 32.00 00
-4
4_ 0 32
Step 4: Think what number digit times digit is less than or equal to the number on bottom.
Next, think what digit could go on the empty line so that
the complete number formed with the new digit times the
new digit would be equal to or less than the number on
that line in the problem. In the example, ask yourself:
What digit should I use so that forty-digit times digit is
less than or equal to 32? In this problem the digit is zero
because 40 > 32.
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2_________
√4 32.00 00
-4
40 0 32
Grade 7 > Mathematics > Number Sense > Enrichment: Solving Roots of Non-Square Integers
Step 5: Multiply, subtract, and bring down.
Write the digit you just found on the top of the square
root sign. Calculate the new number times that digit and
subtract it from the number on the bottom line. Bring
down the next pair of numbers. In this problem, 40 x 0 = 0;
subtract this from 32, and bring down the next two zeroes.
2 0______
√4 32.00 00
-4
40 0 32
-0 00
32 00
Step 6: Repeat doubling and leaving a space.
Repeat the process of doubling the number on top of the
square root sign and writing the doubled number off to
the left with a blank digit. In this problem, 20 doubled is
40, so write 40_ to the left of 3200.
2
√4
-4
40 0
-0
40_
0______
32.00 00
32
00
32 00
Step 7: Think what number digit times digit is less than or equal to the number on bottom.
Repeat the process of filling in the blank with a digit so
that the complete number times the digit is equal to or
less than the number in the current line in the problem. In
this example, ask yourself: What digit do I choose so that
four hundred digit times digit is ≤ 3200? Through trial and
error we see that we need to choose the digit 7. 407 x 7 =
2849. (8 is too large since 408 x 8 = 3264, and 3264 >
3200).
2 0______
√4 32.00 00
-4
40 0 32
-0 00
407
32 00
Step 8: Multiply, subtract, and bring down.
Write the digit you just found over the next pair of
numbers under the square root sign. Place the decimal
point in the correct location in the square root; just carry it
up as you would in long division. Calculate the new
number times that digit and subtract it from the number
on the bottom line. Bring down the next pair of numbers.
In this problem, 407 x 7 = 2849; subtract this from 3200,
and bring down the next two zeroes.
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2 0. 7___
√4 32.00 00
-4
40 0 32
-0 00
407
32 00
-28 49
3 51 00
Grade 7 > Mathematics > Number Sense > Enrichment: Solving Roots of Non-Square Integers
Step 9: Continue process: double, find the digit, multiply, subtract, bring down.
Continue with the same process as already outlined. In this
example, we double 207 and write 414_ to the left of
35,100. (Ignore the decimal point when you double). Then,
we find the blank digit: What digit should I choose so that
414digit times digit is less than or equal to 35,100? We
find 8 to be the correct choice. Write 8 on top of the
square root sign above the next pair of zeroes. You don't
need to do the subtraction (unless you want to continue
with more decimal places).
2 0. 7 8
√4 32.00 00
-4
40 0 32
-0 00
407
32 00
-28 49
4148
3 51 00
-3 31 84
Step 10: Round and check.
Finally, round the result to the accuracy asked for in the
problem, and check your answer. In this example, we
found the square root of 432 to the tenths place to be
20.8. We check our answer by squaring the result; the
example is shown at right.
Summary of Steps:
1. Set up the problem: group digits in pairs from right to
left and add pairs of zeroes as needed.
2. Find the first digit in the square root: choose a number
whose square is less than or equal to the first
pair/number.
3. Double the top number; write this to the left of the
current line and leave a space for the last digit.
4. Find the digit that goes in the space and place in the
square root. Think what numberdigit times digit is less
than or equal to the number on bottom?
5. Multiply, subtract, and bring down.
6. Repeat steps 3, 4, 5 until you have the square root to
the desired accuracy.
7. Round and check.
___
√432 = 20.8
Check: 20.8² = 432 ?
20.8 x 20.8 = 432.64
____ Another example:
Find √7456 to the tenths place.
8 6. 3 4
√74 56.00 00
-64
166
10 56
-9 96
1723
60 00
-51 69
17264
8 31 00
-6 90 56
Thus, to the tenths place,
_____
√7456 = 86.3
Check: 86.3² = 7456 ?
86.3 x 86.3 = 7447.69 OK
Note: The answer is not exact
because it is rounded.
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Grade 7 > Mathematics > Number Sense > Enrichment: Solving Roots of Non-Square Integers