Rational Numbers
Verbal definition any number that can be
expressed as a fraction with an
integer numerator, and a nonzero integer
denominator
Symbolic definition Q p/q ? p ? J, q ? J, q
? 0 Read the set of all p/q such
that p is in J, q is in J, and q is not
equal to 0.
Irrational Numbers
Verbal definition non-terminating,
non-repeating decimal. It cannot be written
as a fraction with an integer numerator,
denominator.
Symbolic definition H x ? x ? Q Read the
set of all x such that x is not a
member of Q.
Real Numbers
Verbal definition any numbers that are
rational or irrational
Symbolic definition R x ? x ? Q U H
Read the set of all x such that x is a
member of Q union H

4
IIVenn Diagrams
Venn diagrams or set diagrams are diagrams that
show all hypothetically possible logical
relations between a finite collection of sets
(groups of things). Venn diagrams were invented
around 1880 by John Venn. They are used in many
fields, including set theory, probability, logic,
statistics, and computer science
A Venn diagram is a diagram constructed with a
collection of simple closed curves drawn in the
plane. The principle of these diagrams is that
classes be represented by regions in such
relation to one another that all the possible
logical relations of these classes can be
indicated in the same diagram. That is, the
diagram initially leaves room for any possible
relation of the classes, and the actual or given
relation, can then be specified by indicating
that some particular region is null or is not
null.
Source Wikipedia
5
II Venn Diagram of Real Number System
R
Q
H
J
2/3
W
-7
Non-terminating, non-repeating decimals like
p and e
N
-23
7
0
1
3¼
-5/6
6
IIISubset Definition Notation of Subsets

The definition of a subset could be stated A is
a subset of B
(Denoted A ? B) if only if (iff) all elements
(numbers) of A are found in B. (? is analogous
to )
Ex1) J ? Q True
Ex2) J ? N False
Ex3) N ? J True

7
IVPrimes and Composites
A Prime Number is a
natural number greater than 1 that has only
itself and 1 as factors A Composite number is a
natural number greater than 1 that is not
prime Factors ? Numbers that are being multiplied
together (See section VI for two methods of prime
factorization)
The first hundred natural numbers and their
designation
8
VTests For Divisibility

There are three easy ways to test the
divisibility of numbers
To test if a number is divisible by 2, look at
the last digit. If its even, then the number is
divisible by 2.
Test for divisibility by 3 by taking the sum of
the digits. If the sum is divisible by 3, then
the original number is also divisible by 3.
Ex 1.
Test for divisibility by 5, by looking at the
last digit. If its a 5 or a 0, then the number
is divisible by 5.

9
VIWork with Fractions

Mixed numbers are fractions

Ex
When reducing fractions, use prime factorization.
You should use divisibility tests. Prime
factorize then cancel like factors.
Multiplying fractions is a lot easier if you use
the prime factorization method.
First you prime factorize the tops and
bottoms.
Next you will cancel like factors
Last but not least, you will multiply straight
across
Ex of PF method for multiplication
d) There are two methods of prime factorization
in arithmetic
The Tree Method (Use for numbers lt 100)
Ex

Primes in yellow
10
VI Work with Fractions, continued

The Underneath Division Method
(exponential form)
for prime factorization
e) Division of fractions (two steps)
1. Invert the back or bottom fraction depending
on how problem is written.
2. Next you must multiply using the
multiplication rules. (Subheading C)
Ex
f) Addition and Subtraction of Fractions
To add or subtract, you must
1. get the least common denominator (LCD).
In order to do this, you must first prime
factorize (PF) the denominators only.
2. the next step is to Get Jealous. This
means you take any prime factors from one
denominator that arent in the other, and you
multiply the entire fraction by the missing
primes.