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 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 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 zero of a ring R is unique.

• Theorem: If a ring has a unity, the unity is unique.

• 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.

Cancellation Law Of AdditionIf 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) = 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.

Thank You !!