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The Relaunch of Indian Mathematics

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India's true mathematice of negatives, positives and zero appears to have never left India. Watch this fun informative slideshow to discover how simple mathematics really becomes when you learn the mathematical laws of India's Brahmagupta.

Here is what two people who saw me talk at Jadavpur University Kolkata said:

"I must say, today your presentation was beautiful and fascinating. This is the best lecture on elementary and fundamentals of Mathematics I have ever seen.

Thanks for giving us such a magnificent, powerful and thought provoking lecture on Indian Mathematics. It has immense importance in terms of originality and to established truth."

and...

"It was quite an amazing and mind-boggling experience at the same time to get acquainted with the commonplace errors that we always make but still compromises with ourselves in a way that it's okay. Thanks a lot, Prof. JC for pointing out this and your inspirational lecture today is still ringing in my ears."

===
If you like the idea of Relaunching of Indian Mathematics, please share your feedback at http://jonathancrabtree.com/feedback/

Best wishes,
Jonathan J. Crabtree
Elementary Mathematics Historian
Melbourne Australia

Published in: Education
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The Relaunch of Indian Mathematics

  1. 1. 1
  2. 2. © 2018 J. J. Crabtree | www.jonathancrabtree.com 2
  3. 3. In 300 BCE the Greek Euclid of Alexandria defined multiplication as repeated addition. a to the power of b (ab) equals a into itself b times. a into b (a × b ) equals a added to itself b times. © 2018 J. J. Crabtree | www.jonathancrabtree.com 3
  4. 4. Zero is defined as a – a ‘a number subtracted from itself’. The Arabic world embraced India’s ideas on Zero and negative numbers. Arabic maths was then adopted by Leonardo Pisano who brought India’s ideas on Zero to Europe. © 2018 J. J. Crabtree | www.jonathancrabtree.com 4
  5. 5. © 2018 J. J. Crabtree | www.jonathancrabtree.com 5 Your ‘facts’ are false! Everyone is entitled to their own opinion, yet not to their own facts.
  6. 6. Extraordinary Claims Require Extraordinary Evidence © 2018 J. J. Crabtree | www.jonathancrabtree.com 6 ✔
  7. 7. •India’s definition of ZERO never made it to either the ancient Arabic world or Europe. © 2018 J. J. Crabtree | www.jonathancrabtree.com 7 •In the Arabic world, India’s ZERO only came to exist as a placeholder, not as the power tool to solve simple problems like +3 minus +4, or –2 minus –4, or –4 minus +2 Extraordinary Claims Extraordinary Evidence...
  8. 8. Al-Khwārizmī (c. 780-850) “So they made 9 symbols, which are these: 9 8 7 6 5 4 3 2 1. ... Because one is the root of all number and is outside number. It is the root of number because every number is found by it. ... But it [one] is outside number because it is found by itself, I mean, without any other number.” © 2018 J. J. Crabtree | www.jonathancrabtree.com 8
  9. 9. Al-Uqlīdisī (c. 920-980) “Why is zero multiplied by zero equal to zero and zero multiplied by any letter zero? We say that by multiplying zero by zero the aim is only to occupy the place; the same applies for multiplying the letter by zero. We multiply the letter by zero ... to occupy the place, and tell that there is a place and that it is empty. © 2018 J. J. Crabtree | www.jonathancrabtree.com 9
  10. 10. 200 years after Brahmagupta, al-Khwārizmī did not accept 1 as a number. Zero as a number? Never! © 2018 J. J. Crabtree | www.jonathancrabtree.com 10 300 years after Brahmagupta, al-Uqlīdisī accepted India’s ZERO as a placeholder, yet not a number. Why?
  11. 11. © 2018 J. J. Crabtree | www.jonathancrabtree.com 11 Al-Uqlīdisī means ‘the Euclidist’. He was known for his skill in studying the Greek geometry of Euclid and translating it into Arabic. Around 300 BCE, Euclid defined ‘number’ as a multitude of units. So Euclid’s definition of number came before 0 and 1 were numbers.
  12. 12. © 2018 J. J. Crabtree | www.jonathancrabtree.com 12 As we will see, India defined zero as the sum of opposing negative and positive numbers with the same multitude or magnitude. If Arabic and European writers in medieval times really understood India’s zero, where are all the negative numbers in their writings?
  13. 13. © 2018 J. J. Crabtree | www.jonathancrabtree.com 13 “I have read a few dozen medieval Arabic books on arithmetic and algebra, and there is no hint of negative numbers in any of them. Zero, too, was not regarded to be a number, but was merely the place holder for an empty place in the representation of a number in Arabic (Indian) notation.” “All numbers in Arabic arithmetic were positive. No Arabic author to my knowledge ever even contemplated the existence of negative numbers.” By email courtesy of Dr. Jeffrey Oaks, Professor of Mathematics on: Medieval Arabic algebra and the mathematics of Greece and medieval Europe
  14. 14. © 2018 J. J. Crabtree | www.jonathancrabtree.com 14
  15. 15. © 2018 J. J. Crabtree | www.jonathancrabtree.com 15
  16. 16. © 2018 J. J. Crabtree | www.jonathancrabtree.com 16 Back to the future...
  17. 17. © 2018 J. J. Crabtree | www.jonathancrabtree.com 17
  18. 18. 18
  19. 19. 19
  20. 20. Images courtesy of the British Library. For this talk, Brahmagupta’s Laws of Positives Negatives and Zero have been freshly analysed. 20© 2018 J. J. Crabtree | www.jonathancrabtree.com
  21. 21. 21© 2018 J. J. Crabtree | www.jonathancrabtree.com
  22. 22. 22© 2018 J. J. Crabtree | www.jonathancrabtree.com
  23. 23. Brahmagupta’s 5 Addition Laws AL (saṅkalana) © 2018 J. J. Crabtree | www.jonathancrabtree.com 23
  24. 24. positive plus positive is positive AL1 © 2018 J. J. Crabtree | www.jonathancrabtree.com 24
  25. 25. negative plus negative is negative AL2 © 2018 J. J. Crabtree | www.jonathancrabtree.com 25
  26. 26. positive plus negative is the difference between the positive and the negative AL3 © 2018 J. J. Crabtree | www.jonathancrabtree.com 26
  27. 27. when positive and negative are equal the sum is… AL4 © 2018 J. J. Crabtree | www.jonathancrabtree.com 27
  28. 28. ZERO AL4 © 2018 J. J. Crabtree | www.jonathancrabtree.com 28 when positive and negative are equal the sum is…
  29. 29. “…sunya (ZERO) is neither positive nor negative but forms the boundary line between the two kinds, being the sum of two equal but opposite quantities.” Joseph, G. G. (2016). Indian mathematics: Engaging with the world, from ancient to modern times. World Scientific. p. 208 29© 2018 J. J. Crabtree | www.jonathancrabtree.com
  30. 30. positive plus zero is positive AL5 part 1 © 2018 J. J. Crabtree | www.jonathancrabtree.com 30
  31. 31. negative plus zero is negative AL5 part 2 © 2018 J. J. Crabtree | www.jonathancrabtree.com 31
  32. 32. zero plus zero is zero AL5 part 3 © 2018 J. J. Crabtree | www.jonathancrabtree.com 32
  33. 33. positive plus zero is positive negative plus zero is negative AL5 zero plus zero is zero © 2018 J. J. Crabtree | www.jonathancrabtree.com 33
  34. 34. © 2018 J. J. Crabtree | www.jonathancrabtree.com Brahmagupta’s 5 Addition Laws positive plus positive is positiveAL1 negative plus negative is negativeAL2 positive plus negative is the difference between the positive and negativeAL3 when positive and negative are equal the sum is zeroAL4 positive plus zero is positive negative plus zero is negative zero plus zero is zero AL5 34
  35. 35. Following text from: Plofker, K. (2009). Mathematics in India: 500 BCE-1800 CE. p. 151, Princeton, N.J: Princeton University Press. Brahmagupta’s 5 Subtraction Laws SL (vyavakalana). 35© 2018 J. J. Crabtree | www.jonathancrabtree.com
  36. 36. [If] a smaller [positive] is to be subtracted from a larger positive, [the result] is positive. SL1 © 2018 J. J. Crabtree | www.jonathancrabtree.com 36
  37. 37. [If] a smaller negative (is subtracted) from a larger negative, [the result] is negative. SL2 © 2018 J. J. Crabtree | www.jonathancrabtree.com 37
  38. 38. [If] a larger [negative or positive is to be subtracted] from a smaller [negative or positive, the algebraic sign of] their difference is reversed - negative [becomes] positive and positive negative. SL3 © 2018 J. J. Crabtree | www.jonathancrabtree.com 38
  39. 39. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. SL4 © 2018 J. J. Crabtree | www.jonathancrabtree.com 39
  40. 40. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added. SL5 © 2018 J. J. Crabtree | www.jonathancrabtree.com 40
  41. 41. © 2018 J. J. Crabtree | www.jonathancrabtree.com Brahmagupta’s 5 Subtraction Laws A smaller positive subtracted from a larger positive is positive.SL1 A smaller negative subtracted from a larger negative is negative.SL2 If a larger negative or positive is to be subtracted from a smaller negative or positive, the sign of their difference is reversed – negative becomes positive and positive negative. SL3 A negative minus zero is negative, a positive minus zero is positive, zero minus zero is zero. SL4 When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added.SL5 41
  42. 42. Seeing is Believing! Applying Brahmagupta’s 5 Addition Laws AL © 2018 J. J. Crabtree | www.jonathancrabtree.com 42
  43. 43. – negative + positive© 2018 J. J. Crabtree | www.jonathancrabtree.com 43
  44. 44. AL1 positive plus positive is positive © 2018 J. J. Crabtree | www.jonathancrabtree.com 44
  45. 45. AL2 negative plus negative is negative © 2018 J. J. Crabtree | www.jonathancrabtree.com 45
  46. 46. positive plus negative is the difference AL3 © 2018 J. J. Crabtree | www.jonathancrabtree.com 46
  47. 47. 3 holes 7 bumps Podo the Puppy by AFX Animation © 2018 J. J. Crabtree | www.jonathancrabtree.com
  48. 48. 3 negatives 7 positives Podo the Puppy by AFX Animation © 2018 J. J. Crabtree | www.jonathancrabtree.com
  49. 49. 3 negatives 7 positives Podo the Puppy by AFX Animation © 2018 J. J. Crabtree | www.jonathancrabtree.com
  50. 50. So –3 + +7 = +4 3 holes (–3) and 7 bumps (+7 ) leads to 4 bumps (+4 ) Watch the AFX Animation video of Podo the Puppy squaring a circle with rope! © 2018 J. J. Crabtree | www.jonathancrabtree.com
  51. 51. ZERO ZERO when positive and negative are equal the sum is ZEROAL4 © 2018 J. J. Crabtree | www.jonathancrabtree.com 51
  52. 52. plus ZERO AL5 positive plus ZERO negative © 2018 J. J. Crabtree | www.jonathancrabtree.com ZERO + positive = positive, ZERO + negative = negative, ZERO + ZERO = ZERO is positive is negative 52
  53. 53. Seeing is Believing! Applying Brahmagupta’s 5 Subtraction Laws SL © 2018 J. J. Crabtree | www.jonathancrabtree.com 53
  54. 54. If a smaller positive is subtracted from a larger positive the result is positive. SL1 larger positive minus smaller positive is positive © 2018 J. J. Crabtree | www.jonathancrabtree.com 54
  55. 55. If a smaller negative is subtracted from a larger negative the result is negative. SL2 larger negative minus smaller negative is negative © 2018 J. J. Crabtree | www.jonathancrabtree.com 55
  56. 56. If a larger positive is to be subtracted from a smaller positive, the sign of their difference is reversed – positive becomes negative. SL3 We don’t have enough positives. So use AL4 and AL5. 3 positives minus 4 positives? © 2018 J. J. Crabtree | www.jonathancrabtree.com 56
  57. 57. If a larger positive is to be subtracted from a smaller positive… e.g. positive 3 – positive 4 SL3 AL4 “when positive and negative are equal the sum is zero” AL5 “positive with zero is positive” minus Add ZERO © 2018 J. J. Crabtree | www.jonathancrabtree.com 57
  58. 58. SL3 Add ZERO Subtractminus © 2018 J. J. Crabtree | www.jonathancrabtree.com AL4 “when positive and negative are equal the sum is zero” AL5 “positive with zero is positive” 58 If a larger positive is to be subtracted from a smaller positive… e.g. positive 3 – positive 4
  59. 59. SL3 Add ZERO Subtract Simplify © 2018 J. J. Crabtree | www.jonathancrabtree.com AL4 “when positive and negative are equal the sum is zero” AL5 “positive with zero is positive” 59 If a larger positive is to be subtracted from a smaller positive… e.g. positive 3 – positive 4
  60. 60. SL3 Add ZERO Subtract Simplify © 2018 J. J. Crabtree | www.jonathancrabtree.com AL4 “when positive and negative are equal the sum is zero” AL5 “positive with zero is positive” 60 If a larger positive is to be subtracted from a smaller positive… e.g. positive 3 – positive 4
  61. 61. SL3 Add ZERO Subtract Simplify © 2018 J. J. Crabtree | www.jonathancrabtree.com AL4 “when positive and negative are equal the sum is zero” AL5 “positive with zero is positive” 61 If a larger positive is to be subtracted from a smaller positive… e.g. positive 3 – positive 4
  62. 62. If a larger positive is to be subtracted from a smaller positive, the sign of their difference is reversed – positive becomes negative. SL3 3 positives minus 4 positives is 1 negative © 2018 J. J. Crabtree | www.jonathancrabtree.com 62
  63. 63. If a larger negative is to be subtracted from a smaller negative, the sign of their difference is reversed – negative becomes positive. SL3 We don’t have enough negatives. So use AL4 and AL5. 2 negatives minus 4 negatives? © 2018 J. J. Crabtree | www.jonathancrabtree.com 63
  64. 64. If a larger negative is to be subtracted from a smaller negative… e.g. negative 2 – negative 4SL3 AL4 “when positive and negative are equal the sum is zero” AL5 “negative with zero is negative” minus Add ZERO © 2018 J. J. Crabtree | www.jonathancrabtree.com 64
  65. 65. SL3 AL4 “when positive and negative are equal the sum is zero” AL5 “negative with zero is negative” Add ZERO Subtractminus © 2018 J. J. Crabtree | www.jonathancrabtree.com 65 If a larger negative is to be subtracted from a smaller negative… e.g. negative 2 – negative 4
  66. 66. SL3 AL4 “when positive and negative are equal the sum is zero” AL5 “negative with zero is negative” Add ZERO Subtract Simplify © 2018 J. J. Crabtree | www.jonathancrabtree.com 66 If a larger negative is to be subtracted from a smaller negative… e.g. negative 2 – negative 4
  67. 67. SL3 AL4 “when positive and negative are equal the sum is zero” AL5 “negative with zero is negative” Add ZERO Subtract Simplify © 2018 J. J. Crabtree | www.jonathancrabtree.com 67 If a larger negative is to be subtracted from a smaller negative… e.g. negative 2 – negative 4
  68. 68. SL3 AL4 “when positive and negative are equal the sum is zero” AL5 “negative with zero is negative” Add ZERO Subtract Simplify © 2018 J. J. Crabtree | www.jonathancrabtree.com 68 If a larger negative is to be subtracted from a smaller negative… e.g. negative 2 – negative 4
  69. 69. If a larger negative is to be subtracted from a smaller negative… the sign of their difference is reversed – negative becomes positive. SL3 2 negatives minus 4 negatives is 2 positives © 2018 J. J. Crabtree | www.jonathancrabtree.com 69
  70. 70. negative minus zero is negative SL4 4– minus 0– is 4– positive minus zero is positive zero minus zero is zero 3+ minus 0+ is 3+ © 2018 J. J. Crabtree | www.jonathancrabtree.com 70
  71. 71. When a positive is to be subtracted from a negative … then it is to be added. –4 – +2 SL5 © 2018 J. J. Crabtree | www.jonathancrabtree.com Add ZERO –4 – +2 = ? 71
  72. 72. When a positive is to be subtracted from a negative … then it is to be added. –4 – +2 SL5 © 2018 J. J. Crabtree | www.jonathancrabtree.com Add ZERO Subtract –4 – +2 = ? 72
  73. 73. When a positive is to be subtracted from a negative … then it is to be added. –4 – +2 SL5 © 2018 J. J. Crabtree | www.jonathancrabtree.com Add ZERO Subtract –4 – +2 = –6 For you. Show how a negative subtracted from a positive is to be added with +3 – –5 73
  74. 74. Brahmagupta’s 4 Multiplication Laws ML (guṇana) © 2018 J. J. Crabtree | www.jonathancrabtree.com Text from: Plofker, K. (2009). Mathematics in India 74
  75. 75. ML1 The product of a negative and a positive is negative © 2018 J. J. Crabtree | www.jonathancrabtree.com Brahmagupta multiplies positives and negatives by either adding to zero multiple times or subtracting from zero multiple times. –a × +b –a added to zero b times 75 False from 1570: a into b, (a × b), is a added to itself b times
  76. 76. Negative Multiplicand Multiplied by Positive Multiplier –a × +b –a added to zero b times × added to zero times Next, from pictures to equations… –2 × +3 –2 added to zero 3 times 76© 2018 J. J. Crabtree | www.jonathancrabtree.com
  77. 77. Negative Multiplicand Multiplied by Positive Multiplier –2 × +3 = –2 × (0 + 1 + 1 + 1) –2 × +3 = 0 + –2 + –2 + –2 –2 × +3 = 0 + –6 –2 × +3 = –6 –a × +b –a added to zero b times –2 × +3 –2 added to zero 3 times 77© 2018 J. J. Crabtree | www.jonathancrabtree.com
  78. 78. Positive Integer Multiplied by Negative Multiplier +2 × –3 Brahmagupta did not say “The product of a positive and a negative is negative”. However, we can demonstrate this via his Laws. +a × –b +a subtracted from zero b times 78© 2018 J. J. Crabtree | www.jonathancrabtree.com
  79. 79. Positive Multiplicand Multiplied by Negative Multiplier × +a × –b +2 subtracted from zero 3 times +2 × –3 As we don’t have any positive twos to subtract, we first add zero in the form +6 + –6 79© 2018 J. J. Crabtree | www.jonathancrabtree.com
  80. 80. Positive Multiplicand Multiplied by Negative Multiplier × Next, from pictures to equations… +a × –b +2 subtracted from zero 3 times +2 × –3 80© 2018 J. J. Crabtree | www.jonathancrabtree.com
  81. 81. Positive Integer Multiplied by Negative Multiplier +2 × –3 +2 × –3 = +2 × (0 – +1 – +1 – +1) +2 × –3 = 0 – +2 – +2 – +2 +2 × –3 = 0 – +6 +2 × –3 = –6 + +6 – +6 +2 × –3 = –6 + 0 +2 × –3 = –6 81© 2018 J. J. Crabtree | www.jonathancrabtree.com
  82. 82. ML2 The product of two negatives is positive © 2018 J. J. Crabtree | www.jonathancrabtree.com –1 × –1 = +1 A demonstration goes like this… Crabtree,JonathanJ.Snippet:Anew reasonnegativemultipliedbynegativeis positiveVinculum,Vol.52,No.3,Jul2015: 82
  83. 83. ML2 © 2018 J. J. Crabtree | www.jonathancrabtree.com Negative Multiplicand Multiplied by Negative Multiplier –a × –b –a subtracted from zero b times –1 × –1 –1 subtracted from zero 1 times 83
  84. 84. Brahmagupta Defined ZERO in Law AL4 when positive and negative are equal the sum is ZERO © 2018 J. J. Crabtree | www.jonathancrabtree.com –1 × –1 84 –1 subtracted from zero 1 times + –1+1
  85. 85. Brahmagupta Defined ZERO in Law AL4 when positive and negative are equal the sum is ZERO + –1 © 2018 J. J. Crabtree | www.jonathancrabtree.com –1 × –1 –1 subtracted from zero 1 times now prove it equals +1 ∴ –1 × –1 = +1 85 +1
  86. 86. ML3 The product of two positives is positive © 2018 J. J. Crabtree | www.jonathancrabtree.com ML4 The product: - of zero and a negative, - of zero and a positive, - or of two zeros is zero. 86
  87. 87. © 2018 J. J. Crabtree | www.jonathancrabtree.com Brahmagupta’s 4 Multiplication Laws The product of a negative and a positive is negative.ML1 The product of two negatives is positive.ML2 The product of two positives is positive.ML3 The product of zero and a negative, of zero and a positive, or of two zeros is zero. ML4 87
  88. 88. MY ASSESSMENT OF THE WORLD’S PEDAGOGICAL EVOLUTION (628 to Now) © 2018 J. J. Crabtree | www.jonathancrabtree.com 88
  89. 89. Brahmagupta’s 4 [Non-Zero] Division Laws DL (haraṇa) © 2018 J. J. Crabtree | www.jonathancrabtree.com 89
  90. 90. © 2018 J. J. Crabtree | www.jonathancrabtree.com Brahmagupta’s 4 Division Laws A positive divided by a positive is positive.DL1 A negative divided by a negative is positive.DL2 A positive divided by a negative is negative.DL3 A negative divided by a positive is negative.DL4 90
  91. 91. NOTE: Why Non-Zero Division Laws Only? © 2018 J. J. Crabtree | www.jonathancrabtree.com Whilst zero divided by any non-zero number (0/±n), is zero, Brahmagupta says a non-zero number divided by zero(i.e. ±n/0), remains ‘zero divided’. Commentators such as Bhāskara II (1114 –1185) suggested n/0 is infinite or indeterminate. 91
  92. 92. © 2018 J. J. Crabtree | www.jonathancrabtree.com MODELS OF DIVISION Partitive Model Equal Groups +12 ÷ +4 Q. If you have 12 things and 4 equal groups, how many things go in each group? A. 3 things go in each group. 12 ÷ 4 = 3 things Partitive Model ✔ 92
  93. 93. © 2018 J. J. Crabtree | www.jonathancrabtree.com MODELS OF DIVISION Quotitive Model Repeated Subtraction Q. If you have 12 things, how many times can you subtract 4 things? A. You can subtract 4 things 3 times until you get 0 things. 12 ÷ 4 = 3 times Partitive Model ✔ Quotitive Model ✔ +12 ÷ +4 93
  94. 94. © 2018 J. J. Crabtree | www.jonathancrabtree.com METHODS OF DIVISION Equal Groups Repeated Subtraction Equal Groups Equal Groups Repeated Subtraction Equal Groups Repeated Subtraction Repeated Subtraction ✔ ✔ ✔ ✔ ✘ ✘ ✘ ✘ 94
  95. 95. © 2018 J. J. Crabtree | www.jonathancrabtree.com MODELS OF DIVISION Partitive Model Equal Groups +12 ÷ –4 Q. If you have 12 positive things and negative 4 groups, how many things go in each group? A. You can’t have a negative number of groups! Equal Groups ✘ Model Fails 95
  96. 96. © 2018 J. J. Crabtree | www.jonathancrabtree.com MODELS OF DIVISION Quotitive Model Repeated Subtraction Q. If you have 12 positive things, how many times can you take away 4 negative things? A. You can’t subtract 4 negative things from positive things. Try... +12 – –4 = +16, – –4 = +20, – –4 = +24. i.e. You go the wrong way and never get to zero. +12 ÷ –4 Equal Groups✘ Repeated Subtraction✘ 96 Model Fails
  97. 97. © 2018 J. J. Crabtree | www.jonathancrabtree.com PROPORTIONAL COVARIATION, PCV Equal Groups✘ Repeated Subtraction✘ Numbers are counts or measures of standard units. We choose 1. All numbers are measured by the unit called 1. 1 is the multiplicative identity and the divisional identity. 97
  98. 98. © 2018 J. J. Crabtree | www.jonathancrabtree.com Equal Groups✘ Repeated Subtraction✘ Our goal is to convert 12/4 into the form n/1 DENOMINATOR To make –4 be +1 we take one of the 4 parts of –4, which is –1 and subtract it from zero to get +1. PROPORTIONAL COVARIATION, PCV 98
  99. 99. © 2018 J. J. Crabtree | www.jonathancrabtree.com Whatever we have done to the denominator, we must do to the numerator to keep the number (ratio) NUMERATOR We take one of the 4 parts of +12, which is +3 and subtract it from zero to get –3. PROPORTIONAL COVARIATION, PCV Equal Groups✘ Repeated Subtraction✘ 99
  100. 100. © 2018 J. J. Crabtree | www.jonathancrabtree.com Whatever we have done to the denominator, we must do to the numerator to keep the number (ratio) PROPORTIONAL COVARIATION, PCV Equal Groups✘ Repeated Subtraction✘ 100 (Zero Minus Numerator) by (Zero Minus Denominator) –12 +4
  101. 101. Multiplication: Whatever we do to the Unit 1 to make the Multiplier b we do to the Multiplicand a to make the Product c. Division: Whatever we do to the Divisor b to make the Unit 1 we do to the Dividend a to make the Quotient c. © 2018 J. J. Crabtree | www.jonathancrabtree.com PROPORTIONAL COVARIATION, PCV 101
  102. 102. As –4 is to +1 +12 is to... 102© 2018 J. J. Crabtree | www.jonathancrabtree.com
  103. 103. © 2018 J. J. Crabtree | www.jonathancrabtree.com 3 MODELS OF DIVISION Equal Groups Repeated Subtraction Proportional Covariation Proportional Covariation Proportional Covariation Equal Groups Repeated Subtraction✔ ✔ ✔ ✔ ✔ ✔ 103 Proportional Covariation ✔ +12 ÷ –4 ✔
  104. 104. PROPORTIONAL COVARIATION, PCV 104
  105. 105. –2 x –3 = ? (i.e. negative two subtracted three times from zero) What do we do to the standard Unit 1 to make the Multiplier –3? We place three Units and subtract them from zero to make the Multiplier –3. PROPORTIONAL COVARIATION, PCV 105© 2018 J. J. Crabtree | www.jonathancrabtree.com
  106. 106. –2 x –3 = ? (i.e. negative two subtracted three times from zero) PROPORTIONAL COVARIATION, PCV What do we do to the Multiplicand –2 to make the Product ? We place three Multiplicands and subtract them from zero to make the Product +6. 106© 2018 J. J. Crabtree | www.jonathancrabtree.com
  107. 107. +2 –3 ? +2 × –3 = ? a × b = ? We start off with +1 in this square
  108. 108. Then we put b the Multiplier here +2 × –3 = ? a × b = ? We start off with +1 in this square +2 –3 ?
  109. 109. +2 –3 ? Then we put a the Multiplicand here +2 × –3 = ? a × b = ? Then we put b the Multiplier here We start off with +1 in this square
  110. 110. To go from +1 to –3 we took 3 Units & changed their sign by subtracting from 0. So, we take 3 a’s and change their sign by subtracting from 0 to make c. +2 × –3 = ? a × b = ? Then we put b the Multiplier here We start off with +1 in this square +2 –3 ? Then we put a the Multiplicand here
  111. 111. +2 × –3 = ? a × b = ? +2 –3 ?
  112. 112. +2 × –3 = ? a × b = ? +2 –3 ?
  113. 113. +2 × –3 = ? a × b = ? +2 –3 ?
  114. 114. +2 –3 ? +2 × –3 = ? a × b = ?
  115. 115. +2 × –3 = ? a × b = ? +2 –3 ?
  116. 116. +2 × –3 = ? a × b = ? +2 –3 ?
  117. 117. PCVMultiMat PCVDiviMat 117© 2018 J. J. Crabtree | www.jonathancrabtree.com
  118. 118. –2 x –3 As +1 is to –3 so –2 is to +6 PROPORTIONAL COVARIATION PCV b a c 1 118© 2018 J. J. Crabtree | www.jonathancrabtree.com
  119. 119. René Descartes 1596 – 1650 Brahmagupta 598 – 668 Brahmagupta’s ideas were not applied 1000 years later, yet should have been.
  120. 120. Copyright © 2016 Jonathan Crabtree All Rights Reserved “For example, let AB be taken as unity, (1), and let it be required to multiply BD (the multiplicand) by BC (the multiplier), I have only to join the points A and C, and draw DE parallel to AC; and BE is the product of this Multiplication.” Applying Indian Logic to Descartes’s Multiplication
  121. 121. https://www.geogebra.org/m/wprjkwam Let point B be 0 and line segments on the other side of 0 be negative...
  122. 122. https://www.geogebra.org/m/wprjkwam A Negative Multiplicand and a Negative Multiplier result in a Positive Product.
  123. 123. https://www.geogebra.org/m/je3SEyQr www.geogebra.org/m/ZBTZd6AF
  124. 124. www.geogebra.org/m/v62CqVEN
  125. 125. © 2018 J. J. Crabtree | www.jonathancrabtree.com 125 www.bit.ly/New-x-Model
  126. 126. Q. Why no numbers in Greek Geometry? A. Aristotle Greek Philosopher 384–322 BCE Geometry cannot be proven with arithmetic © 2018 J. J. Crabtree | All Rights Reserved www.jonathancrabtree.com
  127. 127. Discussion on Brahmagupta’s Laws © 2018 J. J. Crabtree | www.jonathancrabtree.com A. Even though the numbers are different, both numerators are smaller than their denominators. Q. How can the following fractions be equal? + 1 + 2 = + 4 + 8 127
  128. 128. Discussion on Brahmagupta’s Laws If one side has a smaller number on the top and the other side has a larger number on the top, even if the same numbers are used as before, the two sides can’t be equal. + 1 + 2 ≠ + 8 + 4 © 2018 J. J. Crabtree | www.jonathancrabtree.com 128
  129. 129. Discussion on Brahmagupta’s Laws Q. Is the following equation correct? + 1 + 2 = − 4 − 8 © 2018 J. J. Crabtree | www.jonathancrabtree.com A. Yes. Therefore, just as +1 < +2 we must accept –4 < – 8, to be consistent with Brahmagupta’s laws and the laws of proportion. The British system has –4 > – 8, which is illogical given positives and negatives are equal and opposite. Brahmagupta’s laws of integer ordering are intuitive logical and correct! 129
  130. 130. Discussion on Brahmagupta’s Laws The same number of positives and negatives sum to zero as they are equal and opposite. Therefore today’s integer inequality laws, based on Greek foundations 1000 years older than India’s are out-of-date and sub-optimal. © 2018 J. J. Crabtree | www.jonathancrabtree.com 130
  131. 131. Which numbers are greater? or or or or © 2018 J. J. Crabtree | www.jonathancrabtree.com Discussion on Brahmagupta’s Laws © 2018 J. J. Crabtree | www.jonathancrabtree.com
  132. 132. Which numbers are greater? or or or or x x x xxx xxxx xx xxxx x xxx xx xx xxxxx x © 2018 J. J. Crabtree | www.jonathancrabtree.com Discussion on Brahmagupta’s Laws © 2018 J. J. Crabtree | www.jonathancrabtree.com
  133. 133. Which numbers are greater? or or or or xx xx x x xx xx x xxx x x xx xx x xxx xx 5+ < 7– 9+ > 4– 1+ < 3– 5+ ≹ 5– xx xx © 2018 J. J. Crabtree | www.jonathancrabtree.com Discussion on Brahmagupta’s Laws © 2018 J. J. Crabtree | www.jonathancrabtree.com
  134. 134. © 2018 J. J. Crabtree | www.jonathancrabtree.com 1+< 3+ 1–< 3– 2–= 2– 3–>1+ 3+>1– 1–< 3+ 1+< 3– 2+= 2+ 3–>1– 3+>1+ 2+ 2–o=2+ are equal and opposite to 2– 134
  135. 135. …to multiply a by integral b is to add a to itself b times Collins Dictionary of Mathematics © 2018 J. J. Crabtree | www.jonathancrabtree.com In book VII, Euclid defines multiplication as ‘when that which is multiplied is added to itself as many times as there are units in the other’ The Development of Multiplicative Reasoning in the Learning of Mathematics 1 × 1 = 2? © 2018 J. J. Crabtree | www.jonathancrabtree.com
  136. 136. 136 Paper @ www.bit.ly/LostLogicOfMath
  137. 137. Returning India’s zero reveals unseen patterns. a × +4 = 0 + a + a + a + a a × +3 = 0 + a + a + a a × +2 = 0 + a + a a × +1 = 0 + a a × 0 = 0 a × –1 = 0 – a a × –2 = 0 – a – a a × –3 = 0 – a – a – a a × –4 = 0 – a – a – a – a Integral multiplication involves either repeated addition or repeated subtraction of the multiplicand from zero, depending on the sign of the multiplier. 137© 2018 J. J. Crabtree | www.jonathancrabtree.com The Billingsley ‘virus’of 1570 (BV1570) spread widely! CLICK HERE FOR MORE
  138. 138. When explaining a3, Sir Isaac Newton used the Latin word ‘bis’, meaning ‘twice’, so a3 is a twice into itself. Yet because of Henry Billingsley’s mistranslation of Euclid’s definition of multiplication, upon translating from Newton’s Latin into English, the translator changed Newton’s explanation. To match Billingsley’s approach, (thought to be Euclid’s), Newton now reads ‘… the Number 3 in the Quantity a3bb, does not denote that bb is to be taken thrice, but that a is to be thrice multiplied by itself.’ So, today, we find, nonsense definitions of exponentiation that simply do not work! © 2018 J. J. Crabtree | www.jonathancrabtree.com© 2018 J. J. Crabtree | www.jonathancrabtree.com
  139. 139. © 2018 J. J. Crabtree | www.jonathancrabtree.com 23 ≠ 16 © 2018 J. J. Crabtree | www.jonathancrabtree.com Cube ‘the result of multiplying a number, quantity, or expression by itself three times’. Collins Dictionary of Mathematics If a3 is a into itself three times, we get the sequence a into a, into a, into a. If 23 is 2 into itself three times, we get the sequence 2 into 2, into 2, into 2, which is 16. Just as India’s 0 went missing from definitions of multiplication, we now know 1 went missing from definitions of exponentiation!
  140. 140. Returning the identity element one into exponentiation. a+4 = 1 × a × a × a × a a+3 = 1 × a × a × a a+2 = 1 × a × a a+1 = 1 × a a 0 = 1 a–1 = 1 ÷ a a–2 = 1 ÷ a ÷ a a–3 = 1 ÷ a ÷ a ÷ a a–4 = 1 ÷ a ÷ a ÷ a ÷ a Integral exponentiation involves either repeated multiplication or repeated division of the base from one, depending on the sign of the exponent. © 2018 J. J. Crabtree | www.jonathancrabtree.com 140 a+b = 1 into a, b times 2+3 = 1 into 2, 3 times, 1 × 2 × 2 × 2 a–b = 1 by a, b times 2–3 = 1 by 2, 3 times, 1 ÷ 2 ÷ 2 ÷ 2 There is much confusion about cubes and cube roots. CLICK HERE FOR MORE
  141. 141. South West Left Down Debts Loss Deaths Emigration Cold Decay Below Zero Less Than Enough Below Ground To the hour Deceleration Head Wind (knots) Under Par (golf) North East Right Up Assets Profit Births Immigration Hot Growth Above Zero More Than Enough Above Ground Past the hour Acceleration Tail Wind (knots) Over Par (golf) Counts or measures of negative units Counts or measures of positive units Simple Symmetries of Quantity ©2018J.J.Crabtree|www.jonathancrabtree.com 141
  142. 142. © 2018 J. J. Crabtree | www.jonathancrabtree.com It has been said… “God created the universe from nothing, from Shunya, from Zero” Planet Positron Planet Negatron Wherever opposing quantities or forces are equal you will find zero. 142
  143. 143. It’s as if ZERO was split, creating infinite real number lines from ZERO. © 2018 J. J. Crabtree | www.jonathancrabtree.com
  144. 144. • China had positives and negatives in the 2nd Century BCE, yet their first maths text with India’s zero, (with a symbol 0) came in 1247 CE in The Mathematical Treatise in Nine Sections, by Qin Jiushao. © 2018 J. J. Crabtree | www.jonathancrabtree.com Discussion on Brahmagupta’s Laws • China used positives and negatives in their mathematics for about1400 years without any concept of a number zero. • The Chinese did not define negatives as less than zero. Their positives and negatives were simply equal and opposite, consistent with science and the philosophy of Yin and Yang. 144
  145. 145. Is zero defined as ‘a number subtracted from itself’ (n – n)? No. Brahmagupta AL4 says zero is the sum of equal positive and negative quantities. Zero is defined by addition, (–n + +n) not subtraction, for use with Indian Laws of positives negatives and zero. © 2018 J. J. Crabtree | www.jonathancrabtree.com 145
  146. 146. © 2018 J. J. Crabtree | www.jonathancrabtree.com Are lessons on inequalities correct? e.g. (–4 > –7) No. SL3 says if a larger negative is subtracted from a smaller negative the sign of the difference is reversed and negative becomes positive. We prove – 4 > –7 false by contradiction. Assume –4 > –7 true. We subtract larger from smaller as –7 – –4 and the difference is –3. As the sign of the difference is NOT reversed we cannot assume –4 > –7 true. Thus –4 < –7 as seems obvious. (A debt of ₹400 is less than a debt of ₹700.) Brahmagupta’s SL3 also holds for greater positive from smaller positive as +4 – +7 = –3 and the sign is reversed as SL3 states. 146
  147. 147. Are negative numbers ‘less than zero’? No. Numbers represent count or measures of quantities, even abstract units. The big bang created matter and antimatter. Equal in magnitude, electrons have negative charge and positrons have positive charge. © 2018 J. J. Crabtree | www.jonathancrabtree.com When antimatter and matter meet, they annihilate each other, producing energy. It is said “God made the Integers”. Those integers are consistent with physics. NOTE asymmetry between matter versus antimatter in universe = Baryon asymmetry “If God made the Integers, when it comes to fixing the foundations of elementary mathematics, the Devil is in the detail!” Jonathan J. Crabtree 147
  148. 148. Are negative numbers ‘less than zero’? No. Before the Chinese adopted India’s zero, they used positives and negative for around 1400 years, with no need to move away from the idea negative and positive are simply equal and opposite. © 2018 J. J. Crabtree | www.jonathancrabtree.com The idea negative numbers are less than zero emerged in 1685 via Englishman John Wallis. If after marching East 5 yards, a man was forced back 8 yards, Wallis said in total he advanced 3 yards less then nothing . Chinese and Indians might have said the man retreated 3 yards West. (We don’t say negative East.) 148
  149. 149. John Wallis, p.265, A Treatise of Algebra 1685. © 2018 J. J. Crabtree | www.jonathancrabtree.com149 retreated 3 yards
  150. 150. In our universe, the least quantity you will ever have of anything is ZERO quantity. Numbers are only ever counts or measures of quantities, even if the quantity is a ‘unit’. Maths is about relationships between quantities and the quantities don’t have to be the same kind. Discussion on Brahmagupta’s Laws 150© 2018 J. J. Crabtree | www.jonathancrabtree.com
  151. 151. There’s also quantity of fuel and distance, which generate the maths called fuel consumption. © 2018 J. J. Crabtree | www.jonathancrabtree.com Discussion on Brahmagupta’s Laws For example, if people drive a car, maths looks at the relationships between quantities of distance and time to get average speed. 151
  152. 152. © 2018 J. J. Crabtree | www.jonathancrabtree.com 152
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  154. 154. © 2018 J. J. Crabtree | www.jonathancrabtree.com 154
  155. 155. © 2018 J. J. Crabtree | www.jonathancrabtree.com 155
  156. 156. © 2018 J. J. Crabtree | www.jonathancrabtree.com 156
  157. 157. © 2018 J. J. Crabtree | www.jonathancrabtree.com 157
  158. 158. © 2018 J. J. Crabtree | www.jonathancrabtree.com 158
  159. 159. ZERO (the equilibrium point where opposing quantities are equal) is the starting point of relations between quantities. © 2018 J. J. Crabtree | www.jonathancrabtree.com Numbers count or measure quantities and the study of … … relations between quantities is called mathematical science. 159
  160. 160. Illustration by AFX Animation Ryunosuke Satoro
  161. 161. © 2018 J. J. Crabtree | www.jonathancrabtree.com 161
  162. 162. For lectures, events or interviews with Jonathan J. Crabtree please contact AFX Animation Kolkata India. Podo the Puppy says...
  163. 163. © 2018 J. J. Crabtree | www.jonathancrabtree.com Please give your feedback online www.jonathancrabtree.com/feedback www.jonathancrabtree.com/feedback 163

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