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# WHOLE NUMBERS; INTEGERS - PowerPoint PPT Presentation

WHOLE NUMBERS; INTEGERS. Whole numbers: Z 0,+ = the natural numbers  {0}. Integers:. Properties:. Z 0,+ is closed under addition and multiplication. Z 0,+ is not closed under subtraction and division. Z is closed under addition, subtraction, and multiplication.

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Presentation Transcript

Whole numbers:

Z0,+ = the natural numbers  {0}.

Integers:

• Z0,+is closed under addition and multiplication.

• Z0,+ is not closed under subtraction and division.

• Z is closed under addition, subtraction, and multiplication.

• Z is not closed under division.

Let m, n  Z.

m is a multiple of n if and only if there is an integer k such that

m = k n.

n is a divisor of m if and only if m is a multiple of n.

Let a, b  Z, with a, b  0. The greatest common divisor of a and b, denoted gcd(a,b), is the largest integer that divides both a and b.

The least common multiple of a and b, denoted lcm(a,b), is the smallest positive multiple of both a and b.

THEOREM: Let a, b  Z, a, b  0. Then

and

The set of rational numbers, Q, is given by

NOTE:

The set of rational numbers, Q, is closed under addition, subtraction, and multiplication; Q {0} is closed under division.

Addition (+): Let a, b, c be rational numbers.

A1. a + b = b + a (commutative)

A2. a + (b + c) = (a + b) + c (associative)

A3. a + 0 = 0 + a = a (additive identity)

A4. There exists a unique number ã such that

a + ã = ã +a = 0 (additive inverse)

ã is denoted by – a

• Multiplication (·): Let a, b, c be rational numbers:

• M1. a b = b a (commutative)

• M2. a (b c) = (a b) c (associative)

• M3. a 1 = 1 a = a (multiplicative identity)

• M4. If a  0, then there exists a unique â such

• that

• a â = â a = 1 (multiplicative inverse)

• â is denoted by a-1 or by 1/a.

D. a(b + c) = ab + ac

(a + b)c = ac + bc

Distributive laws (the connection between addition and multiplication).

Let be a rational number.

Use long division to divide p by q. The result is the decimal representation of r.

The set of rational numbers Q is the set of all terminating or (eventually) repeating decimals.

Problem: Given a rational number

The decimal expansion of r either terminates

or repeats. Give a condition that will imply

that the decimal expansion:

a. Terminates;

b. repeats.

The decimal expansion of r terminates if and only if the prime factorization of q has the form

Problems: Write

in the form

Let a and b be any two distinct real numbers with a < b. Then there is rational number r such that

a < r < b.

That is, there is a rational number between any two real numbers. The rational numbers are dense on the real line.