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Unit 1 – Review of the Real Number System Unit 1.1 – Real Number System Sets and Set Notation A set is a collection of objects called the elements or members of the set. We use braces { } to enclose the elements of the set. Sets can be either finite or infinite. {1, 2, 4} {2, 4, 6, 8,…} Set builder notation: {x | x has some property P}. This is read as “the set of all elements x such that x has some given property of P”. Examples: {x | x is a whole number greater than 5} {6, 7, 8,…} {x | x is a natural number between 3 and 15 {4, 5, 6, 7,…, 14} Sets of Numbers: Natural numbers {1, 2, 3, 4,…} (the counting numbers) Whole numbers {0, 1, 2, 3, 4,…} Integers {…,-3, -2, -1, 0, 1, 2, 3,…} p Rational numbers | p and q are integers, q 0 q Irrational numbers {x | x is a real number that is not rational} Real numbers {x | x is represented on a number line} 9 2 In this set 8, 2, , 0,.5, ,1.12, 3, 2 , identify: 64 3 Integers Rational numbers Irrational numbers Real numbers Number Line – A good way to get a picture of a set of numbers is by using a number line. To construct find 0 and then place 1 to the right. Use the scale established by where the 1 is to place other integers, with positive numbers to the right and negative numbers to the left. Each number is the coordinate of the point it labels and the point is the graph of the number. The set of rational and irrational numbers make up the real numbers. Every point on a number line is associated with a real number and every real number is associated with a point on the number line. 582676199 Page 1 of 5 Unit 1 – Review of the Real Number System Additive Inverse For any real number a, the number –a is the additive inverse of a. Also; for any real number a, -(-a) = a. Absolute Value The absolute value of a number a, written a , is the distance on the number line from 0 to a. Because absolute value represents distance, and distance is always positive (or 0), absolute value is always positive (or 0). Absolute value is used to calculate greatest or least change. We don’t care if the change is negative or positive. Inequalities on a Number Line < means “is less than”. > means “is greater than”. On a number line, a < b if a is to the left of b and a > b if a is to the right of b. Other inequality symbols: means “is not equal to”. means “is less than or equal to”. means “is greater than or equal to”. Unit 1.2 – Operations on Real Numbers Adding Real Numbers: The answer of an addition problem is called the sum. Like signs: to add two numbers with the same sign, add their absolute values. The sign of the answer is the same as the sign of the two numbers. Unlike signs: subtract the smaller absolute value from the larger. The sign of the answer is the same as the number with the largest absolute value. Subtracting Real Numbers: For all real numbers a and b, a b a (b ) in words, change the sign of the second number (the subtrahend) and add. (note: the first number is called the minuend.) The answer is called the difference. When problems contain both addition and subtraction, work from left to right. Do inside brackets and parentheses first. Multiplying Real Numbers: The answer to a multiplication problem is called the product. Like signs: the product of two numbers with the same sign is positive. Unlike signs: the product of two numbers with different signs is negative. Reciprocal of a number: 582676199 Page 2 of 5 Unit 1 – Review of the Real Number System (We defined subtraction in terms of addition. We will define division in terms of multiplication.) This is based on the idea of a Multiplicative Inverse or Reciprocal. Two numbers are reciprocals if their product is 1. 1 The reciprocal of a nonzero number a is . Note that a number and its reciprocal have the same a sign. Dividing Real Numbers: For all real numbers a and b ( b 0 ), a b Remember: 0 6 is undefined, but 0 . 6 0 a 1 a b b Equivalent forms of fractions: The fractions x , x , and x are equal (assuming y 0). y y y The fractions x and x are equal. y y Unit 1.3 – Exponents, Roots, and Order of Operations Exponents: Use exponents to write products of repeating factors. 2 2 2 2 2 25 where 5 is the exponent and 2 is the base. Exponential Expression: If a is a real number and n is a natural number, a n a a a a (n factors of a ) Identifying Exponents and Bases: 36 6 is the exponent and 3 is the base. 7 2 2 7 is the exponent and is the base. 5 5 2 3 3 is the exponent and -2 is the base. 2 3 is the exponent and 2 is the base. (Caution: since there are no parentheses, the exponent only applies to the 2 and this exponential expression can be expanded to be 1 2 2 2 1 8 8 Common terminology: 62 is commonly called “6 squared”. 63 is commonly called “6 cubed”. Exponents are also called ‘powers’. (i.e. raised to the 5th power) 3 Square Roots: 582676199 Page 3 of 5 Unit 1 – Review of the Real Number System The opposite of squaring a number (i.e. 62 ) is taking its square root. For example, a square root 2 of 36 is 6 because 62 equals 36. Another square root of 36 is -6 because 6 equals 36. So 36 has two square roots. The positive or principal square root is written with the radical sign . Example: the principal square root of 36 is written 36 6 . The principal square root is usually referred to as the square root. The negative square root is written 36 6 . The square root of any nonzero real number is positive, therefore the square root of a negative number (i.e. 36 ) is not a real number. Order of Operations: 1. Work separately above and below any fraction bar. 2. If grouping symbols such as parentheses, square brackets or absolute value bars are present, start with the innermost set and work outward. 3. Evaluate all powers, roots and absolute values. 4. Do any multiplications or divisions, working left to right. 5. Do any additions or subtractions, working left to right. Algebraic Expressions: Any collection of numbers, variables, operation symbols, and grouping symbols is an algebraic expression. 8 Examples: 6ab , 5xy 12q 4 g , 2 x 3x 3 12y 11 We evaluate algebraic expressions by substituting values for the variables. Different values give different results. Unit 1.4 – Properties of Real Numbers Distributive property. Inverse properties. Identity properties. Commutative property. Associative property. Multiplication property of 0. Distributive property: For any real numbers a, b, and c, a b c ab ac and b c a ba ca . This can also be extended to more than two numbers, a b c d ab ac ad . Inverse properties: For any real number a, there is a single real number –a such that a a 0 and a a 0 . 582676199 Page 4 of 5 Unit 1 – Review of the Real Number System For any nonzero real number a, there is a single real number a 1 1 1 and a 1 . a a 1 such that a Identity properties: For any real number a, a 0 0 a a and a 1 1 a a . Expressions such as 12m, 5x , and 3xyz 3 are examples of terms. A term is a number or the product of a number and one or more variables. Terms with exactly the same variables raised to exactly the same powers are called like terms. Examples: 5p and 21p are like terms. 5x and 5x 2 are unlike terms. The numerical factor in a term is called the numerical coefficient or just the coefficient. Example: in the term 9egt 5 , 9 is the coefficient. When working with algebraic expressions, like terms (and only like terms) can be combined (combining like terms). Doing this frequently requires us to use the commutative and/or associative properties. Commutative property: For any real numbers a, b, and c, a b b a and ab ba . Associative property: For any real numbers a, b, and c, a b c a b c and a bc ab c . Multiplication property of 0: For any real number a, a 0 0 and 0 a 0 . 582676199 Page 5 of 5