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Showing posts with label Duplex Method. Show all posts
Showing posts with label Duplex Method. Show all posts

3/9/13

Finding Squares Easy method...


Squaring of a number is largely used in mathematical calculations. There are so many rules for special cases. But we will discuss a general rule for squaring which is capable of universal application.
From our childhood we all know one common formula.
i.e (a+b)2 = a2+2ab+b2
We can calculate squares using above formula. So let us do some examples
242=???
To calculate squares by using above formula we have to follow some steps.
i.e let us take unit digit as ‘b’ and remaining part as ‘a’.
in 24, a=2 and b=4.
We should start our calculation from the right side of the formula.
Step-1: b2=16
242=       6 and 1 as reminder.
Step-2: 2ab=2x2x4=16
 242=       76 and 1 as reminder. (16+1=17)
Step-3: a2=4
242=   576 (answer).
This model is time taking to explain but it is very easy to calculate. Now we do some exercise.
Calculate squares for the following numbers.
27, 36, 85, 99
Now how are you feeling? Is it hard or easy??
But it has some limits. It is useful up to 99 only. We can calculate squares more than 99 by using this formula but it seems some difficult. So we need to have some easy and it should satisfy all the conditions.
Now we are moving to next method which is very useful to calculate square of any number whether it is 3 digits or 5 digits.
i.e
Duplex Combination Process
Here I’ll explain this method with an example
8972=???
Explanation:
1.    Square of unit digit. i.e 72=49; writte down 9 and carry over 4.
2.    2x9x7=126+4=130; writte down 0 and carry over 13. (ten digit and unit digit)
3.    2x8x7+92=112+81=193+13=206; writte down 6 and carry over20. (ten digit square and multiplication of unit digit and hundred digit.
4.    2x8x9=144+20=164; writte down 4 and carry over 16.
5.    82=64+16=80
So the final answer is 804609.
Lets we do another example
14322=????  Ans: 210253026124
Explanation:
1.    22=4; Write it down.
2.    2x3x2=12; write down 2 and carry over 1
3.    2x4x2+32=25+1=26; write down 6 and carry over 2.
4.    2x1x2+2x4x3=28+2=30; write down 0 and carry over 3.
5.    2x1x3+42=22+3=25; write down 5 and carry over 2.
6.    2x1x4=8+2=10; write down 0 and carry over 1.
7.    12+1=2
Final Answer is 2050624
So now Practice some squares…
123452=??
875322=??
And do your own numbers…
Some special cases….
A)  Square of a number with unit digit as 5
It is very easy to calculate square of a number with unit digit as 5. Lets take an example.
252=???
Ans is 625.
Explanation: 52=25; write down 25
Next number for remaining digit 2 is 3. Multiply 2and 3 we get 6. Now write down.
The final answer is 625…
Lets try following numbers…
35,45,75,95,115

Square of a number which is nearer to 10x
For this type of numbers we use following formula
(a+b)2=a2+2ab+b2
(a-b)2=a2-2ab+b2
Ex:
1)    982=??
98=100-2
(100-2)2=10000-400+4=9604
2)    1072=(100+7)2=10000+1400+49=11449
3)    9982=??
4)    100082=??
Today we end over topic. Tomorrow we discuss about finding cubes…..