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Understanding Real Numbers in Algebra

Learn about sets, integers, operations, graphs, exponents, equations, and solving problems using real numbers in intermediate algebra. Master basic properties and skills essential for further algebraic studies. Check your understanding with helpful tips and examples.

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Understanding Real Numbers in Algebra

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  1. Intermediate Algebraby Gustafson and Frisk Chapter 1 A Review of Basic Algebra

  2. Section 1.1: The Real Number System SETS: collections of objects. Integers Positive Numbers Negative Numbers Even Numbers Odd Numbers • Natural Numbers • Whole Numbers • Rational Numbers • Irrational Numbers • Real Numbers • Use { } {x | x > 5}is read “the set of all x such that x is greater than 5”

  3. Section 1.1: The Real Number System GRAPHS: plot on the number line. Individual numbers are dots -3 -2 -1 0 1 2 3 4

  4. Section 1.1: The Real Number System GRAPHS: plot on the number line. Intervals including end points [ [ ] -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4

  5. Section 1.1: The Real Number System GRAPHS: plot on the number line. Intervals not including end points ( ( ) -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4

  6. Section 1.2: Arithmetic & Properties of Real Numbers OPERATIONS: • Addition • Subtraction (the same as adding a number with the opposite sign) • Multiplication • Division (the same as multiplying by the reciprocal)

  7. Section 1.2: Arithmetic & Properties of Real Numbers ADDITION: • Addends that have the same signs • Add absolute values • Keep the sign of the addends • Addends that have opposite signs • Subtract absolute values • Keep the sign of the addend with the largest absolute value

  8. Section 1.2: Arithmetic & Properties of Real Numbers MULTIPLICATION: • Multiply absolute values • If the factors have the same signs, the product is positive • If the factors have opposite signs, the product is negative

  9. Section 1.2: Arithmetic & Properties of Real Numbers STATISTICS: measures of central tendency • Mean • Median • Mode

  10. Section 1.2: Arithmetic & Properties of Real Numbers Properties: • Associative – addition, multiplication • Commutative – addition, multiplication • Distributive – multiplication is distributed over additiona (b + c) = ab + ac

  11. Section 1.2: Arithmetic & Properties of Real Numbers Identities: • Addition – zero • Multiplication – one Inverses: • Addition – opposites • Multiplication – reciprocals

  12. Section 1.3: Definition of Exponents EXPONENTS: repeated multiplication • In the expression: ana is the base and n is the exponent • Exponents are NOT factors • Means to multiply “a” n times

  13. Section 1.3: Definition of Exponents ORDER OF OPERATIONS: If an algebraic expression has more than one operation, the following order applies: Clear up any grouping. Evaluate exponents. Do multiplication and division from left to right. Do addition and subtraction from left to right.

  14. Section 1.5: Solving Equations Algebraic Expression vs. Equation • Expressions: a combination of numbers and operations • Equation: a statement that two expressions are equal

  15. Section 1.5: Solving Equations EXPRESSIONS: • Terms • Like terms • When multiplying, the terms do not need to be alike • Can only add like terms!

  16. Section 1.5: Solving Equations TO SOLVE AN EQUATION IN ONE VARIABLE: • If you see fractions, multiply both sides by the LCD. This will eliminate the fractions. • Simplify the algebraic expressions on each side of the equal sign (eliminate parentheses and combine like terms). • Use the addition property of equality to isolate the variable terms from the constant termson opposite sides of the equal sign. • Use the multiplication property to make the coefficient of the variable equal to one. • Check your results by evaluating.

  17. Section 1.5: Solving Equations TYPES OF EQUATIONS: • CONDITIONAL: if x equals this, then y equals that. • IDENTITY: always true no matter what numbers you use. • CONTRADICTION: never true no matter what numbers you use. • FORMULAS: conditional equations that model a relationship between the variables.

  18. Section 1.6 & 1.7: Solving Problems, Applications TYPES OF PROBLEMS: • Geometry • Percent • Physics (forces) • Uniform motion • Mixtures • Good ‘ole common sense analysis

  19. Chapter 1: Basic Algebra Review SUMMARY: • KNOW YOUR VOCABULARY! You can’t follow directions if you don’t know what the words in the instructions mean. • Memorize the processes and the properties. • I will provide formulas for your reference. • Ask questions if you are unsure. • Always check your work to make sure that you answered the question, and that your answer is reasonable.

  20. This powerpoint was kindly donated to www.worldofteaching.com http://www.worldofteaching.com Is home to well over a thousand powerpoints submitted by teachers. This a free site. Please visit and I hope it will help in your teaching

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