Transcript Lecture3 - West Virginia University

Whole Numbers
Are the whole numbers with the
property of addition a group?
Extending The Natural Numbers
• Natural or Counting Numbers {1,2,3…}
• Extend to Whole Numbers { 0,1,2,3…} to
get an additive identity.
• Extend to Integers { … -3,-2,-1,0,1,2,3…}
to get additive inverses.
• (Z, +) is a group.
Integer Number Set
Extension of Whole Number Set
1. Natural or counting Numbers {1,2,3…}
2. Additive identity 0
3. Negative Integers {-1,-2,-3,…..}
History of Zero
• In around 500AD Aryabhata devised a
number system which has no zero yet
was a positional system.
• He used the word "kha" for position and
it would be used later as the name for
zero.
• There is evidence that a dot had been
used in earlier Indian manuscripts to
denote an empty place in positional
notation.
• The Indian ideas spread east to China
as well as west to the Islamic countries.
• In 1247 the Chinese mathematician
Ch'in Chiu- Shao wrote Mathematical
treatise in nine sections which uses the
symbol O for zero.
Key Points
• Both the Greeks and Romans had symbolic
zeros but not the concept of zeros
• EXAMPLE: MCVIII = 1000 + 100 + 8 = 1108.
Notice the 0 is used just as a placeholder
• The Babylonians and Mayans also used 0 as
a placeholder in their base 60 and base 10
numbering systems.
• The Hindus originally gave us the modern
day 0.
Claudius Ptolemy
• Ptolemy was of Greek descent and lived
in Egypt .
• The astronomical observations that he
listed as having himself made cover the
period 127-141 AD .
• Ptolemy in the Almagest written around
130
AD
uses
the
Babylonian
sexagesimal system together with the
empty place holder O .
• By this time Ptolemy is using the symbol
both between digits and at the end of a
number.
Negative Numbers History
• The concept of a "negative" number has
often been treated with suspicion.
• The ancient Chinese calculated with
colored rods, red for positive quantities
and black for negative (just the opposite
of our accounting practices today) .
• But, like their European counter-parts,
they would not accept a negative number
as a solution of a problem or equation.
• Instead, they would always re-state a
problem so the result was a positive
quantity.
• This is why they often had to treat many
different "cases" of what was essentially a
single problem.
Example (s)
• The Ancient Egyptians used forms such as these
to express negative numbers:
• If line 61 is more than line 54, subtract line 54
from line 61. This is the [positive] amount you
OVERPAID.
• If line 54 is more than line 61, subtract line 61
from line 54. This is the [positive] amount you
OWE.
• Interestingly, the above form does not
provide any guidance on how to proceed if
line 61 EQUALS line 54.
• This may suggest that the concept of zero
has not yet been fully assimilated.
• In fact, many ancient cultures did not even
regard "1" as a number (let alone 0),
because the concept of "number" implied
plurality.
• As recently as the 1500s there were
European mathematicians who argued
against the "existence" of negative
numbers by saying :
• Zero signifies "nothing", and it's
impossible for anything to be less than
nothing.
• On the other hand, the Indian
Brahmagupta (7th century AD) explicitly
and freely used negative numbers, as
well as zero, in his algebraic work.
• He even gave the rules for arithmetic,
e.g., "a negative number divided by a
negative number is a positive number",
and so on.
• This is considered to be the earliest
[known] systemization of negative
numbers as entities in themselves.
(Z, )
• Are the integers with the property of
multiplication a group?
Rings
Let R be a nonempty set on which there
are defined two binary operations of
addition and multiplication such that the
following properties hold:
For all a, b, c R
Addition Properties:
• Closure: a + b  R
• Commutative: a + b = b + a
• Associative: a + (b + c) = (a + b ) + c
• Identity (Zero):  0  R such that
a + 0 = 0 + a = a for all a  R
• Inverse:  a  R  x  R such that
a + x = x+ a = 0
Multiplication Properties
• Closure: a b  R
• Associative: a  (b  c) = (a  b)  c
• Distributive Property Of Multiplication over
addition: a  (b + c) = ab + ac
Ring of Integers
• (Z, + ,  ) is a ring
• Let E be the even integers. Is (E, + ,  ) a
ring?
Ring Types
• Commutative Ring: A ring (R,+,) with the
commutative law of multiplication.
 a, b  R , a  b = b  a.
• Rings with unity: A ring (R,+,) with a
Multiplicative Identity (called unity)
 e R  a  e = e  a = a  a R.
Exploration
• Is (Z,+,) a commutative ring with unity?
• Is (E,+,) a commutative ring with unity?
Exploration
Let T = {0, e} with binary operations
defined by the tables:
+ 0 e
0 0 e
e e 0
 0 e
0 0 0
e 0 e
• Is (T,+,) a ring?
• Is it commutative ring?
• Is it a ring with unity?
Power Set
Let P=(A) with binary operation
a + b = (a  b) \ (a  b)
ab=ab
Is (P,+,) a ring?
Is (P,+,) a commutative ring?
Is (P,+,) a ring with unity?
HINT: Use Venn Diagram to verify the
above.
• Theorem: The zero of a ring R is unique.
• Theorem: If a ring has a unity, the unity is
unique.
• Theorem: The additive inverse of aR is
unique.
• Theorem: If a, b  R , a + x = b has a
unique solution in R x = b - a.
Cancellation Law Of Addition
If a, b, c  ring R and if a + c = b+ c,
then a = b.
Integers
Division
• If a and b are integers with a not equal to 0,
then a divides b (a | b) if there exists an
integer c such that b = a * c, i.e., the quotient
is an integer. If a | b, then a is a factor (or
divisor) of b and b is a multiple of a.
• Examples:
– 2 | 7?
– 4 | 16?
Prime Numbers
• A positive integer p > 1 is called a prime if
the only positive factors of p are 1 and p.
A positive integer > 1 that is not prime is
called composite.
• Examples:
– Primes: 2, 3, 5, 7, 11,…
– Is 19 prime?
– Is 20 prime? No, it is composite. Factors of
20 are: 2 | 20, 4|20, 5 | 20, 10 | 20.
Fundamental Theorem of
Arithmetic
• Every positive integer can be written uniquely as
a product of primes. This is called a
prime factorization.
• Examples:
– 100 = 2 * 2 * 5 * 5 = 22 52
– 79 = 79 (it is prime, and factors are 1 and 79)
– 999 = 33 37
GCD, LCM
•
•
Let a and b be integers, not both 0. The largest
integer d such that d | a and d | b is the greatest
common divisor of a and b, i.e., the gcd(a, b).
Example:
–
What is gcd(12, 48)? 12 because…
– Positive divisors of 12 are 1, 2, 3, 4, 6, 12.
– Positive divisors of 48 are 1, 2, 3, 4, 6, 12, 24, 48.
•
The least common multiple of positive integers a
and b is the smallest positive integer divisible by
both a and b, i.e., the lcm(a, b).
–
–
–
Using prime factorizations,
lcm(a,b) = p1max(a1,b1) … pnmax (an,bn)
Example: lcm(22 33 112, 23 114) = 23 33 114
Division Algorithm
• Let a and d be an integers, with d not = 0. Then there
exist unique integers q and r, with 0 <= r < | d | such
that
a = d * q + r. Here, d is called the
divisor, q the quotient, and r the remainder.
• Examples:
q
+r
– 9=3*3+0
d a
– 11 = 2 * 5 + 1
– 29 = 3 * 9 + 2
5 +1
2 11
The Function “mod”
•
•
Let a be an integer and m a positive integer. We
denote by a mod m the remainder r when a is divided
by m.
(a = q * m + r)
If a and b are integers and m is a positive integer,
then a is congruent to b modulo m if m | (a – b).
We denote this as a = b(mod m).
–
Example: Clock notation
14 = 2(mod 12)
•
It is possible to do modular arithmetic. See Section
2.3 of your text for details. If time permits, we will
study this in class.
Modular Arithmetic and Primes
Used in RSA, one of the most popular cryptographic systems.
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GF8
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Thank You !!