Transcript lesson 2-1

Principles
Principles are basic truths
or laws.
Luke 6:31
“And as ye would that men
should do to you, do ye also
to them likewise.”
Closure Property of
Addition
For all integers a and b,
a + b is an integer.
Integers
Integers are closed for
addition.
The set of odd numbers is
not closed for addition.
Commutative Property
of Addition
For all integers a and b,
a + b = b + a.
−5 + 3
Equivalent
When two expressions
represent the same value,
they are said to be
equivalent.
Example 1
Use the Commutative Property
to write an equivalent
expression for each of the
following. Be sure the answer
is in simplest form.
a. −9 + 4 = 4 + (−9) = 4 − 9
Example 1
b. x + 9 = 9 + x
Associative Property of
Addition
For all integers a, b, and c
a + (b + c) = (a + b) + c.
Example 2
Use the Associative Property
to write an equivalent
expression for each of the
following.
a. (−8 + 17) + 43 =
Example 2
b. −9 + (7 + x) =
Identity Property of
Addition
For any integer a,
a + 0 = 0 + a = a.
Additive Inverse
Additive inverses are two
numbers whose sum is
zero.
A.K.A. “Opposites”
Example 3
What are the additive inverses
of 17 and −14?
Example 4
Use the properties of addition
to find the value of each
variable. Name the properties
used.
a. x = 6 + (−6)
Example 4
b. y + 0 = 19
c. (9 + 2) − 6 = x + (2 − 6)
Property of
Description
Addition
Closure: for
all integers a The sum of any
and b, a + b is two integers is an
integer.
an integer.
Example
5 + 9 = 14;
14 is an
integer.
Commutative: Changing the order 8 + 4 = 4 + 8
a+b=b+a
of the addends
does not change
the sum.
Associative:
Changing the
(−5 + 4) + 6 =
(a + b) + c =
grouping of the
−5 + (4 + 6)
a + (b + c)
addends does not
change the sum.
Property of
Addition
Identity:
a+0=
0+a=a
Inverse:
a + (−a) =
−a + a = 0
Description
Example
The sum of any
integer and zero
equals the original
integer.
The sum of any
integer and its
additive inverse
equals zero, the
identity element of
addition.
−2 + 0 = 0 +
(−2) = −2
6 + (−6) = −6
+6=0
Exercise
+
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
Exercise
Is this set closed? Why?
Exercise
Is this set commutative?
Why?
Exercise
Which element is the identity
element? Why?
Exercise
What is the inverse of a?
Why?
Exercise
A clock has no zero on it.
What number serves as the
identity number for adding
clock times?
Exercise
Based on your answer to the
last question, what is the
inverse of 7 on a clock?