Properties of Real Numbers
Addition and Multiplication ONLY
Property
1.
Commutative Property of Addition
a+b=b+a
2. Commutative Property of Multiplication
a•b=b•a
3.
Associative Property of Addition
a+(b+c)=(a+b)+c
4.
Associative Property of Multiplication
a•(b•c)=(a•b)•c
5. Distributive Property
a•(b+c)=a•b+a•c
6. Additive Identity Property
a+0=a
7.
Multiplicative Identity Property
a• 1=a
8. Additive Inverse Property
a + ( -a ) = 0
9.
Multiplicative Inverse Property
Note: a cannot = 0
10. Zero Property
a•0=0
Example
Rational and Irrational Numbers
Rational number is a number that can be expressed as a fraction or ratio
Irrational number
cannot be expressed as a fraction.
Irrational numbers are non-terminating, non-repeating decimals.
Scientific Notation
... is a way to express very small or very large numbers.
... consists of two parts:
(1) a number between 1 and 10, such that
and
(2) a power of 10.
3.2 x 1013
is correct scientific notation
23.6 x 10-8
is not correct scientific notation
Remember that the first
number MUST BE
greater than or equal to
one and less than 10.
Converting To Scientific Notation ...
(1) Place the decimal point such that there is one non-zero digit to the left of the decimal
point.
(2) Count the number of decimal places the decimal has "moved" from the original number.
This will be the exponent of the 10.
(3) If the original number was less than 1, the exponent is negative; if the original number
was greater than 1, the exponent is positive.
Converting From Scientific Notation ...
(1) Move the decimal point to the right for positive exponents of 10. The exponent tells
you how many places to move.
(2) Move the decimal point to the left for negative exponents of 10. Again, the exponent
tells you how many places to move.
Interval vs. Inequality Notation
Interval Notation:
Visual
Open Interval: (a, b) is interpreted as a < x < b where the
endpoints are NOT included.
Closed Interval: [a, b] is interpreted as a < x < b where
the endpoints are included.
(1, 5)
[1, 5]
INCLUSIVE
Half-Open Interval: (a, b] is interpreted as a < x < b
where a is not included, but b is included.
Half-Open Interval: [a, b) is interpreted as a < x < b where
a is included, but b is not included.
Non-ending Interval:
is interpreted as x > a where a
is not included and infinity is always expressed as being
"open" (not included).
Non-ending Interval:
is interpreted as x < b where b
is included and again, infinity is always expressed as being
"open" (not included).
How you would express the interval "all numbers except 13".
In interval notation:
As an inequality:
x < 13 or x > 13
Notice that the word "or" has been replaced with the
symbol "U", which stands for "union".
(1, 5]
[1, 5)
INEQULITY
SYMBOL
MEANING
less than
greater than
less than or equal to
greater than or equal to
A compound inequality is two simple inequalities joined by "and" or "or".
Solving an "And" Compound Inequality:
3x - 9 < 12 and 3x - 9 > -3
Solving an "Or" Compound Inequality:
2x + 3 < 7 or 5x + 5 > 25
[2x + 3 < 7]
Or written ...
The common statement
is sandwiched between
the two inequalities.
[5x + 5 > 25]
Solve the first inequality
Solve as a single unit or solve
each side separately.
The solution is 2 < x < 7,
which can be read x > 2 and x < 7.
Interval notation: [2, 7]
Solve the second inequality
The solution is x < 2 or x > 4.
Interval notation: