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. Exploration Is (Z,+, ) a commutative ring with unity? Is (E,+, ) a commutative ring with unity? 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 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. 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. 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. 100 = 2 * 2 * 5 * 5 = 2 2 5 2 79 = 79 (it is prime, and factors are 1 and 79) 999 = 3 3 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) = p 1 max(a1,b1) p n max (an,bn) Example: lcm(2 2 3 3 11 2, 2 3 11 4 ) = 2 3 3 3 11 4
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. 9 = 3 * 3 + 0 11 = 2 * 5 + 1 29 = 3 * 9 + 2 d q a + r 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. 435 GF8 Thank You!! 983dygg2 34sxVy3k3 Bb63ew.bl #evkwa