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Squaring tricks for beginners

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It is a compilation of some methods for mentally squaring numbers rapidly

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Squaring tricks for beginners

  1. 1. Squaring Tricks For Beginners: A Compilation SC Chopra sccpharma@gmail.com
  2. 2. Plan of Presentaion ■ Introduction ■ Squaring Numbers – ending with 5 – ending with 1 – in 100-130 range – in 70-100 range – in 50-80 range – in 30-50 range – in 150-180 range – in 120-150 range – close to n x 100 – any number ■ Parting tips
  3. 3. Introduction
  4. 4. General Information ■ Presenter is a retired Medical Professor ■ Presentation based on dozens of YouTube videos by expert mathematics teachers. ■ They have simplified the application of algebraic formulas in calculations and presented them in the easiest form. Hats off to them! ■ Presentation is meant for beginners. They should keep practising to remain proficient
  5. 5. Conventions Followed ■ 2-digit given numbers are referred to as XY and 3-digit as XYZ ■ (XY)2 will give a 3-digit answer if the value of XY is 11-31 and a 4-digit answer if it is 32-99 ■ A 3-digit no. (XYZ) in the range 100-316 will give a 5-digit square and 317-999 will give a 6 digit square ■ Answers are referred to as LMR (Left, Middle & Right Digits) for 3-digit answers, LLMR for 4-Digit answers and so on ■ Symbol “│” is used for “join with” without adding e.g. 3 │ 4 │ 5 = 345
  6. 6. Suggestions for Squaring - Mental vs Paper ■ Several types of calculations are presented starting with the easiest one. ■ Do it initially on paper for understanding each step. ■ Then do it mentally, slowly at first till accuracy is attained and increase the speed till you can do it in under 5 seconds ■ Ability to recall squares of numbers 1-30 is a must. Table is given at the end of the presentation ■ Practice is a must for mastering any method ■ Master one method before going to the next
  7. 7. Method 1 – Numbers ending with 5
  8. 8. Method 1 – Numbers Ending with 5 (X5) ■ Right two digits (MR) of (X5)2 are always 25 ■ You only have to multiply the left digit(s) of the given number (X) by its successor (X + 1) to get left part of the answer ■ 252 = 2 x 3 │ 25 = 6 │ 25 = 625 (LMR) ■ 752 = 7 x 8 │ 25 = 56 │ 25 = 5,625 (LLMR) ■ 1952 = 19 x 20 │ 25 = 90 │ 25 = 38,025 (LLLMR) ■ 3252 = 32 x 33 │ 25 = 1056 │ 25 = 105,625 (LLLLMR) ■ Easy! No big deal! Master it and then go to Method 2
  9. 9. Method 2 – Numbers ending with 1
  10. 10. Method 2: Squaring X1 Numbers ■ R = 1 (Constant for ALL X1 and X9) numbers ■ M = 2 x X. If answer is in 2-digits, drop the right digit and carry the left digit to L/LL ■ L/LL = X2 + carry from M (if any) ■ Then join them as L/LL │ M │ R as shown in next slide
  11. 11. Method 2: Squaring X1 Numbers ■ 312: R = 1, M = 3 x 2 = 6, L = 32 = 9. Answer = 9 │ 6 │ 1 = 961 ■ 512: R = 1, M = 5 x 2 = 10 (drop 0, carry 1), L = 52 = 25 + 1 carry from M = 26. Answer = 26 │ 0 │ 1 = 2,601 ■ 712: R = 1, M = 7 x 2 = 14 (drop 4, carry 1), L = 72 = 49 + 1 carry from M = 50. Answer = 50 │ 4 │ 1 = 5,041 ■ Master this method before proceeding further
  12. 12. Method 3: Squaring Numbers 100 - 130
  13. 13. Method 3: Squaring Numbers 100 - 130 ■ Here we assume a base of 100. Difference between the given number and the base is important ■ Formula: a2 - b2 = (a + b) (a - b). Therefore, a2 = (a + b) (a - b) + b2 ■ a = Given number and b (difference) = a – assumed base = a - 100 ■ You have to add the difference (b) to “a” i.e. (a + b) and also subtract it from “a” (a – b) to get 100. Then you multiply the two values and add the square of the difference (b2) to get the answer (a2) ■ It is far easier to do it by the modified method (next slide)
  14. 14. Method 3: Squaring Numbers 100 - 130 ■ (a – b) will always be 100 by the previous method. You can ignore it and get the answer by joining (a + b) and b2 ■ If given number is 102 ■ a = 102; b = 102 – 100 = 2; b2 = 22 = 4 = 04 (2-digits for MR) ■ a + b = 102 + 2 = 104 ■ a2 = (a+b) │ b2 = 104 + 04 = 10404 ■ Takes just 2-3 seconds to spot that b = 4 (right 1-2 digits) of the given number. The right part of the answer is its square and the left part is the sum of the given number and b. Just join them togeth
  15. 15. Method 3: Squaring Numbers 100 - 130 ■ Given number (a) = 104. b = 104 – 100 = 4 ■ MR = b2 = 42 = 16 (If > 2 digits, drop right 2 digits and carry the left digit(s) to the LLL part ■ LLL = a + b = 104 + 4 = 108 ■ Answer = LLL │ MR = 108 │ 16 = 10816 ■ Takes just 2-3 seconds to spot that b = 04 (right 2 digits of the given number). The right part of the answer is its square and the left part is the sum of the given number and b. Just join them together
  16. 16. Method 3: Squaring Numbers 100 - 130 ■ Given Number (a) = 114 ■ b = 114 – 100 = 14 or right 2 digits of “a” ■ MR = 142 = 196 (drop 96 and carry 1 to LLL) ■ LLL = 114 + 14 = 128 + 1 carry = 129 ■ Answer = LLL │ MR = 129 │ 96 = 12,996
  17. 17. Method 3: Squaring Numbers 100 - 130 ■ If given no. (a) = 126 ■ b = 26 (Right 2 digits of a) ■ MR = 262 = 676 (drop 76 and carry 6 to LLL) ■ LLL = 126 + 26 = 152 + 6 carry = 158 ■ Answer = 158 │ 76 = 15,876
  18. 18. Method 4: Squaring Numbers 70-100
  19. 19. Method 4: Squaring Numbers 70-100 ■ Same formula as for Method 3 but with minor variations ■ Here, difference would be 100 – a since a < 100 ■ (a + b) would be 100, hence ignored ■ (a – b) = LL part ■ b2 = MR part ■ a2 = (a – b) │ b2 = LL │ MR ■ Note LLL is (a – b) and NOT (a + b)
  20. 20. Method 4: Squaring Numbers 70-100 ■ Given number (a) = 94 ■ b = 100 – 94 = 6 (It is NOT the right digits of the given number a) ■ MR = 62 = 36 (In > 100 numbers, you could square the digits to the right of 1 but here, b is different from the right digit of the given number which is 4 in this case) ■ LL = 94 – 6 = 88 ■ Answer = 88 │ 36 = 8,836
  21. 21. Method 4: Squaring Numbers 70-100 ■ Given Number (a) = 78 ■ b = 100 – 78 = 22 ■ MR = 222 = 484 (Drop 84, carry 4 to LL) ■ LL = 78 – 22 = 56 + 4 carry = 60 ■ Answer = 60 │ 84 = 6,084
  22. 22. Method 5: Squaring Numbers 50-80
  23. 23. Method 5: Squaring Numbers 50-80 ■ Assumed base = 50, b = a – 50 ■ Therefore, a – b = 50 and a= 50 + b ■ (a + b)(a – b) would result in a number multiplied by 50 ■ Multiplying a number with 50 is difficult. It is made easy by multiplying one factor of the term by 2 and dividing the other by 2 ■ We multiply 50 by 2 to make it 100 (easy to multiply) ■ Since (a – b) is multiplied by 2, (a + b) is divided by 2 to maintain mathematical balance. ■ Since a = (50 + b), a + b = 50 + 2b. Hence (50 + 2b)/2 = 25 + b ■ Therefore, join 25 + b with b2
  24. 24. Method 5: Squaring Numbers 50-80 ■ Given No. (a) = 62 ■ Base = 50, ■ b = 62 – 50 = 12 ■ b2 = 122 = 144 (MR = 44, carry 1). ■ LL = 25 + 12 = 37 + 1 carry = 38 ■ Answer = LL │ MR = 38 │ 44 = 3,844
  25. 25. Method 5: Squaring Numbers 50-80 ■ Given Number (a) = 722 ■ Base = 50 ■ b = 72 – 50 = 22 ■ MR = 222 = 484 (drop 84 and carry 4) ■ LL = 25 + b = 25 + 22 = 47 + 4 carry = 51 ■ Answer = LL│ MR = 5,184
  26. 26. Method 5: Squaring Numbers 50-80 ■ Given Number (a) = 792 ■ Base = 50 ■ b = 79 – 50 = 29 ■ MR = 292 = 841 (drop 41 and carry 8) ■ LL = 25 + b = 25 + 29 = 54 + 8 carry = 62 ■ Answer = LL│ MR = 6,241
  27. 27. Method 6 – Squaring Numbers 30-50
  28. 28. Method 6 – Squaring Numbers 30-50 ■ Same procedure as for Method 5 but b is subtracted from 25 (in place of adding) to get LL as numbers < 50 are involved ■ Join it with b2 to get the answer
  29. 29. Method 6 – Squaring Numbers 30-50 ■ Given Number (a) = 46 ■ b = 50 – 46 = 4. ■ Remember b is 4 i.e. the difference between base and a and NOT the 6 which is the right digit of a (46 in this case) ■ MR = b2 = 42 = 16 ■ LL = 25 – b = 25 - 4 = 21 ■ Answer = 21 │ 16 = 2,116
  30. 30. Method 6 – Squaring Numbers 30-50 ■ Given Number (a) = 34 ■ b = 50 – 34 = 16. ■ MR = b2 = 162 = 256 (drop 56, carry 2) ■ LL = 25 – b = 25 - 16 = 9 + 2 = 11 ■ Answer = 11 │ 56 = 1,156 ■ It is very easy spot the difference between base and a (16 in this case). You only have to square it to get MR and subtract it from 25 to get L/LL. Rest is merely carry and joining LLMR
  31. 31. Method 7 – Squaring Numbers 150- 180
  32. 32. Method 7 – Squaring Numbers 150-180 ■ Same Formula: a2 = (a + b) (a – b) + b2 but 150 is taken as base ■ a = 156; b = 156 – 150 = 6 and b2 = 62 = 36 ■ (a + b) (a – b) = (156 + 6) (156 – 6) = 162 x 150 ■ Multiplying with 150 can be tedious. Hence, use the same method as in method 5 i.e. multiply 150 by 2 to get 300 and divide a + b (162) by 2 to make it 81. a2 = 81 x 300 = 24300. Discard 00 to get LLL or simply multiply 81 by 3 to get 243 ■ Simply add b to a and multiply by 3 to get LLL and use b2 as MR ■ Join LLL with b2 to get 24,336 as the answer
  33. 33. Method 7 – Squaring Numbers 150-180 ■ Given Number (a) = 167 ■ Base = 150 and multiplication factor (MF) = 150/100 x 2 = 3 ■ b = 167 – 150 = 17 ■ MR = b2 = 172 = 289 (drop 89, carry 2) ■ a + b = 167 + 17 = 184 ■ LLL = (a + b) (a – b) = (a + b)/2 x MF (used in place of 150) =184/2 x 3 = 92 x 3 = 276 + 2 carry = 278 ■ Answer = LLL │ MR = 278 │ 89 = 27,889
  34. 34. Method 7 – Squaring Numbers 150-180 ■ a = 174 ■ Base = 150, MF = 3 ■ b = 174 – 150 = 24 ■ MR = b2 = 242 = 576 (drop 76, carry 5) ■ a2 = (174 + 24)/2 x3 = 198/2 x 3 = 99 X 3 = 297+ 5 carry = 302 (LLL) ■ Answer = 302 │ 76 = 30,276
  35. 35. Method 8 – Squaring Numbers 120-150
  36. 36. Method 8 – Squaring Numbers 120-150 ■ Same Formula: a2 = (a + b) (a – b) + b2 ■ a = 146 ■ Base = 150 and MF = 3 ■ b = 150 – 146 = 4 ■ b2 = 42 = 16 ■ LL = MF x (a – b)/2 = 3 x (146 – 4)/2 = 2 x 142/2 = 3 x 71= 213. ■ Answer = 213 │ 16 = 21,316
  37. 37. Method 8 – Squaring Numbers 120-150 ■ a = 126 ■ Base = 150 and MF = 3 ■ b = 150 – 126 = 24 ■ b2/ = 242 = 576 (drop 76, carry 5) ■ LL = MF x (a – b) /2= 3 x (126 – 24)/2 = 3 x 102/2 = 3 x 51 = 153 + 5 carry = 158 ■ Answer = 158 │ 76 = 15,876
  38. 38. Method 9: Squaring Numbers close to n X 100
  39. 39. Method 9: Squaring Numbers close to n x 100 ■ Given number = 777 ■ Let the base be 800 and MF = 8 ■ b = 800 – 777 = 23 ■ MR = b2 = 232 = 529 (drop 29, carry 5) ■ LLLL = 777 – 23 = 754 x 8 = 6032 + 5 carry = 6037 ■ Answer = 6037 │ 29 = 6,03,729
  40. 40. Method 9: Squaring Numbers close to n x 100 ■ Given Number = 918 ■ Let the base be 900 and MF = 9 ■ b = 918 – 900 = 18 ■ MR = 182 = 324 (drop 24, carry 3 to LLLL) ■ LLLL =918 + 18 = 936 x 9 = 8424 + 3 carry = 8427 ■ Answer = 8427 │ 24 = 842,724
  41. 41. Method 9: Short cut for Squaring when Base is used ■ Add b to a and multiply by base/100 and join it with b2 ■ Given Number = 206 ■ b = 6 ■ Assumed base = 200 and MF = 2 ■ 206 + 6 = 212 x 2 = 424. ■ Join it with 62 to get 42,436, which is the right answer
  42. 42. Method 9: Short cut for Squaring when Base is used ■ If given number = 721, b = 21 and b2 = 441 (drop 41, carry 4). Base = 700 and MF = 7 ■ LLLL 721 + 21= 742 x 7 = 5194 + 4 = 5198 ■ Answer = LLLL │ MR = 5198 │ 41 = 519,841
  43. 43. Method 10. Generic method for Squaring any Number
  44. 44. Method 10: Generic method for Squaring any Number ■ Formula: (a + b)2 = a2 + 2 ab + b2 ■ Let the number be 36. Split it by taking at least one term which has terminal zero(es). In this case, (30 + 6) will ne useful ■ (a + b)2 = (30 + 6)2 = 900 + 360 + 36 = 1296 ■ Inspection tells us that it is LLMR pattern for 2 digit numbers squared ■ R = b2. If > 1 digit, drop right digit and carry the left digit to M ■ M = 2 ab + carry, if any from R. If > 1 digit, drop right and carry left digit ■ L = a2 + carry from M, if any ■ As in other methods, drop zerog(es) and use “join with”
  45. 45. Method 10: Generic method for Squaring any Number ■ Set table as shown ■ Given Number = 84. a = 8, b = 4 ■ b2 = 42 = 16 ■ a2 = 82= 64 ■ Write both values as shown ■ 2ab = 2 x 8 x 4 = 64. Add a cross (x) to it and write as shown ■ Sum up to get the answer (7,056) a2 b2 a2 & b2 64 16 2ab 6 4x Sum 70 56
  46. 46. Method 10: Generic method for Squaring any Number ■ Given Number = 47. a = 4, b = 7 ■ b2 = 72 = 49 ■ a2 = 42= 16 ■ Write both values as shown ■ 2ab = 2 x 4 x 7 = 56. Add a cross (x) to it and write as shown ■ Sum up to get the answer (2,209) a2 b2 a2 & b2 16 49 2ab 5 6x Sum 22 09
  47. 47. Method 10: Generic method for Squaring any Number ■ Given Number = 112. a = 11, b = 2 ■ b2 = 22 = 04 (2-digits) ■ a2 = 112= 121 ■ Write both values as shown ■ 2ab = 2 x 11 x 2 = 44. Add a cross (x) to it and write as shown ■ Sum up to get the answer (1,25,44) a2 b2 a2 & b2 121 04 2ab 4 4x Sum 125 44
  48. 48. The Parting Tips
  49. 49. Parting Tip - 1 ■ If you know the square of a number (a) and want the square of its successor (a + 1), use the under-mentioned formula (a + 1)2 = a2 + 2 x a x 1 + 12 = a2 + 2a + 1 ■ Hence, to the square of the known number, add twice that number and 1. Alternately, you can add the number and its successor. ■ 262 = 252 + 2 x 25 + 1 = 625 + 50 + 1 = 676 ■ 332 = 322 + 2 x 32 + 1 = 1024 + 64 + 1 = 1,089 ■ 1012 = 1002 + 2 x 100 + 1 = 10000 + 200 + 1 = 10,201
  50. 50. Parting Tip - 2 ■ If you know the square of a number (a) and want the square of its predecessor (a - 1), use the under- mentioned formula (a - 1)2 = X2 - 2 x a x 1 + 12 = a2 – 2a + 1 ■ Hence, from the square of the known number, subtract double that known number and then add 1. ■ 242 = 252 - 2 x 25 + 1 = 625 - 50 + 1 = 576 ■ 492 = 502 - 2 x 50 + 1 = 2500 - 100 + 1 = 2,401 ■ 992 = 1002 - 2 x 100 + 1 = 10000 - 200 + 1 = 9,801
  51. 51. Parting Tip 3: Rough Checks ■ Squares of numbers ending with 1 or 9, ends with 1 ■ Squares of numbers ending with 2 or 4, ends with 4 ■ Squares of numbers ending with 3 or 7, ends with 9 ■ Squares of numbers ending with 4 or 6, ends with 6 ■ Squares of numbers ending with 5, ends with 25 ■ Squares of numbers ending with zero(es), ends with double the no. of zeroes
  52. 52. Parting Tip 5: Rough Checks ■ Squares of 1-10: lie between 1 and 100 ■ Squares of 10-20: lie between 100 and 400 ■ Squares of 20-30: lie between 400 and 900 ■ Squares of 30-40: lie between 900 and 1600 ■ Squares of 40-50: lie between 1600 and 2500 ■ Squares of 50-60: lie between 2500 and 3600 and so on ■ Topic of No. of digits in the square is discussed in the beginning of the presentation
  53. 53. 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
  54. 54. Thank you

It is a compilation of some methods for mentally squaring numbers rapidly

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