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PROPERTIES OF REAL NUMBERS

Learn about subsets of real numbers, rational and irrational numbers, ordering numbers, properties of addition and multiplication, and absolute value. Includes examples and explanations to help improve your understanding.

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PROPERTIES OF REAL NUMBERS

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  1. PROPERTIES OF REAL NUMBERS ¾ .215 -7 PI 1

  2. Subsets of real numbers – REVIEW Natural numbers numbers used for counting 1, 2, 3, 4, 5, …. Whole numbers the natural numbers plus zero 0, 1, 2, 3, 4, 5, … Integers the natural numbers ( positive integers ), zero, plus the negative integers

  3. …,-4, -3, -2, -1, 0, 1, 2, 3, 4, … Rational numbers numbers that can be written as fractions decimal representations can either terminate or repeat Examples: fractions: 7/5 -3/2 -4/5 Any whole number can be written as a fraction by placing it over the number 1 8 = 8/1 100 = 100/1

  4. terminating decimals ¼ = .25 2/5 = .4 Repeating decimals 1/3 = .3 2/3 = .6 These will always have a bar over the repeating section. Irrational numbers Cannot be written as fractions Decimal representations do not terminate or repeat

  5. if the positive rational number is not a perfect square, then its square root is irrational Examples: Pi - non-repeating decimal 2 - not a perfect square

  6. THE REAL NUMBERS Rational numbers Irrational numbers Integers Whole numbers Natural numbers

  7. Graphing on a number line - 2 .3 -2 ¼ Tip: Best to put them as all decimals Put the square root in the calculator and find its equivalent -1.414… .333……… -2.25 -3 -2 -1 0 1 2 3

  8. Ordering numbers • Use the < , >, and = symbols • Compare - .08 and - .1 • Here again for square roots put them in the calculator and get their equivalents • .08 = -.282842712475 - .1 = -.316227766017 • So: - .1 < - .08 or - .08 > - .1

  9. Properties of Real Numbers Opposite or additive inverse sum of opposites or additive inverses is 0 Examples: 400 4 1/5 - .002 - 4/9 -400 Additive inverse of any number a is -a 4/9 - 4 1/5 . 002

  10. Reciprocal or multiplicative inverse • product of reciprocals equal 1 • Examples: • 4 1/5 - .002 - 4/9 • 1/400 • Multiplicative inverse of any number a is 1/a - 9/4 5/21 - 500

  11. Other Properties: Addition: Closure a + b is a real number Commutative a + b = b + a 4 + 3 = 7` 3 + 4 = 7 numbers can be moved in addition Associative (a + b) + c = a + (b + c) (1 + 2) + 3 = 6 1+ (2 + 3) = 6 3 + 3 = 6 1 + 5 = 6 the order in which we add the numbers does not matter in addition

  12. Identity a + 0 = a 7 + 0 = 7 when you add nothing to a number you still only have that number Inverse a + -a = 0 7 + -7 = 0

  13. Multiplication Closure ab is a real number Commutative ab = ba 6(4) = 24 4 (6) = 24 When multiplying the numbers may be switched around, will not affect product Associative (ab)c = a(bc) The order in which they are multiplied does not affect the outcome of the product

  14. (3*4)5 = 60 3(4*5) = 60 12(5) = 60 3(20) = 60 Identity a * 1 = a One times any number is the number itself 7 * 1 = 7 Inverse a * 1/a = 1 Product of reciprocals is one 7 * 1/7 = 7/7 = 1

  15. DISTRIBUTIVE Property Combines addition and multiplication a(b + c) = ab + ac 2(3 + 4) = 2(3) + 2(4) 6 + 8 14

  16. ABSOLUTE VALUE Absolute value is its distance from zero on the number line. Absolute value is always positive because distance is always positive Examples: -4 = 0 = -1 * -2 = 4 0 2

  17. Assignment Page 8 – 9 Problems 34 – 60 even

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