Download 1. Give an example of a. a negative rational number that is

August 19, 2011
1. Give an example of
a. a negative rational number that
is not an integer.
b. a positive irrational number that is not
transcendental.
c. an imaginary number.
2. Prove that 4.22222...... is a rational number.
August 19, 2011
August 19, 2011
1.2 The Field Axioms
August 19, 2011
Commutativity:
Addition and multiplication of real
numbers are commutative operations.
That is, if x and y are real,then
a) x+y = y+x
b) xy = yx
August 19, 2011
August 19, 2011
Associativity:
Addition and multiplication of real
numbers are associative operations.
That is if x,y,and z are real then,
a) (x+y) + z = x + (y + z)
b) (xy)z = x(yz)
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Distributivity:
Multiplication distributes over addition.
That is if x,y,z are real numbers, then
x(y+z) = xy + xz
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Identity Elements:
{real numbers} contains:
a) A unique identity element for addition,
namely 0.
(Because x+0 = x for any real number x.)
b) A unique identity element for
multiplication, namely 1.
(Because 1x = x for any real number x.)
August 19, 2011
Inverses:
{real numbers} contains:
a) A unique additive inverse for every real
number x. (Meaning that every real
number x has a real number -x such that
x + (-x) = 0. )
b) A unique multiplicative inverse for every
real number x except zero. (Meaning
that every non-zero number x has a real
number 1/x such that x(1/x)=1.)
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Closure:
{real numbers} is closed under addition
and under multiplication.
That is, if x and y are real, then
a) x + y is a real number
b) x•y is a real number
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Any set of numbers that obeys all
11 of these axioms is a field.
1. Is the set of integers a field?
2. Is the set of rationals a field?
3. Is the set {0,1} a field?
August 19, 2011
HW: p.7-8 #1-10