Easy methods to calculate perfect and non-perfect squares mentally

1. Square Root
Calculation Mentally
SC Chopra
sccpharma@gmail.com
Square Root: Mental
Calculation

2. General
• Presentation based on several YouTube videos
• Square roots of 3-5 digit numbers discussed in the presentation
• 3-digit numbers have square roots from 10 to 31
• 4-digit numbers have square roots from 32 to 99
• 5-digit numbers have square roots from 100 to 316

3. Square Root of Numbers ending
with 25 (X25)

4. Square Root of Numbers ending with 25
• Easiest to calculate
• Numbers ending with 5 (X25 numbers), which is not
preceded by 2, are NEVER perfect squares.
• X25 numbers MAY or MAY NOT be perfect squares
• Perfect Squaares: 225. 625, 1225, 2025, 3025 etc
• Non-perfect squarea: 125, 425, 925, 1425, 2525 etc

5. Square Root of X25 Numbers
• X is the part of the number other than 25 (yellow digits below)
625, 1225, 3025
• Find a perfect square (S) which is just smaller than X
• Left part (L) of the Square Root = √S
• Right part (R) of the Square Root = √25 = 5
•  √X25 = L │ R = LR ( “│” used for “join with” instead of “add”)

6. Square Root of 1225
• Given Number = 1225 and hence, X = 12
• Perfect square 9 (32) < 12 < Perfect square 16 (42)
•  S = 9 and hence, L = √9 = 3
• R is always 5
• Final Answer = L │ R = 3 │ 5 = 35
• Verification: X-part of 352 = L (L + 1) = 3 x 4 = 12
• R-part of 352 = 52 = 25
• LR = 12 │ 25 = 1225, hence verified (Used only for X25
numbers)

7. 4225
42 25
36 (62)< 42 < 49 (72)
 L = √36 = 6
R = √25 = 5
Ans. = 6 │ 5 = 65
Verify 652 = 4225
6 x 7 = 42 52 = 25
652 = 4225
(Verified)
Square Root and Verification

8. 625
4 < 6 < 9
 L = √4 = 2
R = √25 = 5
 √625 = 25
18225
169 < 182 < 196
 L = √169 = 13
R = √25 = 5
 √18225 = 135
2025
16 < 20 < 25
 L = √16 = 4
R = √25 = 5
 √2025 = 45
Square Roots of Three X25 Numbers

9. Verification of √X25 Answers
210 25
196 < 210 < 225
 L = √196 = 14
5
14 x 15 = 210
 √21025 = 145
280 25
256 < 280 < 289
 L = √256 = 16
5
16 x 17 = 272 ≠ 280
 √28025 ≠ 165
529 25
529 = 232
 L = √529 = 23 5
23 x 24 = 552 ≠ 529
 √52925 ≠ 235

10. Square Roots of All Numbers

11. Square Root of All Numbers (XYZ)
•Given number is divided into a right part of 2
digits (YZ) and a left part of all other digits (Yellow
digits below)
169 1225 12544
• X Clue to L-part of Sq. Root
• XYZ
YZ Clue to R-part of Sq. Root
•

12. Determination of Left Part (L) from X
• Just smaller perfect Square (S) < X < Larger Perfect Square (LS) and L =√S
• If X is 7, then 4 (22) < 7 (X) < 9 (32). Here S = 4 and L = √S = √4 = 2
• If X is 68, then 64 (82) < 68 < 81 (92). Here S = 64 and L = √S = √64 = 8
• If X is 564, then 529 (232) < 564 < 576 (242). Here S = 529 and L = √S =
√529 = 23

13. Determination of Right Part (R) from Z
• Based on the Z-digit, the right part (R) can have either of a pair of two
digits as under:
• If the Z-digit is 00, R-part of square root has to be 0
• If the Z-digit is 1, R-part of square root has to be either 1 or 9
• If the Z-digit is 4, R-part of square root has to be either 2 or 8
• If the Z-digit is 25 R-part of square root has to be 5 only
• If the Z-digit is 6, R-part of square root has to be either 4 or 6
• If the Z-digit is 9, R-part of square root has to be either 3 or 7
• 0 (single), 2, 3, 7 and 8 are NOT seen at the end of a perfect square

14. Unit Digit of Perfect Squares from 1-10
• Squares from 1 to 9 have only 1, 4, 5,
6 and 9 as unit digits
• Except 5, each digit occurs with two
numbers, one below and the other
above 5 & their sum is 10
• 5 occurs only once i.e. with 5
• Zeroes occur in even numbers (00,
0000 etc) in perfect squares

15. R-part of the Square Root
• The Z-digit gives the option of choosing one of the two possible digits
• One of the two digits will be below and the other above five.
• If X >= L(L + 1), the square root will be more than (L5)2 and hence, L is
joined with > 5 digit to take the place of R
• If X < L(L + 1), the < 5 digit is used as R (Example on next slide).

16. Square Root of XYZ when X >= P
• Given No = 676
• L-calculation: 4 < 6 < 9. Hence, S = 4, L = √S = √4 = 2
• Candidates for R-digit when Z = 6 are 4 and 6
• Choosing R-digit out of the two candidates: L (L + 1) = 2 x 3 = 6 (Variable
P for product, named so for convenience)
• If X >= P, choose the larger candidate and if X < P, choose the smaller one
• X = 6 = P, hence larger of the two candidayes i.e. 6 should be chosen.
• Answer: √676 = 2 │ 6 = 26

17. Square Root of XYZ when X < P
• Given No = 576
• L-calculation: 4 < 5 < 9. Hence, S = 4, L = √S = √4 = 2
• Candidates for R-digit when Z = 6 are 4 and 6
• Choosing R-digit out of the two candidates: L (L + 1) = 2 x 3 =
6 (Variable P for product)
• X < P, hence select smaller of the candidates (4 and 6) i.e. 4
• Answer: √576 = 2 │ 4 = 24

18. Square Roots of Three XYZ Numbers
7 29
4 < 7 < 9
 L = √4= 2
3 or
7
2 x 3 = 6
6 < 7 R > 5
√729 = 27
62 41
49 < 62 < 64
 L = √49 = 7
1
or
9
7 x 8 = 56
56 < 62 R > 5
√6241 = 79
973 44
961 < 973 < 1024
 L = √961 = 31
2
or
8
31 x 32 = 992
992 > 973  R < 5
√97344 = 312

19. Newer Method for Calculation of
Square Root of any Number

20. Newer Method for Square Root Calculation
• Take the example of 1521
• We know that for all numbers > 900 (302) and < 1600 (402),
the square root will be >=30 and < 40
• Unit’s place in 1521 has 1, hence R-digit can only be 1 or 9.
• Since, 1 < 5 < 9, if 1521 > 352, its square root will be 39 and if
1521 < 352, its square root will be 31

21. Crucial Step: (X5)2
• (X5)2 = X (X + 1) joined with 25
• Therefore, 352 = 3 x 4 joined with 25 = 12 │ 25 = 1225
• Since 1521 > 352, its square root > 35 i.e. 39 and NOT 31
• For √961, again the options are 31 and 39
• However, 961 < 352 (1225) and hence, √961 = 31
• Next slide shows localization of a number within the range of
100-10,000 and further localisation below or above 5-level in
two intervals

22. 100 ( 102)
400 (202)
900 ( 302)
1600 ( 402)
2500 (502)
3600 (602
0
4900 (702)
6400 (802)
8100 (902)
400 (202)
625 (252)
900 (302)
6400 (802)
7225 (852)
8100 (902)
144
484
1156
2304
2809
4761
5625
7396
8649
22
28
84
86

23. Not All Numbers ending with 0, 1, 4, 5, 6 0r 9
are Perfect Squares
• √100 = 10 but √1000 = 31.6, hence 1000 is NOT a perfect square
• Same holds true for innumerable other numbers as well
• Hence, we must NOT blindly accept the results obtained with these methods
• The answers must be verified by calculating the squares, which should be the
same as the given number. If they are not, the given number must be treated as
a non-perfect square i.e. a rational number instead of a whole number
• Calculation of square root by this method coupled with verification by a short
cut method is still much faster than calculating square root by the conventional
factor or division methods

24. Squaring of Answer for Verification
• Set table as shown
• Given √576 = 24. a = 2, b = 4
• b2 = 42 = 16
• a2 = 22= 4
• Write both values as shown
• 2ab = 2 x 2 x 4 = 16. Add a cross (x) to it and write as shown
• Sum up to get the answer (576), which is the number whose square
root was taken
a2 b2
a2 & b2 4 16
2ab 1 6x
Sum 5 76

25. Squaring of Answer for Verification
• Set table as shown
• Given √3364 = 58. a = 5, b = 8
• b2 = 82 = 64
• a2 = 52= 25
• Write both values as shown
• 2ab = 2 x 5 x 8 = 80. Add a cross (x) to it and write as shown
• Sum up to get the answer (3364), which is the number whose square
root was taken
a2 b2
a2 & b2 25 64
2ab 8 0x
Sum 33 64

26. Squaring of Answer for Verification
• Set table as shown
• Given √45796 = 214. a = 21, b = 4
• b2 = 42 = 16
• a2 = 212= 441
• Write both values as shown
• 2ab = 2 x 21 x 4 = 168. Add a cross (x) to it and write as shown
• Sum up to get the answer (45796), which is the number whose
square root was taken
a2 b2
a2 & b2 441 16
2ab 16 8x
Sum 457 96

27. Squaring of Answer for Verification
• Set table as shown
• Given √725 = 25. a = 2, b = 5
• b2 = 52 = 25
• a2 = 22= 4
• Write both values as shown
• 2ab = 2 x 2 x 5 = 20. Add a cross (x) to it and write as shown
• Sum up to get the answer (625). OOPS! It is NOT the number whose
square root was taken.
• Treat it as a non-perfect square
a2 b2
a2 & b2 4 25
2ab 2 0x
Sum 6 25

28. Dealing with Non-perfect Squares

29. Dealing with Non-perfect Squares
• Square root of a non-perfect square is a rational number
• For its determination, square root of a perfect square just
smaller or just larger than the given number is needed
• √n = √(x + y) or √(x – y) where n = given number, x = perfect
square closest to n and y = Difference between n and x
• √(x + y) = √x + y/(2 x √x) and √(x - y) = √x - y/(2 x √x)

30. Square Root of Non-perfect Squares
• Let given number be 150
• 150 = √{144 + 6) = √144 + 6/(2x√144)} = 12 + 6/(2 x 12) = 12 + 6/24 = 12 +
¼ = 12¼ or 12.25
• Another Example: Let given number be 200
• 200 = √{225 - 25) = √225 - 25/(2x√225)} = 15 – 25/(2 x 15) = 15 – 25/30 =
15 – 5/6= 14 1/6 = 14.16.

31. Square Root of Non-perfect Squares
• Given number = 1190, by the addition method
• √1176 = √{1156 + 34} = {34 + 34/(2 x 34)} = 34 + ½ = 34½ = 34.5
• Another Example: Let given number be 1980
• √1980 = √{2025 - 45} = {45 – 45/(2 x 45)} = 45 – 45/90 = 45 – ½ =
44½ = 44.5

32. Parting Gift

33. Squares from 1 to 30
X X2 X X2 X X2 X X2 X X2
1 1 7 49 13 169 19 361 25 625
2 4 8 64 14 196 20 400 26 676
3 9 9 81 15 225 21 441 27 729
4 16 10 100 `16 256 22 484 28 784
5 25 11 121 17 289 23 529 29 841
6 36 12 144 18 324 24 576 30 900