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PRACTICE PROBLEMS FOR MATH 112 MIDTERM

(1) Prove that the only regular polyhedron with square faces is the cube.

(2) Give the continued fraction expansion for 77/21 and draw the associated

anthyphairesis picture.

(3) What irrational number is represented by the continued fraction [4; 4, 4, 4, 4, ...]?

Explain why a repeating continued fraction expression always represents an

irrational.

(4) Use Egyptian algorithms to do 14 × 15, 121/11, 4/5.

(5) Show the steps to convert BABA16 to base 2.

(6) Explain the difference between accuracy and efficiency for a rational approximation to an irrational number.

(7) Define the following: algorithm, greedy algorithm, lexicographic order.

(8) How does Brahmagupta’s method for finding solutions to Pell’s equation

work? Is it an algorithm?

(9) What is the difference between the chord of an angle and its sine?

(10) Explain (generally) how al-Khwarizmi solves a problem like ax2 + bx = c.

(11) If p(n) is the partition function, which is bigger, p(10) or 2 · p(5)?

(12) Suppose F (x) is the generating function for a sequence (an ). Suppose

G(x) = x5 ·F (x), and suppose that is the generating function for a sequence

(bn ). What is the relationship between an and bn ?

(13) Given a unit length, explain how to construct

ion.

√

1+ 5

2

in the Euclidean fash-

(14) How many solutions are there in the quaternions to x2 = −1?

(15) What does it mean for one infinite set to be bigger than another?

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