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M9 Nombor Nisbah

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  • Nombor Nisbah/Rational NumbersMINGGU 9

  • Nombor NisbahThe need for a closed set for division (or closure for division) before subtractionHow do the Egyptians & Romans avoided fractions?Is Z closed for addition? Subtraction? Multiplication? Division?What does closure for an operation mean?

  • ContohAre both sets of numbers found in Z ?Does closure for division exist for Z ?Hence, the need for a closed set for division

  • Ciri-ciri Asas Nombor NisbahNombor Nyata ( Real Numbers ) terdiri daripada : Nombor Bulat ( Whole Numbers )Nombor Asli ( Natural Number )Nombor Integer ( Integers )Nombor Nisbah ( Rational Numbers )Nombor Bukan Nisbah ( Irrational Numbers )

  • Nombor NisbahQ = { set of all numbers that can be written as , where a and b are integers and b 0 }Recall division by 0 is undefined.Q = {sebarang nombor yang dapat ditulis sebagai nisbah / pecahan (ratio) dua integer , dengan keadaan penyebut, tidak sama dengan 0 }

  • Nombor NisbahWritten in the fractional forma/b; a numerator; b denominatorIf both a & b are positive; is called aProper fraction if a < b;Improper fraction if a > b; andWhole number if b divides evenly into a.

  • Nombor Nisbah1.5 adalah Nombor Nisbah, boleh ditulis sebagai nisbah 3/2.7 adalah Nombor Nisbah, boleh ditulis sebagai nisbah 7/1.0.317 adalah Nombor Nisbah, boleh ditulis sebagai nisbah 317/1000.

  • Sifat Asas/Fundamental PropertyIf the greatest common factor of the numerator & denominator of a given fraction is 1, then we say the fraction is in the lowest terms or reduced. If the greatest common factor (gcf) is not 1, then divide both the numerator & denominator by this gcf using the fundamental property of fractions.

  • If is any rational number and x is any non-zero integer, then Sifat Asas / Fundamental Property of Fractionswhere x can be the gcf

  • Contoh:Reduce the given fractions

    Find the gcfDivide the fraction by the gcf

  • 24 = 23. 31 . 5030 = 21. 31. 51g.c.f. = 21. 31. 50 = 6

  • Operasi dengan Nombor NisbahSubtraction:Addition:

  • Operasi dengan Nombor NisbahDivision/Pembahagian:Multiplication/Pendaraban:

  • Contoh 2:1.2.3.4.5.

  • Contoh 3 : Kaedah PolyaJustify the rule for division of rational numbersUnderstand the problem:We dont say b 0 or d 0, because the definitionof rational numbers exclude these possibilities, But does not exclude c = 0, so this condition must be stated.

  • We are looking for the value of .

  • That means we invert the fraction we areDividing by, and then we multiply. Refer to page 191 devise & carry out the plan & look back

  • Example 4:(a)(b)

  • Example 5 (page 192)(a), (b), (c), (d)

  • Problem Set 4.4 (page 194)Q.10, 12Q19, 21bQ27, 28Q33, 34

  • Rational & Irrational numbersN = { 1, 2, 3, 4, }Z = { , -3, -2, -1, 0, 1, 2, 3, }Q = {set of rational numbers} = set of all numbers that can be written as fractionsR = {set of real numbers} = all numbers which can be represented on a number line

  • Basic properties of the rational numbersTo carry out the 4 operations on fractions:Example: (a) 5/7 + 3/4(b) 5/7 3/4(c) 5/7 x 3/4 (d) 5/7 3/4

  • Exercise 7.1 : Complete the following:a/b + c/d =a/b c/d =a/b x c/d =a/b c/d = , where c 0

  • Equivalent fractionsTwo fractions are equivalent if they represent the same numberWhen are a/b and c/d equivalent?

  • Number lineLocate the following fractions on the number line below:

  • Representation on a Cartesian PlaneRepresent a/b by the ordered pair (a, b)-4-3-2-11234The fraction 1/2 is represented by the point A (1, 2)

  • Complete the following tableRefer to Cartesian plane on page 137

    PointABCDEFCoordinates(1,2)(2,4)(2,3)(-2,-3)(3,-2)(-1,-2)Fractions1/2

  • The cardinality of the rational numbersHow many rational numbers are there?Are there as many integers as there are rational numbers?

  • One-to-one correspondenceThere is a one-to-one correspondence between the elements of two sets if we can exactly match the elements of one set with those of the other (and vice versa)If such a one-to-one correspondence exists, we say that the sets have the same cardinality.

  • Example:The sets { 1, 2, 3, 4 } and { a, b, c, d } have the same cardinality since a one-to-one correspondence can be established by pairing off their elements as follows:

    1 a 2 b 3 c 4 dCardinality is denoted by the number of elements in the set. For example the set {a, b, c, d} has a cardinality of 4

  • Question?What is the cardinality of the set of rational numbers?Refer to Figure 7.2 by using the beading methodThe cardinality of the set of rational numbers Q, is N0

  • Countable setsA set which can be put in one-to-one correspondence with the set of natural numbers N = { 1, 2, 3, 4, } is said to have cardinality aleph zero (or aleph nought), N0. (Aleph is the first letter of the Hebrew alphabet.)

  • A set with cardinality N0 is somehow not very big, since its elements can be put in one-to-one correspondence with the natural or counting numbers.Therefore, a set which is finite or has a cardinality of N0 is called countable.

  • The set of integers, Z, has cardinality N0 ,Can you explain?

  • The cardinality of the set of rational numbers, Q, is also N0 Can you explain why?

  • Irrational numbersAre there any numbers which are not rational?

  • Do Activity 7.1 (Tutorial)Refer to page 138

  • ISLRead the story of Cantor.