Solutions to First Exam, Math 170, Section
002 Spring 2012
Multiple choice questions.
Question 1.
You have 11 pairs of socks, 4 black, 5 white, and 2 blue, but they are not
paired up. Instead, they are all mixed up in a drawer. Its early in the morning, a
MATH 170
RACTICE QUIZ 5 SOLUTIONS
SPRING 2010
1. Draw the two lines 2 x + 3 y = 12 and 3 x + 2 y = 8 and find their point of intersection.
5
4
3
2x + 3y = 12
2
1
3x + 2y = 8
2
4
6
The point of intersection is (0, 4). Here is the derivation:
2 x + 3 y = 12
C ONGRUENCES
1. Congruence numbers
(1.1) D EFINITION Let n Z be an integer.
(a) Two integers a, b are said to be congruent modulo n if (and only if) a b is a multiple of n. The
standard notation for congruence modulo n is (mod n). In other words
ab
(mod n
Geometry
1. Axioms in the Elements
Euclids Elements, is by far the most successful textbook every written, and second only to the Bible
in the number of editions published. It was written around 300 BCE and contains the best results from
600 BCE to 300 BC
Limits and innity
1. What is the square root of 2
(1.1) The Greeks proved that the square root of 2 is not a rational number. They also showed that the
(positive) square root of 2 is a number which can be constructed using a ruler and a compass. But just
Innity
1. Cardinality
(1.1) One form of the idea about innity is easy enough: a set is innite if it is not nite. It is also
easy to formulate rigorously what one means by nited:
A set S is nite if and only if there exists a one-to-one and onto map from S
Solutions to the Second Midterm Exam,
Math 170, Section 002 Spring 2012
Multiple choice questions.
Question 1.
Suppose we have a rectangle with one side of length 5 and a diagonal of
length 13. What is the length of the segment joining the midpoint of the
Math 170, Section 002
Spring
2012
Practice Exam 1 with Solutions
Contents
1 Problems
2
2 Solution key
8
3 Solutions
9
1
1
Problems
Question 1: After a calculus test the instructor announces that no student
got more than four problems wrong. If thirty ve p
Math 170, Section 002
Spring
2012
Practice Exam 2 with Solutions
Contents
1 Problems
2
2 Solution key
10
3 Solutions
11
1
1
Problems
Question 1: A right triangle has hypothenuse of length 25 in and an altitude
to the hypothenuse of length 12 in. What is t
Math 170, Section 002
Spring
2012
Practice Final Exam with Solutions
Contents
1 Problems
2
2 Solution key
11
3 Solutions
12
1
1
Problems
Question 1: Consider all natural numbers that can be written only with
the digits 1 and 2 and such that the sum of the
Math 170, Section 002
Spring
2012
Practice Exam 1 with Solutions
Contents
1 Problems
2
2 Solution key
8
3 Solutions
9
1
1
Problems
Question 1: After a calculus test the instructor announces that no student
got more than four problems wrong. If thirty ve p
Solutions to the Second Midterm Exam,
Math 170, Section 002 Spring 2012
Multiple choice questions.
Question 1.
Suppose we have a rectangle with one side of length 5 and a diagonal of
length 13. What is the length of the segment joining the midpoint of the
Solutions to First Exam, Math 170, Section
002 Spring 2012
Multiple choice questions.
Question 1.
You have 11 pairs of socks, 4 black, 5 white, and 2 blue, but they are not
paired up. Instead, they are all mixed up in a drawer. Its early in the morning, a
Math 170, Section 002
Spring
2012
Practice Final Exam with Solutions
Contents
1 Problems
2
2 Solution key
11
3 Solutions
12
1
1
Problems
Question 1: Consider all natural numbers that can be written only with
the digits 1 and 2 and such that the sum of the
Math 170, Section 002
Spring
2012
Practice Exam 2 with Solutions
Contents
1 Problems
2
2 Solution key
10
3 Solutions
11
1
1
Problems
Question 1: A right triangle has hypothenuse of length 25 in and an altitude
to the hypothenuse of length 12 in. What is t
W HOLE NUMBERS
1. Axioms for positive integers
We give a list of axioms for positive integers, following the tradition. The number 0 and negative
integers will be introduced later.
The properties proved here are of course all familiar to everyone. The poi
P ROOFS
1. Proofs: why and what
The credit of inventing the concept of proofs must go to the Greeks. This transformative idea changed
the very essence of mathematics.
A proof in mathematics is a logically correct argument which establishes a (permissible)
S TATEMENTS AND LOGIC IN MATHEMATICS
1. The mathematical language
Mathematicians write and speak in a corrupted form of English; we will call it mathematical English.
Motivations, remarks and comments in mathematical writings tend to be closer to natural
MATH 170, Section 002, Syllabus, Spring 2014
COURSE GOALS:
Experience mathematics as a methodology for eective and critical thinking.
Understand the science and art of applied thinking. Develop techniques for exploration, analysis, and discovery.
Emplo
MATH 170-002: IDEAS IN MATHEMATICS
UNIVERSITY OF PENNSYLVANIA
SPRING 2014
Instructor: Dr. Camelia Pop
Lectures: Tuesday-Thursday 1:30-3:00 pm, David Rittenhouse Laboratory A1
Oce: 4N53 David Rittenhouse Laboratory
Oce Hours: Tuesday 5:30-6:30 pm and by ap
MATH 170, Section 002, Review Midterm 2
The second midterm exam will take place on Tuesday 3/25 during our regular lecture.
The exam is cumulative. In addition to the topics covered by the rst midterm exam,
we will also have sections 2.4, 2.5 and 2.6 of t
MATH 170, Section 002, Review Midterm 1
The rst midterm exam will cover the topics below. To practice problems you can use as
a guide the problems listed below. They are all taken from the 3rd edition of our textbook.
You can nd an electronic version of t
MATH 170, Section 002, Review Final Exam
The nal exam will take place on Tuesday 5/6 from 9-11 am in Towne 100. The exam
is cumulative and puts the same emphasis on all topics covered during the semester.
The topics covered and practice problems are liste