Discussion and Homework
Seventh Reading Assignment
Notes for the Readings
Note the following from the Elements for your reading (See Reading 7):
Definition VII.13: A composite number is that which is measured by some number.
Definition VII.14: Numbers composite to one another are those which are measured by some number as a common measure.
Questions for Discussion
Reading 45
- Diophantus knows the rules for multiplying a "minus" by a "minus" and a "minus" by a "plus", yet the biography of Diophantus states that he accepted only positive rational roots. Why might you need to know the rules rules for multiplying a "minus" by a "minus" and a "minus" by a "plus" to solve equations even if you only accept positive solutions?
- What do you think Diophantus would call x7, x8, x9and x10?
Reading 46
- In IV. 29,
does x22 + x+ 1/4 come out of nowhere? Why does Diophantus use this particular perfect square trinomial?
- In IV. 30,
does x2 - x + 1/4 come out of nowhere? Why does Diophantus use this particular perfect square trinomial?
Homework Problems
- Divide 25 and 36 into two squares using Diophantus' method.
- Use Diophantus' method to find four square numbers such that their sum added to the sum of their sides is 73.
- Use Diophantus' method to find four square numbers such that their sum minus the sum of their sides is 15.
- Prove that if 3 measures an odd number k times, then 3 measures the odd number "two places removed" k + 2 times.
- Prove that if 5 measures an odd number k times, then 3 measures the odd number "four places apart" k + 2 times.
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