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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