If you will notice, 2.00'00'00'00 (or simply 2.00), is in-between the square of 1 (or 12 ) and the square of 2 (or 22 )
22 = 4.00
12 = 1.00
The middle half square of 1.02 (which is also equivalent to 12) and 2.02 (another way of representing 22) is 1.52
2.02 = 4.00 Upper Limit
1.52 = 2.25 Middle Square
1.02 = 1.00 Lower Limit
MIDDLE SQUARE: Phase 1
In
this new method of 'taking the square root of a number', we can 'use'
the method of getting the squares of numbers ending in five (Review Here)
as a 'tool' to make it easier for us to 'detect' where the next digit
could be located. To be candid, I thought learning the technique on how
to get the squares of two-digit numbers ending in five is just a 'fancy'
way of showing off. I was wondering what is the significance or use of
it. Now, I realized that it is indeed, very important.
Comparing the square of 1.52 (which is equivalent to 2.25) to 2.00 (given problem), we can say...
2.25 > 2.00
The
value 2.00 is above 1.00, which is the square of 1 but below 4.00 (the
square of 2). Determining the middle square between 1.0 and 2.0 will
give us instantly, an idea of where the value 2.00 could be located.
Actually, there are nine digits to select (from 1, 2, 3... up to 9) and
it is not practical that we randomly choose any of the these digits to
suspect as the next digit to consider. Determining that the 'middle
square' is above (higher or greater than 2.00), we eliminated four
digits above 5 (these are the digits 6, 7, 8 and 9). We then limited our
selection into only four digits (4, 3, 2 and 1) and only one of them is
the required digit. (See the shaded area below).
2.02 = 4.00 .
1.92 = ?
1.82 = ?
1.72 = ?
1.62 = ?
1.52 = 2.25 .
1.42 = ? .
1.32 = ? . (2.00 is possibly located in this shaded area)
1.22 = ? .
1.12 = ? .
1.02 = 1.00 .
WE still have four digits to select to know the next digit. But we can still trim it down into a more
comfortable set of selection.
QUARTER SQUARE
If
you will notice, 2.00 is in-between 2.25 ( the square of 1.5) and 1.00
(the square of 1). The 'middle' value between them is the 'square of
1.25'...
1.52 = 2.25 .
1.42 = ? .
1.32 = ? .
1.252 = ??? Middle portion between 1.52 and 1.002
1.22 = ? .
1.12 = ? .
1.02 = 1.00 .
There are three ways to get the square value of 1.25 ...
1) Digit-per-Digit SSQ
1.252 = 1.44'25
12x10 = 12'0
.............. 1.56'25
2) Ending in Five SSQ
Next to 12 is 13 (that is, 12 + 1 = 13)
1.252 = (12x13)'25
......... = 1.56'25 (Don't forget the proper place for the decimal point)
Square Averaging Method
The
third method is somehow, the more practical way we can use. Doing this, we will further limit
the possible next digit to a much manageable shaded area.
Activity 1: Add the square of 1.5 and the square of 1 (or written as 1.02)
1.52 = 2.25
1.02 = 1.00 .
........... 3.25
Activity 2: Divide the sum by 2
1.52 = 2.25
1.02 = 1.00 .
........... 3.25/ 2
........... 1.62 remainder 1
Activity 3:
Ignore the remainder and subtract 6 to the floating value of 1.62
(Note: When we say floating value, we must ignore temporarily the
decimal point as part of the true value of 1.62)
162 - 6 = 156 or 1.56 (don't forget to insert back the decimal point at its proper place)
Activity 4: If you want, just paste 25 after 1.56 so that value we got is exactly equivalent to the square of 1.25
1.52 = 2.25
1.252 = 1.56'25
1.02 = 1.00 .
........... 3.25/ 2
........... 1.62 remainder 1 (Take note: Ignore the remainder then 162 - 6 = 156 => 1.56)
"The Always Minus Six" Notation
In
ESR (Easy Square Rooting), it is more convenient to use the third
method (Square Averaging Method), to get the 'quarter square value' of
the numbers involved. But simply getting the average between two square
values involved (this time, 1.00 and 2.25), will not reflect the actual
true value of what we're trying to determine. So we must always follow
the rule that, "we must always subtract six" to the qoutient we got from
dividing the sum of the two numbers involved.
REVIEW
Let's review back the steps in doing ESR:
Question: What is square root of 2 ?
......
√2 = ?
Step 1: Instead of putting two zeroes after 2 (see this), let's put 8 zeroes grouped by in twos
.......... .
√2.00'00'00'00
Step 2 : Find a number that when "being multiplied by itself" will bring out a product that is near but lesser to that asked number
..1...... . (Put a decimal point after 1)
√2.00'00'00'00
we choose 1 because 1 x 1 = 1 ( or 12 = 1)
Step 3: Determine the 'middle square value" between the upper limit (that is, 22 ) and the lower limit (that is, 12 )
2.02 = 4.00 Upper Limit
1.52 = 2.25 Middle Square
1.02 = 1.00 Lower Limit
Step 4: Check if the 'middle square' is higher or lower than to the given 'number' problem.
2.25 > 2.00, so the next digit could be, either 1,2,3 or 4
Step 5: Determine the 'quarter square value' by doing the 'square averaging method'...
1.52 = 2.25
1.252 = 1.56
1.02 = 1.00 .
........... 3.25/ 2
........... 1.62 (Take note: 162 - 6 = 156 => 1.56)
SAMPLING
If you will notice, there are four digits that is in-between 1.02 and 1.52 (see table below)
1.52 = 2.25
.. 4 = ?
.. 3 = ?
.. 2 = ?
.. 1 = ?
1.02 = 1.00
Now, determining the value of the quarter-square, our selection from four digits is now cut down into two digits only
1.52 = 2.25
.. 4 = ? (2.00 is in-between 2.25 and 1.56'25, therefore,
.. 3 = ? it is either the digit 3 or 4 is the possible next digit to consider)
2.25 = 1.56'25
.. 2 = ?
.. 1 = ?
1.02 = 1.00
To become sure of which of the two (between the digits 3 and 4), is the rightful or precise digit to consider, we need to 'sample' them one at a time using the Two-Digit SSQ
1.42 = 1.16
1x8 = 8
........< 1.96 > (Note: Enclose your answer in < > to indicate that it is the nearest value)
1.32 = 1.09
1x6 = 6 .
.......... 1.69 (cannot be considered nearest to 2.00)
In reality, there's no need to do the squaring for 1.32, for reason that 1.42 can be considered the 'nearest' and yet lesser value than 2.00. So we choose 4 as the next digit to consider.
..1. 4.. . (Put a decimal point after 1)
√2.00'00'00'00
