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```Table of Properties
Let a, b, and c be real numbers, variables, or algebraic expressions.
(These are properties you need to know.)
Property
1.
Example
a+b=b+a
2+3=3+2
2. Commutative Property of Multiplication
a • b = b • a
2•(3)=3•(2)
a + ( b + c ) = ( a + b ) + c
4.
Associative Property of Multiplication
a • ( b • c ) = ( a • b ) • c
5. Distributive Property of Multiplication over Addition
a • ( b + c ) = a • b + a • c
a + 0 = a
7.
Multiplicative Identity Property
a • 1 = a
a + ( -a ) = 0
9.
2+(3+4)=(2+3)+4
2•(3•4)=(2•3)•4
2•(3+4)=2• 3+2•4
3+0=3
3•1 =3
3 + (-3) = 0
Multiplicative Inverse Property
Note: a cannot = 0
10. Zero Product Property
a •0=0
5•0=0
Above are properties you already know but another one is the property of closure.
The Closure Property states that when you perform an operation (such as addition, multiplication,
etc.) on any two numbers in a set, the result of the computation is another number in the same
set. For example, when you multiply two integers you will always get an integer, so the integers
are closed with respect to multiplication. However, when you multiply two irrational numbers, you
don’t always get an irrational number ( 2  18  36  6 ), so the irrational numbers are not
closed with respect to multiplication. Furthermore, when you divide two integers you don’t always
get an integer (7
 2 = 3.5), so the integers are not closed with respect to division.
Accelerated CCGPS…Math II
Name______________________
“Proofs” of Properties of Numbers
Period______Date____________
To prove a mathematical statement, we must show the statement to be true in all cases, but to
disprove a mathematical statement, we only need 1 counterexample. For each of the following
statements, find an example that verifies the statement and an example that contradicts the
statement, if possible. If no counterexample can be found, write a “proof” of the statement. If
no example can be found, rewrite the statement so that it is always true. The first one is done
for you.
Statement
The sum of two
rational numbers is
rational.
The product of two
irrational numbers is
irrational.
The quotient of two
natural number is
not natural.
The area of a
30˚- 60˚- 90˚
triangle is
irrational.
The perimeter of a
45˚- 45˚- 90˚
triangle is rational.
The area of a circle
is irrational.
Example
Counterexample
2 7 29
 
3 4 12
none
which is rational.
“Proof” or Rationale
If x 
a
c
where a, b, c, d
and y 
b
d
are integers and bd ≠ 0, then
which will
x y   
b d
bd
still be rational.
More Review Questions:
I. Simplify completely:

1.  6


4.

4
3



3
2
2 3
2.

2
x 3k 2
7.
x
10.
3
40  7 3 5  3 8
3
4
5.

8.
p 
p p 
2 8
2x
11.
6
3

2
3.
81x 
6.
 
3
9.
x
x
5
n5
2
n

108
3
4
5
2

12. 2  3 2  3
27

13. Without using a calculator, approximate the following values to the nearest tenth:
80
= _____________,
2
(Check your approximations using a calculator.)
97 = _____________,
14. Use your calculator to approximate
1
17  33 = ____________.
= _________ and
3 5
What do you think must be true about the above values?
3 5
= _________.
4
Can you prove it?
```
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