Tomorrow: Questions and answers, on chapters 4–6,
and the current home works. (I
need to attend a WeBWorK demo at 9.)
Friday: Exam 2, closed books and notes, no calculators. I won’t be able
to answer
questions during the exam. On your way out pick up an answer sheet. Do check
your
answers.
Understand every term, phrase, and concept menti oned in the following notes.
Go over them today, and ask any questions tomorrow. Everything you see
here, except for the game of Nim, is covered in the textbook.
• Vocabulary of arithmetic operations. add, multiply, subtract, divide,
addition, multipli-
cation, division, subtraction, sum, product, difference , quotient, term, addend,
minuend,
subtrahend, divisor, dividend, plus, minus, times, divided by , +, −, /, ×,* , a
missing
operator means multiplication. Closure, commutative, associative, distributive
laws .
• Written algorithms for base 10, addition, subtraction, multiplication,
division. Trading or
ex changing some powers of 10 for others .
• The word “prime” is a noun meaning “prime number”, where “prime” is an
adjective. A
natural number is prime, or a prime number, if it has precisely two divisors , 1
and itself.
1 has only one divisor, itself, and is not prime. A natural number with more
than two
divisors is “composite”.
• Thus there are three types of natural numbers: 1, prime numbers, composite
numbers.
Note that the number 1 is neither prime nor composite. To see if p is prime
check divisibility
by prime factors up to 
• The prime factorization of a natural number is unique.
• Finding prime factorizations. Conventionally sort the prime factors by
increasing size, and
use exponents . For example,
• Finding all factors of a number. They come in pairs. Abundant, deficient, and
perfect
numbers.
• The greatest common factor, gcf, or GCF, of two numbers m and n, is the
largest natural
number that is a factor of both m and n. The definition can be extended in an
obvious
way to the gcf of more than two numbers . There is always a common factor, but it
may
be 1.
• Ways to find the gcf:
− List all factors, and find the gcf by inspection.
− Look at the prime factorizations of the two numbers. The gcf is the product of
all prime
powers that are common to both.
− Use the Euclidean Algorithm. Keep dividing with remainder, until the remainder
is zero.
The gcf is the last divisor.
• Least common multiple, lcm or LCM, of two numbers m and
n, the smallest number that
is a multiple of both m and n, with a similar definition for more than two
numbers. The
product, mn, is always a common multiple, but there may be smaller ones.
• Finding the lcm. We discussed three ways:
− Making lists of the first few multiples for each number, and finding the lcm
by inspection.
− Considering the prime factorization. The lcm must have as factors all prime
powers present
in either number.
− Exploit relationship below:
• Relationship between gcf and lcm:
Thus another way of finding the lcm is via first finding the gcf and then
computing
• Divisibility criteria for base 10: divisibility by 2, 3, 4, 5, 6, 8, 9, 19,
11.
• There are infinitely many prime numbers (Euclid, approx 300BC). Suppose there
are only
finitely many,
2, 3, 5, 7, 11, . . . , P
While we don’t know P, we assume the list is complete and contains all prime
numbers. Define
N = 2 × 3 × 5 × 7 × . . . × P + 1.
N is certainly greater than P. Thus it is not on the list. It is not divisible
by any of the
prime numbers on the list. Thus it must itself be prime, or it must have a prime
factor
that’s not on the list. In either case we are compelled to conclude that there
is some prime
number not on our list. But we assumed the list is complete. This is a
contradiction, our
assumption can’t be right, there can be no such list, there must be infinitely
many prime
numbers.
• The game of Nim: To win play such that there are even numbers of all powers of
2.
• Terminology of fractions : numerator, denominator, common denominator, least
common
denominator, simplest form, lowest terms, addition, multiplication, subtraction,
division
of fractions.
• The (least) common denominator of two fractions is a (the least) common
multiple of the
two denominators.
• Arithmetic Ope rations with fractions : addition and subtraction , find a common
denom-
inator and add or subtract the numerators ; multiply by multiplying numerators
and de-
nominators; divide by multiplying with the reciprocal; convert mixed numbers to
improper
fractions before operating on them.
• Ways of introducing the rules for adding , subtracting, multiplying, and
dividing fractions.
• graph paper rules.
