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# The Lost Logic of Elementary Mathematics

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New ideas on zero, negative numbers, multiplication and more!

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### The Lost Logic of Elementary Mathematics

1. 1. Thanks Jonathan and Podo. What’s on the agenda? The Lost Logic of Elementary Mathematics JonathanCrabtree La TrobeUniversity Melbourne Campus, Australia December2016 www.jonathancrabtree.com/LLEM research@jonathancrabtree.com
2. 2. Background | Zero | Negative Numbers | Multiplication | Exponentiation?
3. 3. Hello. I’m Jonathan. Podo is my super puppy. My story is... Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
5. 5. My name is Little Math. So what’s on the agenda? Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
6. 6. It’s elementary what’s on! The simple ideas of...
7. 7. Euclid, Liu Hui Brahmagupta, Newton and others, have…
8. 8. murdered! been…
9. 9. Mathematics most hated subject in school Science 149 10.9% Math 499 36.6% History 218 16.0% English 282 20.7% Phys.Ed 214 15.7% TOTAL 1362 100.0% Source: www.quibblo.com/quiz/1lE5Q15/Whats-your-most-hated-subject-in-school (Nov. 2016) Background | Zero | Negative Numbers | Multiplication
10. 10. Background | Zero | Negative Numbers | Multiplication
12. 12. Two multipliedby three is two added to itselfthree times and that equals six. But that’s crazy. Two added to itself three times is eight. Grade 2C 1968 Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
13. 13. Of course2 multiplied by 3 equals 2 added to itself3 times. It’s Euclid’smultiplicationdefinition from 300 BC! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
14. 14. …to multiply a by integral b is to add a to itself b times Collins Dictionary of Mathematics Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved 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
15. 15. How can 2 added to 1 three times be seven? 1 + 2 + 2 + 2 Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved and 2 added to itself or 2 three times be six? 2 + 2 + 2 + 2
16. 16. With 2 multiplied by 3, the three hops of 2 drawn on the number line start at zero. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
17. 17. So 2 multiplied by 3 equals two added to zero three times, not itself. 0 + 2 + 2 + 2 Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
19. 19. Yet Jonathan felt stupid. Later, he failed mathematics and stayed down a year at school. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
20. 20. Then, in 1983, on Friday March 18, he broke his back! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
21. 21. Let me walk and I promise to make maths simpler! A few years later... Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
23. 23. Zero is what you get when you subtract a number from itself. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
24. 24. 4 – 4 = 0 123 – 123 = 0 Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
25. 25. –3 – –3 = 0 – – – = 0 Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
26. 26. The person most responsible for modern mathematics is the Indian, Brahmagupta. In 628 CE he gave rules for adding and subtracting integers. In his rules for subtraction, Brahmagupta does not define zero as any number subtracted from itself! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
27. 27. Brahmagupta defined zero, not in his vyakalana rules for subtraction, but in his saṅkalana rules for addition, as: the sum of a positive number and negative number of equal magnitude, सम-ऐक्यम् खम् (Brāhma Sphuta-siddhānta, Chapter 18:30a). So zero was defined as: +n + –n Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
28. 28. So zero in mathematics is also what you get when you add any equal number of opposites together. Why is a –ve subtracted from zero equal to a +ve? +n + –n Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
29. 29. For example, start at zero, go right n steps, then left n steps, (or vice-versa) and you’re back where you started, at zero! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
30. 30. So, North/South, East/West, Forwards/Backwards and Left/Right are all opposites that cancel each other out. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
31. 31. Assets/Debts, Revenues/Expenses, Surpluses/Deficits and so on, can also cancel each other out to make zero. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
32. 32. If you use the correct unit, from any two opposing units, you don’t need to use the ‘adjective’ negative. (Nouns are simpler!) Background | Zero | Negative Numbers | Multiplication Get ready... Copyright © 2016 Jonathan Crabtree All Rights Reserved
33. 33. Zero was split and the real number line was the result! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
34. 34. The real number line is symmetrical about zero. The mirror-image numbers and magnitudes on either side of zero sum to ZERO. Next? Negative numbers... Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
35. 35. Collins Dictionary “A negative number, quantity, or measurement is less than zero.” Negative numbers are complex abstract ideas, involving numbers less than zero. So we avoid negatives until kids are age 12 to 13. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
36. 36. Oxford English Dictionary “A number or amount that is lower than zero; a negative quantity; spec. (with of) the quantity obtained from a given (positive) quantity by subtracting it from zero or multiplying it by −1.” Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
38. 38. What’s negative seven minus negative four? Let’s dig deep for the answer from ground level zero! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
39. 39. Ground Level Zero No Bumps and No Holes! Background | Zero | Negative Numbers | Multiplication
40. 40. From Ground Level Zero, Podo makes ... ... bumps and holes! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
50. 50. Yet the first Chinese math text with a zero was in 1247. The Mathematical Treatise in Nine Sections, by Qin Jiushao Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
51. 51. So negative numbers were not less than zero in China or India! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
57. 57. Like holes and bumps, or go away and come here! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
58. 58. All opposite things that cancel each other out can get counted with the same numbers! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
59. 59. DIRECTION FINANCIAL POSITION POPULATION TEMPERATURE SUFFICIENCY SEA LEVEL TIME North/South, East/West, Left/Right, Up/Down Assets/Debts, Profit/Loss Births/Deaths, Immigration/Emigration Hot/Cold, Above Zero/Below Zero More Than Enough/Less Than Enough Above/Below To the hour/Past the hour Q. How +ve result bad & –ve result good? Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved A. Your cancer test! So, in mathematics, ‘negative’ is not ‘bad’ and ‘positive’ is not ‘good’!
60. 60. I can imagine lots of negatives as holes and lots of positives as bumps. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
61. 61. Separate, or altogether, all my bumps and holes give me zero! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
63. 63. Yet three holes for –3 and seven bumps for +7 is lots of fun! Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
67. 67. So –3 + +7 = +4 3 holes (–3) and 7 bumps (+7 ) leads to 4 bumps (+4 ) Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
68. 68. You have 3 holes! ©2011JonathanCrabtree|AllRightsReserved Now, if you have 7 holes and 4 holes are taken away... Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
71. 71. Background | Zero | Negative Numbers | Multiplication “Brahmagupta (598 – 670 CE) was a Hindu mathematician and astronomer who lived in the first century.” “He used negative integers to represent debts and positive integers to represent assets.” “The product ... of two debts is one fortune.”
72. 72. Background | Zero | Negative Numbers | Multiplication “Brahmagupta... defined zero as the result of subtraction of a number from itself.” “He also gave the following rules for operations on what he called ‘fortunes’ (positive numbers) and ‘debts’ (negative numbers).” “The product... of two debts is one fortune.”
73. 73. Background | Zero | Negative Numbers | Multiplication Debt × Debt = Fortune! Really? Credit Card Debt × Mortgage Debt = Fortune! Really? If it looks like nonsense it probably is! Yet authors keep writing it and teachers keep teaching it! No Indian mathematician would have been that stupid. “The product or quotient of two negatives is one positive.” – Brahmagupta!
74. 74. Background | Zero | Negative Numbers | Multiplication Brahmagupta "The product of a positive and a negative (number) is negative; of two negatives is positive; positive multiplied by 'positive is positive.“ Mahāvīra "In the multiplication of two negative or two positive numbers the result is positive; but it is negative in the case of (the multiplication of) a positive and a negative number." Śrīpati "On multiplying two negative or two positive numbers (the product is) positive; in the multiplication of positive and negative (the result is) negative." Bhāskara II "The product of two positive or two negative (numbers) is positive; the product of positive and negative is negative.” The same rule is stated by Nārāyaṇa. SOURCE: History of Hindu Mathematics: A Source Book, Part II, Algebra, Bibhutibhusan Datta and Bidyāraṇya Avadesh Narayan Singh, pp. 22-23, Motilal Banarsidass, Lahore, 1938.
75. 75. John Wallis Paraphrase (Note: cipher = zero) But in case the multiplier is a negative number; suppose –2; then instead of adding the multiplicand to cipher 2 times, it will signify so many times to subtract the multiplicand from cipher. For as A × +2 implies twice adding A to cipher; 0 + A + A, to arrive at + 2A, so A × –2 implies twice subtracting A from cipher; 0 – A – A, to arrive at –2A. A Treatise of Algebra, both Historical and Practical, John Wallis, p. 74, Printed by John Playford, for Richard Davis, Bookseller, in the University of Oxford, 1685. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
76. 76. Colin Maclaurin Paraphrase Multiplication by a positive number implies a repeated addition of the multiplicand to cipher: But multiplication by a negative number implies a repeated subtraction of the multiplicand from cipher. And when positive a is to be multiplied by negative n, (+a × –n), the meaning is that +a it to be subtracted as many times from cipher as there are units in n: Therefore the product is negative, being –na. A treatise of algebra : in three parts, Colin Maclaurin, pp. 12-13, Printed for A. Millar, and J. Nourse, London, 1748. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
77. 77. In 263 CE, Liu Hui wrote a commentary on the ancient The Nine Chapters on the Mathematical Art (九章算术 Jiǔzhāng Suànshù circa 100 CE). Liu Hui said: I read the Nine Chapters as a boy, and studied it in full detail when I was older. I observed the division between the dual natures of Yin and Yang [the negative and positive aspects] which sum up the fundamentals of mathematics. The Nine Chapters on the Mathematical Art: Companion and Commentary, Shen Kangshen, John N. Crossley and Anthony W. C. Lun, Oxford University Press, 2000. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved Now we reveal what –n × –n REALLY means!
81. 81. +2 x –3 = –6 Positive Integer Multiplied by Minus Multiplier +2 x –3 Background | Zero | Negative Numbers | Multiplication
82. 82. Negative Integer Multiplied by Minus Multiplier – 2 x –3 Background | Zero | Negative Numbers | Multiplication
83. 83. Negative Integer Multiplied by Minus Multiplier – 2 x –3 Background | Zero | Negative Numbers | Multiplication
84. 84. Negative Integer Multiplied by Minus Multiplier – 2 x –3 Background | Zero | Negative Numbers | Multiplication
85. 85. – 2 x –3 = +6 Negative Integer Multiplied by Minus Multiplier – 2 x –3 Background | Zero | Negative Numbers | Multiplication Use plastic bottle caps for classrooms
86. 86. If you want to interpret the Sanskrit of Brahmagupta with a financial analogy for –n × –n , it is A debt repeatedly subtracted makes a fortune! If Bill Gates paid your \$1000 mortgage debt for the next 10 months, it would have the same net effect on your financial position as being given a \$1000 fortune for the next 10 months. Both debt repeatedly subtracted AND fortune repeatedly added result in a fortune, which is the reason why –n × –n = +n × +n Negative Integer Multiplied by Minus Multiplier – 2 x –3 Background | Zero | Negative Numbers | Multiplication
87. 87. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved Plastic bottle caps work great to model integer arithmetic. HINT! If you ever run out of caps to subtract, just ‘add zero’ eg. 5 white caps and 5 black caps and keep going!
88. 88. So… For integral multiplication, a × ±b, according to the sign of b, we can either add a to zero b times in succession or subtract a from zero b times in succession. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved
89. 89. Returning the identity element zero reveals patterns not often seen. 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 So, integral multiplication involves either repeated addition or repeated subtraction, from zero, depending on the sign of the multiplier. Background | Zero | Negative Numbers | Multiplication Copyright © 2016 Jonathan Crabtree All Rights Reserved