I already created two blogs about this topic but I realized that my approach or method of explaining them to my readers seems to be difficult to comprehend (or a little bit complicated to understand). So, I decided to create a better, simple presentation of taking the square root of any number, much better than the usual method taught in schools.
Example, let’s take the square root of 7
√7 = ?
The number 7 is not a ‘perfect square’ number. It is in-between the square of 2, (22 = 4) and the square of 3 (32 = 9).
But we choose 2 as our first digit. (It the nearest but lesser than 7)
_2
√7
But to say that the square root of 7 is 2 is wrong (since we know that the 2 is the square root of 4). It is more acceptable to say that the square root of 7 is about “two point something” (or written as 2.- - - ). But the question is, what is that “point something”?
THE NEXT DIGIT
Okay, put another two zeroes after 7, indicating that we’re intended to know ‘what’ that “point something” is.
_2.___
√7.00
If you will notice, 7.00 is in-between 22 (which is, 4) and 32 (which is, 9).
32 = 9.00
?2 = 7.00
22 = 4.00
Now, in-between 2 and 3 are the following –
2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, and 2.9
Or we can represent them this way:
(Table 1)
3.02 = 9.00
2.92 = …
2.82 = …
2.72 = …
2.62 = … (The square root of seven or 7.00 is “within” these
2.5 2 = … numbers in-between 3.002 and 2.02)
2.4 2 = …
2.32 = …
2.22 = …
2.12 = …
2.0 2 = 4.00
We have nine choices and only one of them is the ‘nearest’ square root of 7.
Middle-Half Square
One way to know the next digit after the first digit “2” in our attempt to take the square root of 7.00 is to know first, the middle- half square of 2.0 and 3.0
3.02 = 9.00
2.52 = …?
2.02 = 4.00
There is a technique in knowing the “
"square of a two-digit number ending in five”
.3.02 = 9.00
2.52 = 6.25
2.02 = 4.00
Now, 7.00 is above the middle half square 6.25 but below 9.00. So, our choices for the nearest possible square root of 7.00 (in a two-digit representation), are the following:
2.6, 2.7, 2.8, and 2.9
Or we can represent them this way:
(Table 2)
3.02 = 9.00
2.92 = …
2.82 = … (The square root of seven or 7.00 is “within” these
2.72 = … numbers in-between 3.002 and 2.52)
2.62 = …
2.5 2 = 6.25
We cut down our choices from nine into four and ‘only one’ is the nearest possible square root of 7.00
Quarter Part Square
We can still cut down the four choices into two. How? Simply add the lower square “L” (this time, it is 6.25) to the higher square “H” (that is, 9.00) and then divide by two.
H = 9.00
L = 6.25 (Add)
S =15.25 ÷ 2 = 7.625 (The “S” stands for Sum)
Some Adjustments
The quotient 7.625 is a little bit higher than the actual square of 2.752, which is equal to 7.5625.
Why 2.752? The middle-half square value between 2.52 and 3.02 is 2.752, which happens
to be the “upper-quarter part square value” between 2.02 and 3.02.
(Table 3)
3.02 = 9.00
2.92 = …
2.82 = …
2.752 = 7.5625 ← upper-quarter part square or “UQS”
2.72 = …
2.62 = …
2.5 2 = 6.25 ← Middle-Half Square
Some modifications should be made to make our computation (getting the upper part square value), much nearer to the actual value without directly squaring it (remember, 2.752 is a three-digit number, squaring it is a little bit difficult). Using the method of “add then divide by two” as shown above (also known as taking the average or median of two adjacent numbers), two more things you should remember to do:
Step 1: Ignore the last digit 5. In the case of the quotient 7.625, simply ignore or delete the last digit 5. So, the number will appear as 7.62
Step 2: Always subtract by the “adjustable 6”. This step does not follow the strict rule in math of putting the 6 in its proper decimal place. Simply subtract the quotient which in this case equal to 7.62 by ..6 (represented as “two dots before the number six). Simply put the 6 in the same decimal place as to the last digit.
7.62 - ..6 = 7.56
The value 7.56 has an error of only .0025 which is negligible (compare to the actual square value of 2.75 which is exactly equal to 7.5625).
If you look at Table 3, you will notice that the upper-quarter part square 7.56 is above or greater than 7.00. Therefore, the two remaining choices will be 2.62 and 2.72:
(Table 4)
2.752 = 7.5625
2.72 = … (The square root of seven or 7.00 is “within” these
2.62 = … numbers in-between 2.52 and 2.752)
2.5 2 = 6.25
Two-Digit Squaring
Now, let us square the two remaining choices and look which of the two is the nearest possible square root of 7.00 using the
2.72 = 04’49 PSV (Partial Square Value)
2x14= 2’8 . Add SP (Sub-product)
Ans.= 07.29
We cannot take 2.7 as the square root of 7.00 (in two-digit representation) because it is “over” or greater than 7.00 (The more preferred answer is nearest but less than 7.00)
2.62 = 04’36 PSV (Partial Square Value)
2x12= 2’4 . Add SP (Sub-product)
Ans.= 06.76
The answer 6.76 is the nearest but lesser than 7.00. Therefore, we choose 2.6 or simply the digit “6” as our next digit after writing the first digit “2”
_2. 6___
√7.00
THE NEXT DIGIT (PART II)
If you intend to continue, simply put another two zeroes after 7.00.
_2. 6___
√7.00’00
Now, we’re going to represent the square root of 7 (or 7.00’00), into three-digit number (of course, with respect to the decimal point).
Middle-Half Square (Part 2)
Again, 7.00 is in-between 2.72 and 2.62.
2.702 = 7.29 (or 7.29’00)
2.652 = …?
2.602 = 6.76 (or 6.76’00)
There are two ways we can determine the middle-half square (which is 2.652):
First: “Add Then Divide By 2” Method
2.72 = 7.29
2.62 = 6.76 (Add)
Sum=14.05 ÷ 2 = 7.02
2.652 = approx. 7.02
Second: “Innovative” Technique which I called “Paste and Copy” (sounds familiar?)
How it works? You will notice that our latest nearest possible ‘square root’ of 7 is 2.6.
Step 1:
Simply write down again the square of 2.6 (latest nearest square root value)
2.62 = 6.76
Step 2:
Put (or paste) a “5” after 2.6 and paste “25” after 6.79
2.652 = 6.76’25
Step 3:
Copy the digits ‘2’ and ‘6’ or ‘26’ (ignore the decimal point) and put it below 6.79’25,
Placing the last digit of 26 ‘aligned’ to the digit before the last two digits of 6.79’25 (which happens to be ‘25’).
2.652 = 6.76’25 ← paste 25 here
+(copy)= 26 .
Ans.. = 7.02’25
2.702 = 7.29 (or 7.29’00)
2.652 = 7.02’25
2.602 = 6.76 (or 6.76’00)
You will notice that 7.02’25 is above 7.00’00, so the next nearest possible square root of 7 is below 2.65 (It could either be 2.64, 2.63, 2.62 or 2.61)
Quarter Part Square (2nd time)
We do this part to cut down our choices from four into only two choices
Let us apply the ‘Add and Then Divide By Two” Method:
H = 7.02’25
L = 6.76 . ← Add
S =13.78’25 ÷ 2 = 6.89’12’5
2.6252 = 6.89’12’5 (approximate value only)
Note: The actual square value of 2.625 (that is, 2.6252 = 6.89’06’25) is much lower by 0.000625, which can be considered negligible.
We can also call “2.6252 = 6.89’12’5” as LQP Square (Lower-Quarter Part Square)
(Table 5)
2.652 = 7.05’25
2.642 = … (The square root of seven or 7.00 is “within” these
2.632 = … numbers in-between 2.652 and 2.6252)
2.62252 = 6.92’12
2.622 = … (Ignore these lines below 2.62252)
2.612 = …
2.602 = 6.76 (or 6.76’00)
So, we cut down our choices into two and only one of them is ‘acceptable’
To find out the square of 2.64 the easier way, simply do the “
”2.642 = (2.62)’16 ← PSV (partial square value)
26x..8 = ..208 ← Add this SP (provided that it is in its proper decimal place)
TSum = 6.76’16 + ..208 = 6.9696
We can write them down this way:
2.642 = 6.76’16 PSV
26 x..8 = 20’8 . PS (Add)
Ans.. = 6.96’96
There’s no need to ‘test’ the second choice (2.632), because it is obvious that its square value is much lower than 2.642.
Therefore, we choose 2.64 as the nearest possible square root of 7.00 or to be more specific, we choose the digit ‘4’ as our third digit.
_2. 6 4___
√7.00’00
Again, if you wish to continue, just repeat the process above…that is, put another two zeroes, find the middle-half square value, determine the quarter-part square (either UQP or LQP) and do the ‘square sampling’ of the two remaining choices and so on and so on.
