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Square root calculation mentally
- 1. Square Root Calculation Mentally SCChopra [email protected] Square Root: Mental Calculation
- 2. General • Presentation basedon 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 ofNumbers ending with 25 (X25)
- 4. Square Root ofNumbers 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 ofX25 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 of1225 • 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 √X25Answers 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 ofAll Numbers
- 11. Square Root ofAll 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 LeftPart (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 RightPart (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 ofPerfect 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 theSquare 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 ofXYZ 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 ofXYZ 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 ofThree 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 forCalculation of Square Root of any Number
- 20. Newer Method forSquare 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 Numbersending 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 Answerfor 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 Answerfor 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 Answerfor 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 Answerfor 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-perfectSquares
- 29. Dealing with Non-perfectSquares • 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 ofNon-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 ofNon-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.
- 33. Squares from 1to 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