Lim, Steven
ID#102238598
MAE301-Spring 08
1st Lecture Note – January 28th 2008
TOPIC OUTLINE:
I.
NUMBER SYSTEMS
1. Natural Numbers
2. Whole Numbers
3. Integers
4. Rational Numbers
5. Irrational Numbers
6. Real Numbers
II.
PRIME NUMBERS
III.
GROUPS
IV.
RING
Topic: NUMBER SYSTEMS
Sets of numbers that are used in mathematics.
1) The Natural Numbers are the positive whole numbers. (
The set of Natural Numbers (
not closed under subtraction.
EX) 5 – 5 = 0 , 0 •
) = {1, 2, 3, …}
) is closed under addition and multiplication, but
Invention of Zero
 Hindus from India first invented a zero around 200 A.D.
 The Hindus were the first to recognize a mathematical representation
of concept of no quantity.
 With the availability of zero, mathematicians were finally able to
develop our present method of writing whole numbers.
 The invention of zero helped the development of ‘decimal system’
and ‘positional notation’.
2) The Whole Numbers are the set of non-negative integers. = { 0, 1, 2, 3, …}
3) The Integers are the set of numbers consisting of the natural numbers
including 0 and their negatives. ( ) = {…, -2, -1, 0, 1, 2, …}
The set of Integers ( ) is closed under addition, subtraction and multiplication,
but not closed under division.
For example, 5/2 = 2.5, and 2.5 is not an integer.
4) The class definition of a Rational Numbers (of a rational fraction) was that it is a
numbers which can be put in the form a where a,d ¢ and d  0 . More
b
complete definition of a rational number is to put additional information, such that
a  c when ad = cd.
b
d
5) An Irrational Number is any real number that is not a rational number. In other
words, an irrational number can’t be expressed as p q where p, q ¢ and q  0
The order of sets of different numbers is as follows:
¢ § °
-------------------------------------------------------Topic: PRIME NUMBERS
A Prime Number is a natural number that has exactly two divisors; 1 and itself.
*Please note that ‘1’ is not a prime number, because it is the only number that is not
suitable under definition of prime numbers. How can 1 be divisible by 1 and itself?
Since divisible by 1 is another way of saying divisible by itself for this specific case, 1 is
not included in the definition and therefore it is not a prime number.
-------------------------------------------------------Topic: GROUPS
Definition: A set S is a group if there is a binary operation * such that,
S is closed under *, and satisfying three axioms.
1) There exists an identity element such that for any element, a in the set a S that
a*e=e*a=a
2) For each a S , there exists a unique a 1 such that a * a 1 = a 1 * a = e.
3) The operation is associative. In other words, a,b,c S , a*(b*c)=(a*b)*c
Not all groups are commutative, but if they are, they are called abelian groups.
(Examples of Groups)
is an abelian group under addition, because it meets the following properties of a
group:
1) Closure: let a,b ¢ , then a  b ¢
2) Associativity: a,b,c ¢ , then (a  b)  c  a  (b  c) ¢
3) There exist an identity element zero: a  0  0  a  a
4) There exist inverse elements: a  (a)  0
5) Commutativity: let a,b ¢ , then a  b  b  a ¢
does not form a group under multiplication because not every elements has a
multiplicative inverses. For example, the multiplicative inverse of 1 is 1 and
multiplicative inverse of –1 is –1. However, multiplicative inverses of any other integers
will be rational numbers, so won’t work.
-------------------------------------------------------Topic: RING
Definition: A set R with 2 binary operations , such that,
(R,+) is an abelian group with identity element 0, so that a,b,c R , the following
axioms holds:
 Closure: a  b  R
 Associativity: (a  b)  c  a  (b  c)
 Identity: 0  a  a
 Commutativity: a  b  b  a
 Inverse:   a  R such that a  (a)  (a)  a  0
(R,



 ) is a monoid with identity element 1, so that a,b,c R that following holds:
a  b R
(a  b)  c  a  (b  c)
1 a  a 1  a
**monoid means binary operation satisfying associativity, identity element and
closure.
Multiplication distributes over addition
 a  (b  c)  (a  b)  (a  c)
 (a  b)  c  (a  c)  (b  c)
*Note: For some, a ring need not have a multiplicative identity.
[ REFERENCES ]
1) Online Wikipedia: The Free Encyclopedia,
[http://en.wikipedia.org/wiki/Integer], 10 February 2008
2) Niven, Ivan. Numbers: Rational and Irrational. MAA. 2002
3) Marie Milach’s Notes for MAE 301 on September 5, 2007
4) {http://www.geocities.com/borhoo/History.htm], 26
January 2001
5) Online Wikipedia: The Free Encyclopedia,[
http://en.wikipedia.org/wiki/Group_theory],
February 2008
10
6) Online Wikipedia: The Free Encyclopedia,[
http://en.wikipedia.org/wiki/Ring_%28mathematics%2
9],
10 February 2008