Finding square roots using an algorithm
There is also an algorithm that resembles the long division algorithm, and was taught in schools in days before calculators. See the example below to learn it. While learning this algorithm may not be necessary in today's world with calculators, working out some examples can be used as an exercise in basic operations for middle school students, and studying the logic behind it can be a good thinking exercise for high school students.
Example: Find √645 to one decimal place.
First group the numbers under the root in pairs from right to left, leaving either one or two digits on the left (6 in this case). For each pair of numbers you will get one digit in the square root.
To start, find a number whose square is less than or equal to the first pair or first number, and write it above the square root line (2).
| 2 | |
| √6 | .45 |
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| Square the 2, giving 4, write that underneath the 6, and subtract. Bring down the next pair of digits. | Then double the number above the square root symbol line (highlighted), and write it down in parenthesis with an empty line next to it as shown. | Next think what single digit number something could go on the empty line so that forty-something timessomething would be less than or equal to 245. 45 x 5 = 225 46 x 6 = 276, so 5 works. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| Write 5 on top of line. Calculate 5 x 45, write that below 245, subtract, bring down the next pair of digits (in this case the decimal digits 00). | Then double the number above the line (25), and write the doubled number (50) in parenthesis with an empty line next to it as indicated: | Think what single digit number something could go on the empty line so that five hundred-something times something would be less than or equal to 2000. 503 x 3 = 1509 504 x 4 = 2016, so 3 works. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| Calculate 3 x 503, write that below 2000, subtract, bring down the next digits. | Then double the 'number' 253 which is above the line (ignoring the decimal point), and write the doubled number 506 in parenthesis with an empty line next to it as indicated: | 5068 x 8 = 40544 5069 x 9 = 45621, which is less than 49100, so 9 works. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Thus to one decimal place, √645 = 25.4
The trick I am going to explain is called the cross-multiplication technique... but not the one you know.
Lets start with 123 * 456.
^Step 1--- arrange the numbers in order (one on top of the other) your pick.
123
456
----------
start by first multiplying 6*3=18. write down the 8 and carry the 1 mentally.
123
456
---------- CARRY 1
----8
^Step 2---- attempt to form an invisible X
It may not sound simple at first but I'll explain.
Start by observing the 6 & 2, then the 5 & 3. Make a line between those numbers and you will get an X.
Now multiple 6*2=12 PLUS(+) 5*3=15
so 12 + 15 = 27
27 + 1 (the carry)= 28
Now write down the 8 and carry the 2 mentally.
123
456
---------- CARRY 2
---88
^Step 3--- now examine the numbers 6 & 1, 4 & 3, then in the middle 2 & 5.
Multiply them and add em' all up:
6*1=6 + 4*3=12 + 5*2=10
so 6 + 12 +10 = 28 PLUS carry 2 = 30.
Write down the 0 and carry the 3.
123
456
---------- CARRY 3
--088
^Step 4--- Now we will attempt the invisible X for the last time. This time take notice of the numbers 5 & 1, 4 & 2
multiply them and add em' all up:
5*1=5 + 4*2=8
so 5 + 8 = 13 PLUS carry 3 = 16.
Write down the 6 and carry the 1.
123
456
---------- CARRY 1
--6088
^Step 5 (final step)--- In the beginning we started by multiplying 6 & 3. Now we will multiply 4 & 1.
so 4*1=4 + carry 1 = 5
Write it down and the behold the FINAL ANSWER!
123
456
----------
56088
Nothing in this world is easy to learn at first shot, so give it some time like I did and become a master at the skill. It just might help you in the long run because I myself can multiply any two 3-digit numbers in my head in less than 5 seconds (using the method I just explained) and I'm getting better. Good luck!
-- ^Step 1--- arrange the numbers in order (one on top of the other) your pick.
123
456
----------
start by first multiplying 6*3=18. write down the 8 and carry the 1 mentally.
123
456
---------- CARRY 1
----8
^Step 2---- attempt to form an invisible X
It may not sound simple at first but I'll explain.
Start by observing the 6 & 2, then the 5 & 3. Make a line between those numbers and you will get an X.
Now multiple 6*2=12 PLUS(+) 5*3=15
so 12 + 15 = 27
27 + 1 (the carry)= 28
Now write down the 8 and carry the 2 mentally.
123
456
---------- CARRY 2
---88
^Step 3--- now examine the numbers 6 & 1, 4 & 3, then in the middle 2 & 5.
Multiply them and add em' all up:
6*1=6 + 4*3=12 + 5*2=10
so 6 + 12 +10 = 28 PLUS carry 2 = 30.
Write down the 0 and carry the 3.
123
456
---------- CARRY 3
--088
^Step 4--- Now we will attempt the invisible X for the last time. This time take notice of the numbers 5 & 1, 4 & 2
multiply them and add em' all up:
5*1=5 + 4*2=8
so 5 + 8 = 13 PLUS carry 3 = 16.
Write down the 6 and carry the 1.
123
456
---------- CARRY 1
--6088
^Step 5 (final step)--- In the beginning we started by multiplying 6 & 3. Now we will multiply 4 & 1.
so 4*1=4 + carry 1 = 5
Write it down and the behold the FINAL ANSWER!
123
456
----------
56088
Nothing in this world is easy to learn at first shot, so give it some time like I did and become a master at the skill. It just might help you in the long run because I myself can multiply any two 3-digit numbers in my head in less than 5 seconds (using the method I just explained) and I'm getting better. Good luck!
Asha V.P
