MATH601 Spring 2008
Exam 1 Solutions
A. Write nineteen and one hundred in senary, then multiply the results
using the Russian Peasant method.
Nineteen is three times six, plus one: CA. One hundred is two times
thirty six, leaving twenty eight. Twenty eigh
MATH601 Spring 2008
Handout 1: Senary Arithmetic
Unit 1: Natural Numbers
We will do all arithmetic in this unit in base six, where our digits are Z (zero), A (one), B (two), C
(three), D (four), and E (ve). Where necessary, write numbers in word notation;
Math 60 Section 7.8 Solving Applications with Rational Eqns.
Ex1) If $1 = 0.75013 euros, then find the
cost of a camera in U.S. dollars if the
camera was 159 euros.
Try) If an 1800 square foot house is sold
for $150,000 in a particular
neighborhood, then
Math 60 Review
1) Factor out the greatest common factor.
a) 5 z 25 xz 4
b) 3 x( y 5) ( y 5)
2) Factor.
a) x 2 3 x 2
b) x 2 6 x 8
c) x 2 15 x 16
b) 3 x 2 3 x 18
c) x 2 y 2 6 xy 2 8 y 2
3) Factor.
a) 2 x 2 4 x 48
4) Factor.
a) 4 y 2 12 y 9
b) 6 x 2 5 x 6
c)
Math 60 Section 8.1 Graphs
Quadrants
Ex1) Plot the points (-4,3) , (-5, -3) , (0, 4) , (4, -5), (2.5, 0), and state their corresponding quadrants.
Solutions of Equations
Ex 2) Determine whether the pairs (-1,-4) and (2,5) are solutions of the equation y =
Math 60 Section 7.1 Simplifying Rational Expressions
Definition: A rational expression is an expression that consist of a polynomial divided by
a nonzero polynomial.
Ex1) Evaluate the expression
2 a 4b
for a 1 , b 2 , c 3
3c 1
Ex3) Find the values where t
Math 60 Section 7.7 Rational Equations
Ex1) Solve:
a)
3x x 1
4 2 2
c)
5 1
4,
2 z
z0
b)
8
1 11
, p0
p 4p 8
d)
4
1
7 1
, b0
3b 6b 2b 3
Ex2) Solve:
a)
x2
9
x3 x3
b)
x2 4
5
x 1 x 1
b)
3
5
x9
2 x 3 2x 6
Ex3) Solve:
a)
4
7
x 1
3 x 4 3x 12
Ex4) Solve:
a)
5
Math 60 Appendix C2 Systems of Equations in Three Variables
Identifying Solutions
Ex 1)
Ex 2) Solve the system of Equations:
x yz 4
x 2y z 1
2 x y 2 z 1
Practice) Solve the system of Equations:
abc 5
a) 2a 3b c 2
2a 3b 2c 4
x y z 180
b) x z 70
2y z 0
Type
Math 60 Section 7.2 Rational Expression (Multiplying and Dividing)
Example 1) Multiply and simplify:
a)
c)
x 3 5x 35
2
5
x 9
p 2 9 25 p
5
2p 6
b)
d)
7 y 28
8
2
4y
y 2y 8
2x 8
7x 7
x 2 3x 4 6 x 30
Example 2) Multiply and simplify.
m2 n2
10m 5n
a)
10m 2 1
Math 60 Section 7.6 Complex Rational Expressions
Definition: A complex rational expression is a rational expression that contains rational
expressions within its numerator and or its denominator.
Example 1) Simplify.
a)
1 1
5 x
x5
2
x 5
2 x
b)
2x 1
3
Exam
Math 60 Appendix C1 Systems of Equations in Two Variables
Ex 1) Solve by graphing:
a)
2x y 1
x y 2
Ex 2)
b)
x y 1
x y 3
Solving Systems Graphically
When we graph a system of two linear equations in two variables, one of the following three outcomes
will o
MATH601 Spring 2008
Handout 6
Unit 3: Hyperreal Numbers
In this unit, we encounter the hyperreals, an extension of the reals. A calculus text using hyperreals can be
found at: http:/www.math.wisc.edu/~keisler/calc.html. Please download the rst two chapter
MATH601 Spring 2008 Handout 4: Subtraction Unit 1: Natural Numbers
Unless otherwise specied, everything on this handout is using exactly three hands. For example, AB really means ZAB (with the initial Z suppressed). The complement of a number is found by
MATH601 Spring 2008
Handout 3: Divisibility Tests
Unit 1: Natural Numbers
We recall divisibility tests in decimal arithmetic, and try to nd analogous tests in senary.
The last digit tests:
The last digit is a multiple of cfw_two, ve, ten i1 the number is
MATH601 Spring 2008
Exam 2 Solutions
1. Carefully and precisely state the denition of real numbers that we have been using.
x = L|R is a real number precisely when L, R are each subsets of the rationals Q,
such that the following three axioms are satised:
MATH601 Spring 2008
Exam 4 Solutions
1. Let R, S, T be sets. Suppose that |S| < |T | (this means that |S| |T | but |S| = |T |).
Suppose further that |R| = |S|. Prove that |R| < |T |. You may use only the denitions
of and =, and the CSB theorem, but not an
MATH601 Spring 2008 Exam 3 Solutions
1. For H any positive innite hyperreal, compute st it does not exist. We focus on the argument:
(H 2 +H+1)H 2 H 2 +H+1+H
H 2 + H + 1 H , or show that
2 H +H+1+H H 2 +H+1+H
1/H
=
H+1 H 2 +H+1+H 1/H
H2 + H + 1 H = =
1
MATH601 Spring 2008
Exam 5 Solutions
1. Simplify ( 2 + 3 + 2 4 + 2 2 + 7) ( 5 + 2 + 8 + 5) and
place the result into Cantor Normal Form.
We rst note that 2 + 3 = 3, and that 2 4 + 2 2 = 2 6, so
the problem reduces to ( 3 + 2 6 + 7) ( 5 + 2 + 8 + 5).
Becau
MATH601 Spring 2008
Handout 8: Dodgeball
Unit 4: Cardinals
At left is the board to play
Dodgeball. It is a two-player
game, between the Pitcher and
the Dodger. They take turns
writing in the boxes at the end
of the game each box will have
either an X or a
MATH601 Spring 2008 Handout 2: Multiplication Unit 1: Natural Numbers
Recall your techniques for doubling and halving, from the exercises of the previous handout. A good method for halving, if you didnt nd one, is as follows: Go from left to right, hand b
MATH601 Spring 2008
Handout 5
Unit 2: Real Numbers
In this unit, we construct real numbers out of fractions. We take as our starting point the set of fractions
Q = cfw_a/b : a Z, b N, or more precisely the set of equivalence classes, where [a/b] = [a /b ]
MATH601 Spring 2008
Handout 7
Unit 4: Cardinals
A cardinal is a number that measures the size of a set. For set S, we use |S| to denote the cardinal that
measures the size of S. Important comment: Our number sense breaks down with innite cardinals. Beware
Math 60 Section 7.3-7.5 Rational Expression (Adding and Subtracting)
Adding and Subtracting with Like Denominators:
Example 1) Add or subtract:
a)
3 x 4
x
x
b)
4x 5 x 2
x3 x3
c)
a 5b a 7b
ab
ab
d)
1
5
4a 2 b 4a 2b
Example 2) Add or subtract:
4 x 2 5 xy 2