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Review Chapter

Review Chapter. What you Should Learn REALLY – WHAT YOU SHOULD HAVE ALREADY LEARNED If not, then you might be in too high of a course level – decide soon!!!. Henry David Thoreau - author. “It affords me no satisfaction to commence to spring an arch before I have got a solid foundation.”.

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Review Chapter

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  1. Review Chapter • What you Should Learn • REALLY – WHAT YOU SHOULD HAVE ALREADY LEARNED • If not, then you might be in too high of a course level – decide soon!!!

  2. Henry David Thoreau - author • “It affords me no satisfaction to commence to spring an arch before I have got a solid foundation.”

  3. Objective • Understand the structure of algebra including language and symbols.

  4. Objective • Understand the structure of algebra including language and symbols.

  5. Definiton • Expression – a collection of constants, variables, and arithmetic symbols

  6. Definition • Inequality – two expression separated by <, <, >, >, • -2>-3 • 4 < 5 • 4 < 4

  7. Definition • Equation – two expression set equal to each other • 4x + 2 = 3x - 5

  8. Def: evaluate • When we evaluate a numerical expression, we determine the value of the expression by performing the indicated operations.

  9. Definition • Set is a collection of objects • Use capitol letters to represent • Element is one of the items of the collection • Normally use lower case letters to describe

  10. Procedure to describe sets • Listing: Write the members of a set within braces • Use commas between • Use … to mean so on and so forth • Use a sentence • Use a picture

  11. Julia Ward Howe - Poet • “The strokes of the pen need deliberation as much as the sword needs swiftness.”

  12. Examples of Sets • {1, 2, 3} • {1, 2, 3, …, 9, 10} • {1, 2, 3, … } = N = Natural numbers

  13. Set Builder Notation • {x|description} • Example {x|x is a living United States President}

  14. Def: Empty Set or Null set is the set that contains no elements • Symbolism

  15. Symbolism – element “is an element of”

  16. Def: Subset: A is a subset of B if and only if ever element of A is an element of B • Symbolism

  17. Examples of subset • {1, 2} {1, 2, 3} • {1, 2} {1, 2} • { } {1, 2, 3, … }

  18. Def: Union symbolism: A B • A union B is the set of all elements of A or all elements of B.

  19. Example of Union of sets • A = {1, 2, 3} • B = {3, 4, 5} • A B = {1, 2, 3, 4, 5}

  20. Real Numbers • Classify Real Numbers • Naturals = N • Wholes = W • Integers = J • Rationals = Q • Irrationals = H • Reals = R

  21. Def: Sets of Numbers • Natural numbers • N = {1,2,3, … } • Whole numbers • W = {0,1,2,3, … }

  22. Integers • J = {… , -3, -2, -1, 0, 1, 2, 3, …} Naturals Wholes Integers

  23. Def: Rational number • Any number that can be expressed in the form p/q where p and q are integers and q is not equal to 0. • Use Q to represent

  24. Def (2): Rational number • Any number that can be represented by a terminating or repeating decimal expansion.

  25. Examples of rational numbers • Examples: 1/5, -2/3, 0.5, 0.33333… • Write repeating decimals with a bar above • .12121212… =

  26. Def: Irrational Number • H represents the set • A non-repeating infinite decimal expansion

  27. Def: Set of Real Numbers = R • R = the union of the set of rational and irrational numbers

  28. Def: Set of Real Numbers = R • R = the union of the set of rational and irrational numbers

  29. Def: Number line • A number line is a set of points with each point associated with a real number called the coordinate of the point.

  30. Def: origin • The point whose coordinate is 0 is the origin.

  31. Definition of Opposite of opposite • For any real number a, the opposite of the opposite of a number is -(-a) = a

  32. Definition: For All

  33. Def: There exists

  34. Bill Wheeler - artist • “Good writing is clear thinking made visible.”

  35. Def: intuitiveabsolute value • The absolute value of any real number a is the distance between a and 0 on the number line

  36. Def: algebraic absolute value

  37. Calculator notes • TI-84 – APPS • ALG1PRT1 • Useful overview

  38. George Patton • “Accept challenges, so that you may feel the exhilaration of victory.”

  39. Properties of Real Numbers • Closure • Commutative • Associative • Distributive • Identities • Inverses

  40. Commutative for Addition • a + b = b + a • 2+3=3+2

  41. Commutative for Multiplication • ab = ba • 2 x 3 = 3 x 3 • 2 * 3 = 3 * 2

  42. Associativefor Addition • a + (b + c) = (a + b) + c • 2 + (3 + 4) = (2 + 3) + 4

  43. Associative for Multiplication • (ab)c = a(bc) • (2 x 3) x 4 = 2 x (3 x 4)

  44. Distributivemultiplication over addition • a(b + c) = ab + ac • 2(3 + 4) = 2 x 3 + 2 x 4 • X(Y + Z) = XY +XZ

  45. Additive Identity • a + 0 = a • 3 + 0 = 3 • X + 0 = X

  46. Multiplicative Identity • a x 1 = a • 5 x 1 = 5 • 1 x 5 = 5 • Y * 1 = Y

  47. Additive Inverse • a(1/a) = 1 where a not equal to 0 • 3(1/3) = 1

  48. George Simmel - Sociologist • “He is educated who knows how to find out what he doesn’t know.”

  49. Order to Real Numbers • Symbols for inequality • Bounded Interval notation • *** Definition of Absolute Value • Absolute Value Properties • Distance between points on # line

  50. George Simmel - Sociologist • “He is educated who knows how to find out what he doesn’t know.”

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