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,…..}
• 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.
• 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.
• 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.
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 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 Let T = {0, e} with binary operations
defined by the tables: + 0 e 0 e 0 0 e 0 0 0 e e 0 e 0 e
Power SetLet 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 additive inverse of aR is unique.
• Theorem: If a, b R , a + x = b has a unique solution in R x = b - a.
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) = p1
max(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:– 9 = 3 * 3 + 0– 11 = 2 * 5 + 1– 29 = 3 * 9 + 2
d aq + r
2 11
5 + 1
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
• 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.
14 = 2(mod 12)
Modular Arithmetic and Primes
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435GF8
Used in RSA, one of the most popular cryptographic systems.