Download sets of numbers - David Michael Burrow

In Algebra we care about
different sets of numbers and
which numbers are part of
different sets.
Natural Numbers
Natural Numbers
•
1, 2, 3, 4, 5, …
Natural Numbers
•
•
1, 2, 3, 4, 5, …
Numbers we count with
Natural Numbers
•
•
•
1, 2, 3, 4, 5, …
Numbers we count with
Positive whole numbers
Natural Numbers
•
•
•
•
1, 2, 3, 4, 5, …
Numbers we count with
Positive whole numbers
Symbol = N
Whole Numbers
Whole Numbers
•
0, 1, 2, 3, 4, 5, …
Whole Numbers
•
•
0, 1, 2, 3, 4, 5, …
Natural numbers & 0
Whole Numbers
•
•
•
0, 1, 2, 3, 4, 5, …
Natural numbers & 0
Symbol = W
Integers
Integers
•
… , -3, -2, -1, 0, 1, 2, 3, …
Integers
•
•
… , -3, -2, -1, 0, 1, 2, 3, …
Whole numbers and their
opposites
Integers
•
•
•
… , -3, -2, -1, 0, 1, 2, 3, …
Whole numbers and their
opposites
Symbol = Z
Rational Numbers
Rational Numbers
•
Symbol = Q
Rational Numbers
•
•
Symbol = Q
Numbers that can be
written as the quotient of
two integers
Rational Numbers
•
•
•
Symbol = Q
Numbers that can be
written as the quotient of
two integers
“Normal” fractions
Rational Numbers
•
For example …
¾
5/
3
Rational Numbers
•
For example …
¾
-½
5/
3
4
3 /7
Rational Numbers
•
For example …
¾
-½
2.25
5/
3
4
3 /7
-.66666…
Rational Numbers
•
For example …
¾
-½
2.25
42
5/
3
4
3 /7
-.66666…
-11
Rational Numbers
•
•
•
Symbol = Q
Numbers that can be
written as the quotient of
two integers
“Normal” fractions
Irrational Numbers
•
•
•
•
Symbol = I or Ir
Not rational
Can’t be written as a
quotient of integers
“Weird” numbers
Examples of Irrational
Numbers
•
Square roots you can’t
simplify
2
17
143
Examples of Irrational
Numbers
•
Higher roots you can’t
simplify
3
10
5
71
Examples of Irrational
Numbers
•
Special numbers

e

Examples of Irrational
Numbers
•
Most trig function values
sin(52)
tan(107)
Examples of Irrational
Numbers
•
Decimals that don’t end
and don’t repeat
.27227722277722227777…
Examples of Irrational
Numbers
3
2
2
Real Numbers
•
Symbol = R
Real Numbers
•
•
Symbol = R
Rational and irrational
numbers together
Real Numbers
•
•
•
Symbol = R
Rational and irrational
numbers together
Every number on the
number line
Real Numbers
•
•
•
•
Symbol = R
Rational and irrational
numbers together
Every number on the
number line
Every number you know
Properties of Real Numbers
• a.k.a. “Field Properties”
A field is just any set that has
the same properties as the
real numbers.
• The properties of numbers
are essentially the
postulates of algebra.
Properties of
Addition and Multiplication
Commutative Property
3+5=5+3
2(-9) = -9  2
Order doesn’t matter when
you add of multiply.
Commutative Property
Associative Property
-17 + (17 + 39) = (-17 + 17) + 39
(7  4)  9 = 7(4  9)
You can group together what you
want to when you add of multiply.
Associative Property
Identity Property
7+0=4
0+2=2
51=5
1(-4) = -4
When you add 0 or multiply by
1, you get back what you
started with.
Identity
Property
Inverse Property
-4 + 4 = 0
3 5
× =1
5 3
Adding opposites or
multiplying reciprocals cancels
out.
Inverse Property
Closure Property
When you add or multiply two
real numbers, the answer is a
unique real number.
When you add or multiply two
real numbers, the answer is a
unique real number.

There is only one answer
when you add or multiply.
When you add or multiply two
real numbers, the answer is a
unique real number.

You get back the same
kind of thing you started
with.
Closure
Property
Distributive Property
3(2x + 3y – 5) = 6x + 9y – 15
When you multiply a number
times parentheses, take the
number times each term in
parentheses one at a time.
Distributive Property
Properties of
Equality
Reflexive Property of =
2=2
2=2
Everything is equal to itself.
2=2
Everything is equal to itself.
Numbers never change.
Symmetric Property of =
If 2 + 3 = 5, then 5 = 2 + 3
If 2 + 3 = 5, then 5 = 2 + 3
You can flip an equation
around without changing its
meaning.
Example …
129 = 3x – 15
Example …
129 = 3x – 15
same as 3x – 15 = 129
Example …
129 = 3x – 15
same as 3x – 15 = 129
Either way x = 48
Symmetric Property of =
Transitive Property of =
If a list of things are equal,
then the first equals the last.
Addition Property of =
You can add or subtract the
same thing from both sides of
an equation without changing
anything.
x + 14 = 44

x = 30
Multiplication Property of =
You can multiply or divide
both sides of an equation by
the same thing without
changing anything.
5x = 30

x=6
Defined
Operations
Definition of Subtraction
Subtraction means adding the
opposite
4 – (-7) = 4 + 7
Definition of Division
Division means multiplying by
the reciprocal (of the 2nd
number)
4 3 4 7
÷ = ×
5 7 5 3
Sets of numbers
• Natural
• Whole
• Integers
• Rational
• Irrational
• Real
Properties
 Commutative
 Associative
 Identity
 Inverse
 Closure
 Distributive
Properties
 Reflexive
 Symmetric
 Transitive
 + Prop =
 X Prop =
 Def. of –
 Def. of 