434 14. EQUATIONS AND ALGORITHMSQuestions and problems
14.1. Problem 6 of Book 1 of the Arithmetica is to separate a given number into
two numbers such that a given fraction of the first exceeds a given fraction of the
other by a given number. In our terms this is a problem in two unknowns ÷ and
y, and there are four bits of data: the sum of the two numbers, which we denote
by a, the two proper fractions r and s, and the amount b by which rx exceeds sy.
Write down and solve the two equations that this problem involves. Under what
conditions will the solutions be positive rational numbers (assuming that a, b, r,
and s are positive rational numbers)? Compare your statement of this condition
with Diophantus' condition, stated in very complicated language: The last given
number must be less than that which arises when that fraction of the first number
is taken which exceeds the other fraction.
14.2. Carry out the solution of the bundles of wheat problem from the Jiu Zhang
Suanshu. Is it possible to solve this problem without the use of negative numbers?
14.3. Solve the equation for the diameter of a town considered by Li Rui. [Hint:
Since ÷ = — 3 is an obvious solution, this equation can actually be written as
x^3 + 3x^2 = 972.]
14.4. Solve the following legacy problem from al-Khwarizmi's Algebra: A woman
dies and leaves her daughter, her mother, and her husband, and bequeaths to some
person as much as the share of her mother and to another as much as one-ninth
of her entire capital. Find the share of each person. It was understood from legal
principles that the mother's share would be ^ and the husband's ã^.
14.5. Solve the problem of Abu Kamil in the text.
14.6. If you know some modern algebra, explain, by filling in the details of the fol-
lowing argument, why it is not surprising that Omar Khayyam's geometric solution
of the cubic cannot be turned into an algebraic procedure. Consider a cubic equa-
tion with rational coefficients but no rational roots,^20 such as x^3 + x^2 +x = 2. By
Omar Khayyam's method, this equation is replaced with the system y(z + 1) = 2,
z^2 = (y+1)(2 - y), one obvious solution of which is y = 2, æ — 0. The desired value
of ÷ is the y-coordinate of the other solution. The procedure for eliminating one
variable between the two quadratic equations representing the hyperbola and circle
is a rational one, involving only multiplication and addition. Since the coefficients
of the two equations are rational, the result of the elimination will be a polynomial
equation with rational coefficients. If the root is irrational, that polynomial will be
divisible by the minimal polynomial for the root over the rational numbers. How-
ever, a cubic polynomial with rational coefficients but no rational roots is itself the
minimal polynomial for all of its roots. Hence the elimination will only return the
original problem.
14.7. Why did al-Khwarizmi include a complete discussion of the solution of qua-
dratic equations in his treatise when he had no applications for them at all?
14.8. Contrast the modern Western solution of the Islamic legacy problem dis-
cussed in the text with the solution of al-Khwarizmi. Is one solution "fairer" than
the other? Can mathematics make any contribution to deciding what is fair?(^20) If the coefficients are rational, their denominators can be cleared. Then all rational roots will
be found among the finite set of fractions whose numerators divide the constant term and whose
denominators divide the leading coefficient. There is an obvious algorithm for finding these roots.