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REAL
NUMBERS
Objective- To recognize symbols, variables, and types of sentences used in algebra.
Equalities Inequalities
= Equals- is the same as
< Is less than
> Is greater than
Is less than or equal to
Approx. equal to
= Not equal to
What is a variable?
A variable represents an unknown value.
1) 3 + ___ = 10
2) 6 + = 9
3) 8 + x = 12
4) 5 + = 8
These are all variables
Expressions vs. Equations
Numerical
Variable
Expressions Equations Inequalities
2 + 3
5(8) - 4
X + 7
8 - 3y
2 + 3 = 5
4 + 2(3) = 10
X - 4 = 13
11= 3 + 2m
9 - 5 > 3
6y - 4 < 8
Sentences
Open sentences
Open sentences have solutions and can be solved.
Translating English to Math
sum of two numbers
difference between two numbers
The product of two numbers
the quotient of two numbers
is =
ab
a - b
a + b
ba
Multiplication and Division Symbols
Ways to Multiply Ways to Divide
5 x 7
5 7
5(7)
5 7With variables...
5 x
5x
5axy
8 484
8 / 4
4 8
{1, 2, 3, 4, . . . }
If you were asked to count, the numbers you’d say are called counting numbers. These numbers can be expressed using set notation.
These are also called the natural numbers.
{0, 1, 2, 3, 4, . . . }If we include 0 we have the set of whole numbers.
{ …, -3, -2, -1, 0,1, 2, 3, . . . }
Include the opposites of the whole numbers and you have the set of integers.
rational numbers
Whole numbers are a subset of integers and counting numbers are a subset of whole numbers.
integers
whole numbers
counting numbers
If we express a new set of numbers as the quotient of two integers, we have the set of rational numbers
This means to divide one integer by another or “make a fraction”
rational numbers
There are numbers that cannot be expressed as the quotient of two integers. These are called irrational numbers.
integers
whole numbers
counting numbers
2
irrational
numbers
The rational numbers combined with the irrational numbers make up the set of real numbers.
REAL NUMBERS
Sets of Numbers Reals
Rationals Irrationals- any number whichcan be written as afraction.
2
3, 7, -0.4
Fractions/Decimals Integers
61
4 , -0.32, - 2.1 …-3, -2, -1, 0, 1, 2, 3...
Negative Integers Wholes…-3, -2, -1 0, 1, 2, 3...
Zero0
Naturals1, 2, 3...
- non-terminating and non-repeating decimals
3141592. ...
2
ORDER OF OPERATIONS
When there is more than one symbol of operation in an expression, it is agreed to complete the operations in a certain order. A mnemonic to help you remember this order is below.
P E M D A Sarenthesis
xponents
ultiplication
ivisiondditio
nubtra
ction
Do any simplifying possible inside of parenthesis starting with innermost parenthesis and working out
Apply exponentsComplete multiplication and division from left to rightComplete addition and subtraction from left to right
423532 2
PEMDASPEMDAS
parenthesis – combine these first
42322 2
PEMDAS
exponents – apply the exponent now
42342
PEMDAS
complete multiplication and division, left to right
468
PEMDAS
complete addition and subtraction, left to right
10
423532 2
Simplify the expressions.
1) 6 + 5(8 - 2)
2) 2 + 3 5
3) 16 2 4
4) 10 - 3 + 2 4
2
Order of Operations
1) 6 + 5(8 - 2)Parenthesis
Exponents 2) 2 + 3 5
6 + 5(6)6 + 30 = 36
2
2 + 3 252 + 75 = 77
Multiply / Divide 3) 16 2 48 4 = 32
Add / Subtract 4) 10 - 3 + 2 410 - 3 + 8
7 + 8 = 15
COMMUTATIVE PROPERTY
The operations of both addition and multiplication are commutative
abba 3223 When adding, you can “commute” or trade the terms places
abba 3223
When multiplying, you can “commute” or trade the factors places
ASSOCIATIVE PROPERTY
cbacba 321321
When adding, you can “associate” and add any terms first and then add the other term.
cbacba 432432
When multiplying, you can “associate” and multiply any factors first and then multiply the other factor.
The operations of both addition and multiplication are associative
DISTRIBUTIVE PROPERTY
The operation of multiplication distributes over addition
acabcba 4323423
The distributive property also holds for a factor that is multiplied on the left.
acabacb
2423243
abba A positive
times a negative is
NEGATIVE
abba A negative
times a positive is
NEGATIVE
aa The negative of a negative
POSITIVE
CAUTION: Remember that the value for a and/or b could also be positive or negative.
b
a
b
a
b
a
A positive
divided by a negative or
A negative divided by a positive is
NEGATIVE
b
a
b
a
A negative
divided by a negative is POSITIVE
You walk directly east from your house one block. How far from your house are you?
1 block1 block
You walk directly west from your house one block. How far from your house are you?
It didn't matter which direction you walked, you were still 1 block from your house.
This is like absolute value. It is the distance from zero. It doesn't matter whether we are in the positive direction or the negative direction, we just care about how far away we are.
2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8
44 4 units away from 044 4 units away from 0
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