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1.3 FRACTIONSREVIEWVariables-letters that represent numbers

examples: x, y, z, a, b, cMultiplication-can be shown many ways

examples: ab a b a(b) (a)bFactors-numbers (or variables) multiplied

together to get a product

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Fractions a/b Where a is the numerator (top) and b is the

denominator (bottom)The fraction bar means to divide A fraction is said to be reduced or simplified or in

lowest terms when the numerator and denominator have no common factors except one.

To simplify a fraction, find the GCF and divide both numerator and denominator by that number

GCF is the biggest number that goes into the top and bottom

(see appendix B for more info on GCF)

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a nice trick to remember Note: you can only cross cancel across a

multiplication sign-never do this across an add, subtract, or division sign.

Below you can cross cancel the two’s and then multiply. (like reducing before you multiply)

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Forms of fractionsProper fractions: ½ or a/b when a<b

the numerator is smaller than the denominator

Improper fractions: when a>b; the numerator is larger than the denominator

Mixed numbers: 3 ½ or A b/chave two parts – a whole number part and a fraction part

baor

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Changing forms (circle trick)

A nice trick for changing from a mixed number to an improper fraction is:

-multiply the denominator and the whole number-add this to the numerator

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Changing forms

The method for changing from a improper fraction to a mixed number is to divide.

Remember the fraction bar is a divide sign.

Can you see the 3 ½ ?

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Multiplying fractionsTo multiply fractions:Mutliply the numerators, multiply the denominators

and reduceIn other words, take top times top; bottom times

bottom and reduce

bdac

dc

ba

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Dividing fractionsTo divide fractions:By definition, division is multipying by the

reciprocal. So . . . -leave the first fraction as is-flip the second fraction-multiply (take top times top; bottom times bottom)-reduce

bcad

cd

ba

dc

ba

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Adding and subtracting fractions-find a common denominator

(LCD)-rewrite each fraction with new

denominator-add or subtract numerators as

indicated-keep denominator the same-reduceSee appendix b for more info on

LCD

dc

ba

dc

ba

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1.4 REAL NUMBERSIn this section we will be working with set

notation. A set is a collection of elements listed within

bracesExample {a,b,c,d,e} – this set has 5

elements{ } 0 -- this set has no elements. It is called

the empty set or null set.{1,2,3 . . } – this set has an infinite number of

elements

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Natural Numbers {1,2,3, . . .} This set includes the positive numbers-no

decimals or fractions. Also referred to as the counting numbers in some books.

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Whole Numbers {0,1,2,3, . . .} This set includes the natural numbers and

zero; still no decimals or fractions

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Integers { . . . -2,-1,0,1,2, . . .} This set includes the positive and negative

“whole” numbers; again, no decimals or fractions

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Rational NumbersThere are a lot of numbers in this set.

This set includes any number that can be written as a fraction. Fractions; Repeating and Terminating decimals as well; (1/3 = 0.333333…. Or ½= 0.5) And all the whole numbers. (put a 1 under them 5 = 5/1)

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Irrational NumbersWe don’t work with these a lot. Common

examples are and certain square roots.

This set includes any number that can not be written as a fraction. These are non-repeating, non-terminating decimals.

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Real NumbersThis where we spend most of our time.

This set includes natural numbers, whole numbers, integers, rational numbers, irrational numbers. Everything we have talked about so far.

Real numbers are any number that can be represented on a number line.

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Real Number System• Venn diagram

Natural

Whole

IntegersRationals

IrrationalsReals

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1.5 INEQUALITIES

< less than less than or => greater than greater than or == equal to= not equal to

You can use a number line to compare numbers. To the left things get smaller. To the right things get bigger.

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Absolute Value

By definition, absolute value is the distance away from zero on a number line.

Denoted by straight lines or bars on either side of a number or an expression

-3 = 3 3 = 3 0 = 0

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1.6 Addition of Real NumbersAddition is combiningWhen adding two numbers with the same

sign, add the absolute values of the numbers and keep the sign the same.

When adding two numbers with different signs, find the difference of the absolute values of the numbers and take the sign of the number with the larger absolute value.

Note: additive inverse means opposite

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Addition examples:3 + 2 = 5 the numbers have the same sign, add the

numbers and keep the sign the same

-3 + -2 the numbers have the same sign, add the numbers and keep the sign the same

-3 + 2 the numbers have different signs, find the difference of the numbers and take the sign of the number with the larger absolute value

3 + -2 the numbers have different signs, find the difference of the numbers and take the sign of the number with the larger absolute value

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1.7 SubtractionBy definition, subtraction means to

add the opposite You will rewrite every subtraction problem

into an addition problem. Then use the rules for addition that we went over in 1.6

a – b = a + -b 3 – 2 = 3 + -2 -3 – 2 = -3 + -2 -3 - -2 = -3 + 2 3 - -2 = 3 + 2

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1.8 Multiplication/Division

Because multiplication and division are so closely related, the chart below works for both operations

When multiplying or dividing two numbers:If the signs are the same, your answer is

positive.If the signs are different, your answer is

negative.

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In other words+ + = +- - = +

+ - = -- + = -

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1.9 Exponents, Parenthesis, and Order of Operations

Exponents-An exponent tells the number of times the

base appears as a factor.

2 is the exponent or power 4 is the base

is read 4 to the 2nd power or 4 squared

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means take 4 x 4

means take 4 x 4 x 4

If no exponent appears, we assume the exponent is one – not zero.

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Order of Operations

Order of Operations exists because when there is more than one operation involved, if we do not have an agreed upon order to do things, we will not all come up with the same answer. The order of operations ensures that a problem has only one correct answer.

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Order of OperationsParenthesis (or grouping symbols)ExponentsMultiplication or Division from Left to RightAddition or Subtraction from Left to Right

PEMDAS

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In the parenthesis step, you may encounter nested parenthesis. Below you will see the same problem written two ways: once with nested parenthesis and the other with a variety of grouping symbols (including brackets, braces, and parenthesis).

(( 5 x ( 2 + 3 )) + 7 ) – 2OR

{[ 5 x ( 2 + 3 )] + 7 } - 2

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1.10 Properties of Real Numbers

In general, these properties are things that you already know to be true. This just puts a name to the idea that you already understand.

You will need to memorize these (or think of tricks to remember the names of them).

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CommutativeCommutative Property says the order does

not matter when you are adding or multiplying. In other words, you can add or multiply in any order, it does not affect the answer.

Commutative Property of AdditionA + B = B + A

Commutative Property of MultiplicationA x B = B x A

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AssociativeThe Associative Property says when you are

adding or multiplying three or more numbers, grouping symbols can be placed around any two adjacent numbers without changing the result.

Associate Property of Addition( a + b ) + c = a + ( b + c )

Associative Property of Multiplication( a x b ) x c = a x ( b x c )

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The Commutative Property and the Associative Property do not apply to Division or Subtraction.

Distributive Property of Multiplication over Addition

a ( b + c ) = ab + acTake something that is out front of the

parenthesis and distribute it through everything in the parenthesis

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Identity and InverseIdentity PropertyIn the Identity Property, whatever you start

with, you end with the same thing.The additive identity is zero

a + 0 = a 0 + a = aThe multiplicative identity is one

a x 1 = a 1 x a = a

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Identity and InverseInverse PropertyWith the Inverse Property, you end up with

the IDENTITY as the answer. The multiplicative inverse means reciprocal

a x = 1 x a = 1

The additive inverse means oppositea + -a = 0 -a + a = 0

a1

a1