MA 180 Lecture. Chapter 0. College Algebra and Calculus by Larson/Hodgkins. Fundamental Concepts of Algebra

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1 0.) Real Numbers: Order and Absolute Value Definitions: Set: is a collection of objections in mathematics Real Numbers: set of numbers used in arithmetic MA 80 Lecture Chapter 0 College Algebra and Calculus by Larson/Hodgkins Fundamental Concepts of Algebra Subset: a set such that every member is a member of a larger set. Set A is a subset of set B if all members of A are contained in B. Natural Numbers (Positive Integers): a subset of the real numbers that include the values Integers: a subset of the real numbers that include the natural numbers, zero and the negatives of all natural numbers Rational Numbers: a subset of the real numbers that include all integers and all fractions that can be written as a quotient of two integers. This includes decimals that are repeating in nature or terminate. Irrational Numbers: a subset of the real numbers such that the combined sets of rational and irrational numbers makes the set of real numbers. Irrational numbers cannot be expressed as a fraction of two integers. Classify the following numbers as natural, integer, rational, or irrational The Real Number Line and Ordering The picture that is used to represent the real numbers is the real number line. It is a horizontal line centered at zero (the origin) with positive numbers to the right and negative numbers to the left. If we are talking about all positive real numbers and zero we use the term non-negative real numbers.

2 Definition of Order on the Real Number Line If the real number a lies to the left of the real number b on the real number line, a is less than b, which is denoted by. This relationship can also be described by saying that b is greater than a and writing. The inequality means that a is less than or equal to b, and the inequality means that b is greater than or equal to a. The symbols are called inequality symbols. The double inequality less than but not equal to b. is the set of numbers x that are both greater than or equal to a and Absolute Value and Distance The absolute value of a real number is its magnitude, or its value disregarding its sign. Let a be a real number. Then the absolute value of a, is denoted by and is defined by Find the absolute value of the following numbers Distance Between Two Numbers Let a and b be real numbers. The distance between a and b is given by Distance= Find the distance between the following numbers: 9 and -4 and 7 0 and -8-2 and 2 Homework for Section 0.: Pages 8-9 #,2,7,9,40,47,52,65,68

3 0.2) The Basic Rules of Algebra Definition: Algebraic Expressions A collection of letters (called variables) and real numbers (called constants) that are combined using the operations of addition, subtraction, multiplication, and division is an algebraic expression. (Other operations can also be used to form an algebraic expression.) Examples:,, and The terms of an algebraic expression are those parts that are separated by addition. So has three terms. (Note x=+(-x)) The terms with the variable x in them are called the variable terms and the number is called the constant term of the expression. The numerical factor of a variable term is called the coefficient of the variable term. Basic Rules of Algebra Let a, b, and c be real numbers, variables, or algebraic expressions. Property Commutative Property of Addition Example Commutative Property of Multiplication Associative Property of Addition Associative property of Multiplication Distributive Property Additive Identity Property Multiplicative Identity Property Additive Inverse Property Multiplicative Inverse Property Properties of Negation Let a and b be real numbers, variables, or algebraic expressions.

4 Property Example. ( ) a a ( ) 8 2. ( a) a ( ). ( a) b ( ab) a( b) ( 4) 5 4. ( a)( b) ab ( 2)( ) 5. ( a b) ( a) ( b) ( 7 ) Properties of Zero. a 0 a and a 0 a 2. a a 4. a is undefined 0 5. Zero Factor property: If ab=0 then a=0 or b=0. Properties of Fractions a c. Equivalent fractions: if and only if ad bc. b d a a a a a 2. Rules of signs: and b b b b b. General equivalent fractions: a ac where c 0 b bc a c a c 4. Add or subtract with like denominators: b b b a c ad cb 5. Add or subtract with unlike denominators: b d bd a c ac 6. Multiply fractions: b d bd a c a d ad 7. Divide fractions: (assuming no division by zero) b d b c bc Adding and subtracting fractions To add or subtract fractions that do not have like denominators we must first find the least common denominator (LCD). We do this by factoring the denominators into their prime factorization. Then we find the LCD is the products of the prime factors, with each factor given the highest power of its occurrence in any denominator.

5 Exercises: Evaluate the following: x 8x 6 Equations An equation is a statement of equality between two expressions. Properties of Equality. Reflexive: a a 2. Symmetric: If a b then b a.. Transitive: If a b and b c, then a c. A note about calculator use: Be careful when entering negatives and parenthesis. A scientific or graphing calculator will apply order of operations if it is all entered before hitting equals. Note the difference between the minus sign and the negative sign on the calculator. Exercises: Perform the indicated operation(s). Write fractional answers in simplest form Homework for Section 0.2 Pages 8-9, # 7-0, 4-50

UNIT 4 NOTES: PROPERTIES & EXPRESSIONS

UNIT 4 NOTES: PROPERTIES & EXPRESSIONS Vocabulary Mathematics: (from Greek mathema, knowledge, study, learning ) Is the study of quantity, structure, space, and change. Algebra: Is the branch of mathematics