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Two Kinds of Real Numbers

Two Kinds of Real Numbers. Rational Numbers Irrational Numbers. Rational Numbers. A rational number is a real number that can be written as a ratio of two integers. A rational number written in decimal form is terminating or repeating. EXAMPLES OF RATIONAL NUMBERS 16 1/2 3.56 -8

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Two Kinds of Real Numbers

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  1. Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers

  2. Rational Numbers • A rational number is a real number that can be written as a ratio of two integers. • A rational number written in decimal form is terminating or repeating. • EXAMPLES OF RATIONAL NUMBERS • 16 • 1/2 • 3.56 • -8 • 1.3333… • -3/4 A number that can be expressed in the form p/q , where p and q are integers and q can not be 0, is called a rational number.

  3. Irrational Numbers • An irrational number is a number that cannot be written as a ratio of two integers. • Irrational numbers written as decimals are non-terminating and non-repeating. • Square roots of non-perfect “squares” • Pi- īī 17

  4. Naturals, Wholes, Integers, Rationals Real Numbers Rationals Integers Wholes Naturals

  5. Real Numbers Rational numbers Irrational numbers Integers Whole numbers

  6. Rational Numbers Natural counting numbers. Natural Numbers - 1, 2, 3, 4 … Whole Numbers - Natural counting numbers and zero. 0, 1, 2, 3 … Integers - Whole numbers and their opposites. … -3, -2, -1, 0, 1, 2, 3 … Rational Numbers - Integers, fractions, and decimals. -0.76, -6/13, 0.08, 2/3 Ex:

  7. Real numbers are all the positive, negative, fraction, and decimal numbers you have heard of. They are also called Rational Numbers. IRRATIONAL NUMBERS are usually decimals that do not terminate or repeat. They go on forever. Examples: π Reminder

  8. By multiplying the numerator and denominator of a rational number by the same non zero integer, we obtain another rational number equivalent to the given rational number. This is exactly like obtaining equivalent fractions. Just as multiplication, the division of the numerator and denominator by the same non zero integer, also gives equivalent rational numbers

  9. POSITIVE AND NEGATIVE RATIONAL NUMBERS RATIONAL NUMBERS ON A NUMBER LINE RATIONAL NUMBERS IN STANDARD FORM Observe the rational numbers 3 / 5 , 2 / 7 , -3/7 A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1. To reduce the rational number to its standard form, we divide its numerator and denominator by their HCF ignoring the negative sign, if any.

  10. To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order. We can find unlimited number of rational numbers between any two rational numbers. Additive inverse Multiplicative Inverse – Reciprocal Addition and Subtraction ( like and unlike ) Multiplication Divison

  11. Four Properties • Distributive • Commutative • Associative • Identity properties of one and zero

  12. We commute when we go back and forth from work to home.

  13. Algebra terms commute when they trade places

  14. This is a statement of the commutative property for addition:

  15. It also works for multiplication:

  16. Distributive Property A(B + C) = AB + BC 4(3 + 5) = 4x3 + 4x5

  17. Commutative Propertyof addition and multiplication Order doesn’t matter A x B = B x A A + B = B + A

  18. To associate with someone means that we like to be with them.

  19. The tiger and the panther are associating with each other. They are leaving the lion out. ( )

  20. In algebra:

  21. The panther has decided to befriend the lion. The tiger is left out. ( )

  22. In algebra:

  23. This is a statement of the Associative Property: The variables do not change their order.

  24. The Associative Property also works for multiplication:

  25. Associative Property of multiplication and Addition Associative Property  (a · b) · c = a · (b · c) Example: (6 · 4) · 3 = 6 · (4 · 3) Associative Property  (a + b) + c = a + (b + c) Example: (6 + 4) + 3 = 6 + (4 + 3)

  26. The distributive property only has one form. Not one for addition . . .and one for multiplication . . .because both operations are used in one property.

  27. This is an example of the distributive property. 4(2x+3) =8x +12 2x +3 4

  28. Here is the distributive property using variables: y +z x

  29. The identity property makes me think about my identity.

  30. The identity property for addition asks, “What can I add to myself to get myself back again?

  31. is the identity element for addition. The above is the identity property for addition.

  32. The identity property for multiplication asks, “What can I multiply to myself to get myself back again?

  33. is the identity element for multiplication. The above is the identity property for multiplication.

  34. Identity Properties If you add 0 to any number, the number stays the same. A + 0 = A or 5 + 0 = 5 If you multiply any number times 1, the number stays the same. A x 1 = A or 5 x 1 = 5

  35. Example 1: Identifying Properties of Addition and Multiplication Name the property that is illustrated in each equation. A. (–4)  9 = 9  (–4) B. (–4)  9 = 9  (–4) The order of the numbers changed. Commutative Property of Multiplication The factors are grouped differently. Associative Property of Addition

  36. Solving Equations; 5 Properties of Equality

  37. 1)       26 +0 = 26                             a) Reflexive • 2)       22 · 0 = 0                                                     b) Additive Identity             • 3)       3(9 + 2) = 3(9) + 3(2)                                                 c) Multiplicative identity • 4)       If 32 = 64 ¸2, then 64 ¸2 = 32                            d) Associative Property of Mult. • 5)       32 · 1 = 32                                                     e) Transitive • 6)       9 + 8 = 8+ 9                        f) Associative Property of Add. • 7)       If 32 + 4 = 36 and 36 = 62, then 32 + 4 = 62            g) Symmetric • 8)       16 + (13 + 8) = (16 +13) + 8                                  h) Commutative Property of Mult. • 9)       6 · (2 · 12) = (6 · 2) · 12                                         i) Multiplicative property of zero • 10)  6 ∙ 9 = 6 ∙ 9                                j) Distributive • Complete the Matching Column (put the corresponding letter next to the number) • Complete the Matching Column (put the corresponding letter next to the number) • 11)    If 5 + 6 = 11, then 11 = 5 + 6                                a) Reflexive • 12)    22 · 0 = 0                                                                b) Additive Identity             • 13) 3(9 – 2) = 3(9) – 3(2)                                               c) Multiplicative identity • 14)    6 + (3 + 8) = (6 +3) + 8                                          d) Associative Property of Mult. • 15)    54 + 0 = 54                                                             e) Transitive • 16)    16 – 5 = 16 – 5                                                       f) Associative Property of Addition • 17)    If 12 + 4 = 16 and 16 = 42, then 12 + 4 = 42             g) Symmetric • 18)    3 · (22 · 2) = (3 · 22) · 2                                      h) Commutative Property of Addition • 19)    29 · 1 = 29                                                              i) Multiplicative property of zero • 20)  6 +11 = 11+ 6                                                           j) Distributive • C. • 21) Which number is a whole number but not a natural number? • a) – 2               b) 3                  c) ½                 d) 0 • 22) Which number is an integer but not a whole number? • a) – 5               b) ¼                 c) 3                  d) 2.5 • 23) Which number is irrational? • a)                   b) 4                  c) .1875                       d) .33 • 24) Give an example of a number that is rational, but not an integer.       • 25) Give an example of a number that is an integer, but not a whole number. • 26) Give an example of a number that is a whole number, but not a natural number.  • 27) Give an example of a number that is a natural number, but not an integer.

  38. Example 2: Using the Commutative and Associate Properties Simplify each expression. Justify each step. 29 + 37 + 1 Commutative Property of Addition 29 + 37 + 1 = 29 + 1 + 37 Associative Property of Addition = (29 + 1) + 37 = 30 + 37 Add. = 67

  39. Exit Slip! Name the property that is illustrated in each equation. 1. (–3 + 1) + 2 = –3 + (1 + 2) 2. 6 y 7 = 6 ● 7 ●y Simplify the expression. Justify each step. 3. Write each product using the Distributive Property. Then simplify 4. 4(98) 5. 7(32) Associative Property of Add. Commutative Property of Multiplication 22 392 224

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