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Links between al-Khwārizmī’s and Brahmagupta’s arithmetic
Negative and Positive Quantities on a Brahmaguptan Plane
Laws of Sign for Multiplication and Division
Multiply like Descartes and Newton
A New Model of Multiplication
Squaring a Circle with Rope
The Lost Logic of Maths
Representations of Negative and Positive Quantities on a ‘Brahmaguptan Plane’ for India’s Primary Classes
Abstract: Children’s fear of maths is often associated with the introduction of negative numbers. By way of example, asking adult non-mathematicians for the answer to ‘negative seven minus negative four’ usually results in a wrong answer. However, asking the same question to 12-year-old children in the form What does seven negatives minus four negatives equal? usually results in the right answer. Why is the difference in comprehension so dramatic? In the problematic expression negative seven minus negative four the syntactic structure is adjective adjective verb adjective adjective. With the absence of a noun, the meaning of such maths for most children is lost. Instead, children (and adults) cling to rules memorised without meaning, such as ‘two minuses make a plus’. So, what can we do? The answer is simple. We return to 7th Century writings of India, where we discover the astronomer Brahmagupta documented ‘adjective-noun’ style laws of sign, not for abstract numbers, but for positive quantities, negative quantities and zero. With this insight, we depict simple object-oriented representations of integer arithmetic involving positive and negative quantities. Such a quantitative pedagogy is concrete in nature, yet isomorphic to ‘signed numbers.’ Therefore, a solid intuitive foundation of integer arithmetic can be laid. Upon this foundation more abstract structures can be built. The integer teaching model that emerges is called the ‘Brahmaguptan Plane’.
Abstract: Many teachers find it hard to explain the laws of sign of integers. Mathematics is a science of patterns, so ⁺2 × ⁺3 = ⁺6 suggests ⁻2 × ⁻3 = ⁻6, yet this is not so. Multiplicative pedagogies such as repeated addition, equal groups, arrays, and area models fail on the Integers. One cannot add a number of objects a negative number of times, nor have a negative number of groups, nor a negative number of rows, nor an area less than zero. Division has a repeated subtraction model and an equal shares model, yet without citing sign laws, teachers cannot explain how to solve ⁺6 ÷ ⁻3. There are no negative threes in positive six and you cannot divide six into negative three groups. So, I introduce new games in a play featuring a multi-cultural group of famous mathematicians as children. Informed by the original writings of great thinkers, I found and fixed enduring flaws in the teaching of elementary mathematics.
Abstract: For 2500 years, mathematicians had fun trying to square the circle using only a compass and straightedge. Yet in 1882 this classic Greek problem was proven to be impossible. So, what do we do? ANSWER We solve the problem as if we lived in the time of ancient Bharat and follow the rules of rope (like Sulba Sutras).
Abstract: For the past 446 years, many people in the West have been misled by the ‘MIRA’ multiplication myth, ‘Multiplication Is Repeated Addition’. It is thought the Greek mathematician, Euclid of Alexandria, defined multiplication as repeated addition around 300 BCE, yet he didn’t. Words such as ‘add’ and ‘added’ do not appear in any of Euclid’s Book VII propositions reliant on a definition of multiplication. The Greek word συντεθῇ in Euclid’s definition was incorrectly translated in 1570 by Henry Billingsley as ‘added to itself’ instead of ‘placed together’. Thus, illogical definitions of multiplication appear, such as in Collins Dictionary of Mathematics, ‘To multiply a by integral b is to add a to itself b times.’
Abstract: Euclid’s multiplication definition from Elements, (c. 300 BCE), continues to shape mathematics education today. Yet, upon translation into English in 1570 a ‘bug’ was created that slowly evolved into a ‘virus’. Input two numbers into Euclid’s step-by-step definition and it outputs an error. Our multiplication definition, thought to be Euclid’s, is in fact that of London haberdasher, Henry Billingsley who in effect kidnapped kaizen, the process of continuous improvement. With our centuries-old multiplication definition revealed to be false, further curricular and pedagogical research will be required. In accordance with the Scientific Method, the Elements of western mathematics education must now be rebuilt upon firmer foundations.
Abstract: If “the purpose of life is to contribute in some way to making things better”, how might we make mathematics better?1 Teachers often explain multiplication and division with repeated addition and subtraction. Yet such approaches do not extend beyond the positive Integers. By contrast, the ideas of René Descartes and Isaac Newton on multiplication and division can be extended from the Naturals to the Reals. So, I reveal how, if they were alive today, they might explain multiplication and division visually in ways seldom seen in western mathematics curricula.
Links between al-Khwārizmī’s method of multiplying and dividing fractions and Brahmagupta’s arithmetic
Abstract: Muḥammad ibn Mūsā al-Khwārizmī’s treatise on arithmetic is the oldest known Arabic text on the Indian decimal place value system. Written around c. 825 CE after his treatise on algebra, it survives only in Latin translations. Known as Dixit Algorizmi or DA (Al-Khwarizmi said), the Latin manuscript was incomplete. It abruptly ends partway through the 12th of 19 chapters during an explanation of 3 1/2 × 8 3/11 . Presumably long lost, there was no English translation of either the multiplication or division of mixed fractions. Yet, a complete manuscript of Dixit Algorizmi was subsequently found in the museum of the Hispanic Society of America in New York. Translated and published in German in 1997, al-Khwārizmī’s method of multiplying and dividing mixed fractions is now presented in English.