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 ExpressionsEquationsInequalities (8) - 4 X y = (3) = 10 X - 4 = 13 11= 3 + 2m > 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 b a

Multiplication and Division Symbols Ways to MultiplyWays to Divide 5 x (7) 5 7 With variables... 5 x 5axy / 4 48

{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 numberscounting 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 numberscounting numbers irrational numbers The rational numbers combined with the irrational numbers make up the set of real numbers. REAL NUMBERS

Sets of Numbers Reals RationalsIrrationals - any number which can be written as a fraction., 7, Fractions/Decimals Integers, , … -3, -2, -1, 0, 1, 2, 3... Negative IntegersWholes … -3, -2, -10, 1, 2, 3... Zero 0 Naturals 1, 2, non-terminating and non-repeating decimals

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 SP E M D A S arenthesis xponentsultiplicationivision dditionubtraction 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

PEMDAS parenthesis – combine these first PEMDAS exponents – apply the exponent now PEMDAS complete multiplication and division, left to right PEMDAS complete addition and subtraction, left to right

Simplify the expressions. 1) 6 + 5(8 - 2) 2) ) )

Order of Operations 1) 6 + 5(8 - 2) Parenthesis Exponents 2) (6) = = 77 Multiply / Divide 3) = 32 Add / Subtract 4) = 15

COMMUTATIVE PROPERTY The operations of both addition and multiplication are commutative When adding, you can “commute” or trade the terms places When multiplying, you can “commute” or trade the factors places

ASSOCIATIVE PROPERTY When adding, you can “associate” and add any terms first and then add the other term. 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 The distributive property also holds for a factor that is multiplied on the left.

A positive times a negative is NEGATIVE A negative times a positive is NEGATIVE The negative of a negative POSITIVE CAUTION: Remember that the value for a and/or b could also be positive or negative. A positive divided by a negative or A negative divided by a positive is NEGATIVE 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 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 units away from 0

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