Brahmasphutasiddhanta by Brahmagupta
HS / Mathematics
Mathematics, Pattern, Quantity, Zero
Create on one wall of your classroom a “Word Wall” where all of the math terms your
students would need to write successfully about your curriculum can be displayed with
definitions (use an interactive white board display if this makes it easier to add terms
and revise definitions). Have students search their textbooks (if you use a textbook) for
terms (at least one each) that are specific to math that should be added to the wall.
Spend a class period working in small groups to add the definitions to the list. Consider
having students create individual math glossaries that they are responsible for
maintaining.
Distribute the text and have students read the opening paragraph (an introduction) and
skim the laws that follow without reading them carefully. Discuss what they think this
text will consist of—given the introduction and the appearance of the rest of the text.
Have students number the laws (1-11) on their copies of the text.
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Share as appropriate: Bramagupta (598-c. 670) was an Indian mathematician and
astronomer who wrote two early and important works on Mathematics and Astronomy.
Brahmagupta was the first to give rules for computing with zero. The texts composed by
Brahmagupta were composed in Sanskrit poetry, as was common in Indian
mathematics. Discuss what it might mean to translate from Sanskrit into English and
poetry into prose.
The Brāhmasphuṭasiddhānta (Correctly translated the “Whole Doctrine of Brahma") is
the main work of Brahmagupta, written c. 628. The text is notable for its mathematical
content, as it contains ideas including the role of zero, rules for manipulating both
negative and positive numbers, a method for computing square roots, methods of
solving linear and quadratic equations, and rules for summing series. Note that during
the seminar, we will refer to the text as the “Doctrine of Brahma.”
Have student work in pairs to highlight all of the important math vocabulary contained in
the 11 laws. Have them share their lists of terms (law by law) and add any that don’t
already exist to the math Word Wall that was created in the Launch Activity. Discuss
how to signify Bramagupta’s terms fortunes and debts since those terms are no longer
in common use. Divide the class into teams to write definitions of the new terms and
add them to the Word Wall.
Divide the class into 11 groups and assign one law to each group. Have them design
one or two math “sentences” to illustrate each law. (For example, for Law # 1: -9 – 0 = 9) Have them write the illustrations on the board. Discuss any law or illustration that is
unclear.
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 Which of these eleven laws is most important in terms of understanding
the other ten? (vote by show of hands)
 How does the law you chose illuminate the other laws? (spontaneous
discussion)
 Based on the first seven laws, how would you define the concept of zero?
 Could these laws exist without the concept of zero? Why or why not?
 Why would the product of two debts be a fortune (Law #9)?
 How are Laws #10 and #11 alike? How and why are they different?
 These laws were first written down in India over 1,300 years ago. How and
why do you think Bramagupta “discovered” them?
 What is the most surprising thing you learned about the study of
mathematics from this text and our discussion of it?
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Have students turn over their copies of the text and brainstorm in writing the most
important ideas about mathematics that they said, heard, and thought during the
seminar.
After reading an excerpt from the Brahmasphutasiddhanta on the Laws governing zero
and the negative numbers, write an Annotation to one of the eleven Laws in which you
relate how and why it holds true. Support your discussion with evidence from the text.
(Informational or Explanatory/Procedural-Sequential)
(LDC Task#: 16 )
Explain that the whole class will work together to write an annotated version of
Bramagupta’s 11 Laws for the modern high school student. Divide the class up into 11
writing teams and assign one Law to each team at random. Assign the teams the task
of translating the Law into modern mathematical terms and then explaining how and
why it works. Have the teams meet to brainstorm the content of their answers.
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Discuss with the class a common template for their Annotations: beginning with a
restating of the Law using modern mathematical terms; followed by a simple English
explanation of why the law works, and several examples (see the Analytical Reading
stage above).
Assign each team to work together to produce a first draft of their Annotation following
the template example above. Note the value of the Math Word Wall created during the
Launch Activity and the Vocabulary stages. Encourage students to make use of the
terms defined there.
Have two teams whose Laws are related meet together with first one team and then the
other reading the first draft of their Annotation aloud. Listening team members say back
one point heard clearly and ask one question for clarification. Switch roles. Give time for
full revisions resulting in a second draft.
Once the second draft is complete, have the paired teams work together again and this
time take turns reading copies of each other’s second drafts slowly and silently, marking
any spelling or grammar errors they find. (Have dictionaries and grammar handbooks
available for reference.) Take this opportunity to again stress the correct use of the
math vocabulary recorded on the Word Wall. Add new words to the Wall if needed. Give
time for full revisions resulting in a third and final draft.
Publish the resulting Annotated Version of eleven Laws from the
Brahmasphutasiddhanta on the class website and invite comment from a wide variety of
international scholars and students.
Terry Roberts
National Paideia Center
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Brahmasphutasiddhanta (early Eastern text on zero and negative numbers)
In India, negative numbers did not appear until about 620 CE in the work of
Brahmagupta (598 - 670) who used the ideas of 'fortunes' and 'debts' for positive and
negative. By this time a system based on place-value was established in India, with
zero being used in the Indian number sytem. Brahmagupta used a special sign for
negatives and stated the rules for dealing with positive and negative quantities as
follows:
A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multiplied by zero is zero.
The product or quotient of two fortunes is one fortune
.
The product or quotient of two debts is one fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.
(Source - http://nrich.maths.org/5961 )
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