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.