MONT 108N – Mathematics Through Time Problem Set 1 due: in class on September 20, 2010

I. “Refresher” on arithmetic with fractions (rational numbers) and radicals. Calculate: A) \frac{3}{7} + \frac{4}{9} B) \frac{4}{19} × \frac{5}{12} (and simplify) C) \frac{5}{6} ÷ \frac{3}{4} (and simplify)

D) \frac{1}{4} + \frac{\frac{1}{4}}{1− \frac{1}{3}} (and simplify)

E) Simplify: \begin{matrix}√ & \overline{200} \\ √ & \overline{8} \end{matrix}. F) Write the number \frac{1}{\sqrt{3}−1} in the form a+b\sqrt{3} with a, b rational numbers by multiplying

by the same number in the numerator and denominator:

\frac{1}{\sqrt{3} − 1} × \frac{\sqrt{3} + 1}{\sqrt{3} + 1}

and simplifying.

II. For us (and all mathematicians), an integer is even if it equals 2 time another integer – so the even numbers are

…, −8 = 2 × (−4), −6 = 2 × (−3), −4, −2, 0 = 2 × 0, 2, 4, 6, 8, ….

All the other integers are odd. A) Show that if an integer n is even, then n^{2} is also even. B) Show that if an integer n is odd, then n^{2} is also odd. (Hint: The odd numbers all

have 1’s digit equal to 1,3,5,7, or 9.) C) Show that if n is an integer and n^{2} is even, then n is also even. (This is the “converse”

of the statement in part A. You may use part B here if it helps!)

III. A) Express the following base 10 numbers using base 2 and base 8: 12, 46, 100. B) Express the following base 2 numbers in base 10: (10011)_{2}, (111111)_{2}. C) Express the following base 8 numbers in base 10: (4006)_{8}, (123)_{8}. D) Suppose we wanted to use a base 60 positional number system. How many base 60

digits would we need? How would you write the base 60 digits as base 10 numbers? E) Short answer: Why is having the 0 digit (almost) a necessity for a positional number

system with base b (b = any positive integer)? Explain.

IV. Solve by factoring (you may need to experiment a bit for the later ones!) A) x^{2} − 5x + 6 = 0 B) x^{2} + 4x + 4 = 0 C) x^{2} + 12x − 108 = 0 D) 3x^{2} + 28x + 32 = 0 E) t^{6} − 6t^{3} + 9 = 0

V. Solve using the quadratic formula A) x^{2} + x − 1 = 0 B) x^{2} + 8x = 3 C) 2x^{2} + 7x + 3 = 0