Download real numbers

WARM UP
1
The least common denominator of the fractions
and
is 24
2
The variable in the expression 5x – 3 is _________
X
3
Fill in the blank with an inequality symbol 6 ___
> 4
4
In the fraction
denominator
________
5
According to the order of operations, what is the first
step in simplifying the expression (3 +4) – 8?
5 is the numerator
_______ and 12 is the
Add 3 + 4 (PEMDAS)
PROPERTIES OF REAL
NUMBERS
INTEGERS AND RATIONAL NUMBERS
OBJECTIVES
Performing operations with real numbers
Applying properties of real numbers
Classifying and reasoning with real numbers.
To add positive and negative numbers.
To subtract positive and negative numbers.
To use real number in solving problems.
VOCABULARY
Absolute value
Additive inverse
Differenc
e
Distributive property
Equivalent expressions
Integers
Irrational numbers
Multiplicative inverse
Natural numbers
Number line
Opposite
Rational numbers
Real numbers
Subtraction
Whole numbers
Using integers & rational
numbers
Whole numbers are numbers 0, 1, 2,
3,…………..
Integers are the numbers ….-3, -2, -1, 0, 1, 2, 3,………
Positive integers are the integers that are greater
than 0 and negative integers are less than 0.
The integer 0 is neither negative nor positive.
REAL NUMBERS
The most important set of numbers in algebra is the
set of real numbers.
The sets of natural numbers [1, 2, 3, 4,……..], whole
numbers [0, 1, 2, 3, 4,……] and integers [-2, -1, 0, 1, 2, 3,….]
are all subsets of the set of real numbers.
RATIONAL NUMBERS
A number
where a and b are integers and b is not
equal to zero. For example - is a rational number.
The rational number belongs to the set of numbers
called real numbers.
NUMBER LINE
Negative integers
Positive integers
left side of zero
right side of zero
Zero is the center of
the number line
EXAMPLE
Graph and compare integers
Graph -3 and -4 on the number line. Then tell which
number is greater.
- 6
-5
-4
-3
-2
-1 0 1
2
3
4
5
6
OPPOSITES
The numbers that are the same distance from 0 on a number
line but are on opposite sides of 0 are called opposites.
For example, 4 and -4 are opposites because they are both 4
units from 0 but are opposite sides of 0.
4 units
to the
left of
zero
4 units
to the
right of
zero
ABSOLUTE VALUE
The absolute value of a number a is the distance
between a and 0 on a number line. The symbol
represents the absolute value of a.
If a is positive, then
If a is 0, then
=a
=0
If a is negative, then
= -a
Example:
=2
=0
= -(-2) = 2
EXAMPLE
Tell whether each of the following numbers is a whole
number, integer, or rational number: 5, 0.6, -2 ,
and -24.
Number
Whole Number?
Integer?
Rational Number?
5
Yes
Yes
Yes
0.6
No
No
Yes
2
No
No
Yes
24
Yes
Yes
Yes
REAL NUMBERS ADDITION
•
One way to add or subtract two real numbers is to use
the number line.
To add a positive number move right
- 6
-5
-4
-3
-2
-1 0 1
To add a negative number, move left
2
3
4
5
6
REAL NUMBER SUBTRACTION
Subtraction rule
To subtract b from a, add the opposite of b to a.
a – b = a + (-b)
Example: 14 – 8 = 14 + (-8)
Find the difference:
-12 – 19 = -12 + (-19)
You can then use the number line to start at negative 12 and go
left 19 units because you are adding a negative number.
REAL NUMBER MULTIPLICATION
The product of two real numbers with the same sign is
positive.
Examples: 3(4) = 12
-6(-3) = 18
The product of two real numbers with the different signs is
negative.
Examples: 2(-5) = -10
-7(2) = -14
More examples:
Find the product -3(6) = -18 Different signs; product is negative
2(-5)(-4) = (-10)(-4) = 40
Multiply 2 & -5, same sign;
product is positive
PROPERTIES OF MULTIPLICATION
Commutative Property : The order in which you multiply two
numbers does not change the product.
axb=bxa
4 x (-5) = -5 x 4
Associative Property: The way you group three numbers in a
product does not change the product:
(a x b) x c = a x (b x c)
(-2 x 7) x 4 = -2 x (7 x 2)
Identity Property: The product of a number and 1 is that number.
ax1=1xa=1
(-5) x 1 = 1 x (-5) = -5
Property of Zero: The product of a number and 0 is 0
ax0=0 0xa=0
3x0=0 0x3=0
Property of negative one (-1): The product of a number and -1
is the opposite of the number.
a x (-1) = -1 x a = -a -2 x (-1) = 2
REAL NUMBER DIVISION
Reciprocals are opposite fractions that when multiplied
equal
=1
The reciprocal of a nonzero number a written , is
called the multiplicative inverse of a. Zero does not
have a multiplicative inverse because there is no
number a such that 0 a = 1 (not true)
Example: a
8
= 1, when a is not equal to 0
=1
Division Rule: to divide a number a by a nonzero number
b, multiply a by the multiplicative inverse
a
b=a
, when b
0
5
2=5
DIVISION AND ZERO
Division by zero is impossible.
We can divide zero by any nonzero number . The answer is always zero.
On the other hand, we can never divide by zero. By definition of division
such that
. But
for any
number c, so the only possible number that n could be is 0.
Let’s consider what
might be.
might be 5 because 0 = 0 5
might be 267 because 0 = 0 267
It looks as if could be any
number. Thus, we cannot
define and must exclude
division by 0. Zero is the only
real number that does not have
a reciprocal.
FINDING
RECIPROCALS
Many scientific calculators have a reciprocal
key:
1/x
Find the
reciprocal of
63
0.015873
SUMMARY OF OPERATIONS
Addition of Real Numbers: To add when there are like signs,
add the absolute values. The sum has the same sign as the
addends.
Subtraction of Real Numbers: Every real number has exactly
one additive inverse or opposite. It is the number added to it
to get 0 and is symbolized by -x
Multiplication of Real Numbers: If numbers are both positive or
both negative, their product is positive. If one number is positive
and the other negative, their product is negative.
Division of Real Numbers: Every nonzero real number has a
multiplicative inverse or reciprocal. The reciprocal of a number
is the number we multiply to get 1.
If numbers are both
positive or both negative, the quotient is positive. If one number
is positive and the other negative, the quotient is negative