Squareing Techniques

8/6/2019 Squareing Techniques

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Easy mental squaring tricks

Squaring numbers between 40 and 60.

Prerequisites: One should remember squares of numbers up to 9

1, 4, 9, 16, 25, 36, 49, 64, 81,

Steps

Note 1: Take 50 as the reference

1. Subtract 50 from the given number.

Note2: This number is +ve or negative as the number is more or less

than 50.

2. Find the square of this number and get the least significant two

digits.

Note 3: If the difference is less than 4 then the two occupied digits

shoulld be 01, 04 or 09.

3. Add or subtract the difference to 25 and get the last two digits of

the square.

Example1: Find the square of 47

47

Step1 gives the difference as -3

Step 2 gives the first two digits of the square as 09

Step 3 gives the last two digits of the square as 25 -3 = 22

And the result is 2209.


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472 = 2209


Step1 gives the difference as +6


Step 3 gives the last two digits of the square as 25 +6 = 31


562 = 3136

Squaring numbers less than 40 and more than 60

Prerequisites: One should remember squares from 10 to 25.

The same steps are to be followed but this time the difference is more

than 10 and the square of the difference is a three digit number. Just

take the hundreds digit as carry and add to the result of step 3.


Step 1 gives the difference as -12

(-12)2 = 144

First two digits are 44

the last two digits are 25 -12 +1 ( carry) = 14

the result is 1444

382 = 1444


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Step 1 gives the difference as 13

132

= 169


the last two digits are 25 +13 +1 ( carry) = 39

the result is 3969

632 = 3969

Squaring numbers close to 100

Numbers between 90 and 110




than 100.


digits.


should be 01, 04 or 09.

3. Add or subtract the difference to the number itself ( unlike 25 in

the previous case) and get the last two digits of the square.



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Step1 gives the difference as -7




The square of 93 is 8649

Example6: 104

Step1 gives the difference as 4





Example7: 86

Step 1 gives the difference with 100 as -14

142 = 196


the last two digits are 86-14 +1 ( carry) = 73

the result is 7396

862 = 7396


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than 200.


digits.



3. Add or subtract the difference to the number itself multiply the

result by 2 and get the remaining digits of the square.


Step1 gives the difference with 200 as -7


Step 3 gives the remaining digits of the square as 2*(193 -7) = 372



Example9: 204

Step1 gives the difference with 200 as 4


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Step 3 gives the remaining digits of the square as 2*(204 +4) = 416



Example10: 186


142 = 196


the remaining digits are 2*(186-14) +1 ( carry) = 345

the result is 34596

1862 = 34596






than 300.


digits.


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3. Add or subtract the difference to the number itself multiply the

result by 3 and get the remaining digits of the square.




Step 3 gives the remaining digits of the square as 3*(293 -7) = 858



Example9: 304

Step1 gives the difference with 300 as 4


Step 3 gives the remaining digits of the square as 3*(304 +4) = 924





142 = 196


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the remaining digits are 3*(286-14) +1 ( carry) = 3*272 +1 = 817

the result is 81796

2862 = 81796

This leads to a simple generalisation for numbers close to nth

100.

e.g. : with 400 as the base the multiplier is 4. With 600 as the

base 6 is the multiplier. Students should try to generalise the

rule and verify the result with the calculator for themselves.

For Number close to 500 a method similar to the one for 50 can

be extended.

Squaring numbers between 490 and 510.

Prerequisites: One should remember squares of numbers upto 9

1, 4, 9, 16, 25, 36, 49, 64, 81,

Steps




than 50.

2. Find the square of this number and get the least significant three

digits.


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Note 3: If the difference is less than 4 then the three occupied

digits should be 001, 004 or 009.

3. Add or subtract the difference to 250 and get the remaining digits

of the square.


497


Step 2 gives the first three digits of the square as 009



4972 = 247009


Step1 gives the difference as +6




5062 = 256036

Squaring numbers less than 490 and more than 510

Prerequisites: One should remember squares from 10 to 25.


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The same steps are to be followed but this time the difference is more

than 10 and the square of the difference is a three digit number.


Step 1 gives the difference as -12

(-12)2 = 144

First three digits are 144

the remaining digits are 250 -12 = 238

the result is 238144

4882 = 238144


Step 1 gives the difference as 13

132 = 169

First three digits are 169

the remaining digits are 250 +13 = 263

the result is 263169

5132 = 263169

***************************************************************

Some general techniques

Start with a very simple case of squaring numbers of the form


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10x +5

The square is simply 100x(x+1) +25 which helps writing the square

instantly.

Examples

152 = 225 obtained by writing 25 and prefixing it with 1*(1+1) =2



Having found an easy trick for squaring numbers ending in 5 we can

find the square of other numbers as follows like for numbers ending in 6.

(10x +6)2 = (10x +5)2 + (10x +5) + (10x +6)

The first part on the right side can be evaluated by the previous trick and

then what remains is to add (10x +5) and then (10x +6).

Example: 362 = 352 +35 +36 = 1225 +35 +36 = 1296

Example: 962 = 952 +95 +96 = 9025 +95 +96 = 9216



(10x +4)2 = (10x +5)2 - (10x +5) - (10x +4)


then what remains is to subtract (10x +5) and then (10x +4).

Example: 342 = 352 -35 -34 = 1225 -35 -34 = 1156

Example: 942 = 952 -95 -94 = 9025 -189 = 8836


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Note: There are better tricks to find square of numbers close to 100

explained in coming pages.


also find the square of other numbers as follows like for numbers ending

in 3.

(10x +3)2 = (10x +5)2 -4*(10x+4)


then what remains is to subtract twice (10x +4) .

Example: 332 = 352 -4*34 = 1225 -136 = 1089

Example: 932 = 952 -4*94 = 9025 -476 = 8649



(10x +7)

2

= (10x +5)

2

+ 4* (10x +6)

The first part on the right side can be evaluated easily and then what

remains is to add 4 times (10x +6).

Example: 372 = 352 + 4*36 = 1225 + 144 = 1369

Example: 972 = 952 + 4*96 = 9025 + 384 = 9409

For numbers of the type 10x +1 the method is

(10x+1)2 = 100x2 + 10x + (10x +1)

312 = 302 + 30 + 31 = 961

912 = 902 + 90 + 91 = 8100 + 181 = 8281

For numbers of the type 10x +2


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(10x+2)2 = 100x2 + 4(10x +1)

422 = 402 + 4*(41) = 1600 + 164 = 1764

822

= 802

+ 4*(81) = 6400 + 324 = 6724

For numbers of the type 10x -1 the method is

(10x-1)2 = 100x2 -10x -(10x +1)

292 = 302 - 30 - 29 = 841

892 = 902 - 90 - 89 = 8100 - 179 = 7921

For numbers of the type 10x -2

(10x-2)2 = 100x2 - 4(10x +1)

382 = 402 - 4*(39) = 1600 -156 = 1444

782 = 802 - 4*(79) = 6400 - 316 = 6084

Note: Suitable trick is chosen as per whether the unit place is

smaller than 5 and close to 5 or 0 else the unit place is larger

than 5 and close to 5 or 10.

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Squareing Techniques