REAL NUMBERS

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. {1, 2, 3, 4, . . . } If we include 0 we have the set of whole numbers. {0, 1, 2, 3, 4, . . . } { …, -3, -2, -1, 0,1, 2, 3, . . . } Include the opposites of the whole numbers and you have the set of integers.

Whole numbers are a subset of integers and counting numbers are a subset of whole numbers. rational numbers integers whole numbers counting numbers This means to divide one integer by another or “make a fraction” If we express a new set of numbers as the quotient of two integers, we have the set of rational numbers

There are numbers that cannot be expressed as the quotient of two integers. These are called irrational numbers. REAL NUMBERS rational numbers integers irrational numbers whole numbers counting numbers The rational numbers combined with the irrational numbers make up the set of real numbers.

Translating English to Math sum of two numbers difference between two numbers The product of two numbers the quotient of two numbers is a + b a - b ab b a =

P E M D A S ORDER OF OPERATIONS arenthesis ultiplication xponents 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. Complete multiplication and division from left to right Complete addition and subtraction from left to right Apply exponents Do any simplifying possible inside of parenthesis starting with innermost parenthesis and working out arenthesis xponents ultiplication ivision ddition ubtraction P E M D A S

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

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 The operations of both addition and multiplication are associative 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.

DISTRIBUTIVE PROPERTY The operation of multiplication distributes over addition The distributive property also holds for a factor that is multiplied on the left.

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