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Transcript
MATHEMATICS 3rd ESO NUMBERS: RATIONAL NUMBERS OBJECTIVES: Know the necessity of the different sets of numbers Know the definition, utility and representation of rational numbers Find equivalent fractions, simplify a given one and compare several ones • Operate with fractions •
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CONTENTS: •
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Rational numbers, fractions, rational numbers representation Equivalent fractions and methods to obtain them. Irreducible fraction Reciprocal of a fraction VOCABULARY: Conjunto: set Números naturales: whole numbers Números enteros: integers Números racionales: rational numbers Números irracionales: irrational numbers Fracción: fraction Numerador: numerator Denominador: denominator Simplificar: simplify Fracciones equivalents: equivalent fractions Metro: metre Máximo común múltiplo: Greatest Common Factor (GCF) Mínimo común múltiplo: Least Common Multiple (LCM) Mínimo común denominador: Least Common Denominator (LCD) Fracción inversa: reciprocal of a fraction The need for numbers
1. The necessity of counting The first numbers appeared as an answer to the necessity of counting. These numbers are the set of whole numbers: {1, 2, 3, 4, …}. At first, the number 0 did not belong to this set. 2. The necessity of subtracting Afterwards, another necessity related to accounting operations appeared. The addition and multiplication of whole numbers are always possible but it is not always possible to subtract them. For instance, the operation b-‐a is only possible if b>a. Otherwise, a number which is not a whole number appears, a negative number. Another example is the operation a-‐a, where we find no result in the set of whole numbers, number 0. IES GONZALO NAZARENO Author: José Antonio Santos-‐Mora Murillo 1 MATHEMATICS 3rd ESO Therefore, another set of numbers had to appear in order to fulfil these necessities. This is the set of integers: {..., -‐4, -‐3, -‐2, -‐1, 0, 1, 2, 3, 4, …}. 3. The necessity of measuring There was still another necessity, which could not be met using the existing numbers: the necessity of measuring lengths, surfaces, volumes, etc. The problem of measuring is solved taking a measuring unit and counting how many times this unit is the length of what we want to measure. As an example, let’s consider a piece of wood of the following length: Let’s suppose the following length is our measuring unit, a metre: Therefore, the piece of wood measures three metres: In case we want to measure a piece of wood smaller than the metre: We divide the metre into smaller equal divisions, for instance, n divisions. And then, we count how many of them are necessary to measure the piece of wood. The line measures three of the four divisions of the metre. This is expressed as follows: ¾ metres Many centuries after this way of measuring was invented, the symbol p/q was accepted as a number and not just as a way of measuring quantities. 4. Irrational numbers: Irrational numbers are those that cannot be expressed like a rational number, p/q. For instance 2 What is a rational number?
The set of rational numbers consists of all numbers that can be expressed as !
, where p and q are integers and q ≠ 0 !
IES GONZALO NAZARENO Author: José Antonio Santos-‐Mora Murillo 2 MATHEMATICS 3rd ESO ! !
!
For instance: ! , !. However, ! is not a rational number because the denominator cannot be 0. If the rational number is written like that, it is called a fraction, where the top number is called numerator and the bottom number is called denominator. What is a fraction for? Fractions representation on a number
line
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Representation of a fraction As previously said, a fraction like p/q means that we take p parts of q equal divisions of something. For instance, if we divide a cake into two halves and take one of them, this would be represented as: !
!
!
And if we want to have of a cake, we have to cut two cakes into two halves and !
take 3 of them: !
!
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Representation of a fraction on the number line We find two cases like the previous ones: !
Let’s use as an example of fraction where numerator < denominator: !
1. The given number is positive so it will lie to the right. 2. As it is smaller than 1, we divide the space between 0 and 1 into 2 equal parts. Each part represents ½, so we take the first of them: -‐ -‐2 -‐1 0 +1 +2 + -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 1/2 2/2 !
Let’s use as an example of fraction where numerator > denominator: !
1. The given number is positive so it will lie to the right. IES GONZALO NAZARENO Author: José Antonio Santos-‐Mora Murillo 3 MATHEMATICS 3rd ESO 2. As it is bigger than 1, we write it as an integer plus a fraction smaller than one, 3/2 = 1 + ½. Now, we divide the space between 1 and 2 into 2 equal parts. Each part represents ½, so we take the first of them: -‐ -‐2 -‐1 0 +1 +2 + -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐|-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 3/2 4/2 In case the number is negative the number will lie to the left of 0. Exercises on web page: http://recursostic.educacion.es/secundaria/edad/3esomatematicas/3quincena1/
index_3quincena1.htm Simplifying fractions
Equivalent fractions •
Two fractions are equivalent when they represent the same value. For instance, let’s have a look at the following divisions of the same length 1/3 2/3 3/3 1/6 2/6 3/6 4/6 5/6 6/6 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12 12/12 If we look at the previous divisions we see that 1/3 = 2/6 =4/12 because they represent the same length. The same happens with 2/3, 4/6 and 8/12 or 3/3, 6/6 and 12/12 and 6/6. Those pairs are equivalent fractions. Therefore, we see how different rational numbers can be expressed with different fractions, in fact, with an infinite number of fractions. •
How to know if two fractions are equivalent To know if two fractions are equivalent, •
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Multiply the numerator of the first one by the denominator of the second one Multiply the denominator of the first one by the numerator of the second IES GONZALO NAZARENO Author: José Antonio Santos-‐Mora Murillo 4 MATHEMATICS 3rd ESO one If both results are the same, the fractions are equivalent. For instance, 3 6
= !"#$%&" 3 · 4 = 2 · 6 2 4
Exercises on web page: http://recursostic.educacion.es/secundaria/edad/3esomatematicas/3quincena1/
index_3quincena1.htm •
How to find a fraction equivalent to a given one Looking at the previous example, it is easy to see that: Multiplying or dividing both the numerator and the denominator of a fraction by the same number, we find an equivalent fraction. Exercises on web page: http://recursostic.educacion.es/secundaria/edad/3esomatematicas/3quincena1/
index_3quincena1.htm •
How to simplify a fraction To simplify a fraction divide both the numerator and denominator of the fraction by the Greatest Common Factor (GCF) of both of them. For instance, let’s simplify the fraction 24/108. The GCF of 24 and 108 is 12, so: 24
2
= 108 9
The simplest equivalent fraction of 24/108 is 2/9. This fraction is called irreducible fraction. Exercises on web page: http://recursostic.educacion.es/secundaria/edad/3esomatematicas/3quincena1/
index_3quincena1.htm Comparing fractions
IES GONZALO NAZARENO Author: José Antonio Santos-‐Mora Murillo 5 MATHEMATICS 3rd ESO Given several fractions, how can we order them from the smaller to the larger one? Let’s try and solve the following problem. Our mother offers us 3/4 or 4/5 of a cake. We want to choose the best option, the more cake we have, the better. At first, it is hard to know if dividing the cake into 4 equal parts and taking 3 is better than dividing the cake into 5 equal parts and taking 4. To solve the problem, we can find two equivalent fractions with the same denominator: 2 4 6
8
10
= = =
=
= ⋯ 3 6 9 12 15
4
8
12
=
=
= ⋯ 5 10 15
Taking the fractions with the same denominator (which is the Least Common Multiple, LCM, of the original denominators), it is obvious 10/15 is smaller than 12/15. Therefore, 2/3 < 4/5. To compare fractions: 1. Find the Least Common Multiple of the denominators 2. Make equivalent fractions with it 3. Compare the numerators Exercises on web page: http://recursostic.educacion.es/secundaria/edad/3esomatematicas/3quincena1/
index_3quincena1.htm Operations with fractions
•
Rule for addition of fractions To add fractions, find equivalent ones with the same denominator (the LCD) and add the numerators. For instance, 3 2 15 4
19
+ =
+
= 2 5 10 10 10
Exercises on web page: IES GONZALO NAZARENO Author: José Antonio Santos-‐Mora Murillo 6 MATHEMATICS 3rd ESO http://recursostic.educacion.es/secundaria/edad/3esomatematicas/3quincena1/
index_3quincena1.htm •
Rule for subtraction of fractions To subtract fractions, find equivalent ones with the same denominator (the LCD) and subtract the numerators. For instance, 3 2 15 4
11
− =
−
= 2 5 10 10 10
Exercises on web page: http://recursostic.educacion.es/secundaria/edad/3esomatematicas/3quincena1/
index_3quincena1.htm •
Rule for multiplying of fractions To multiply fractions, multiply the numerators and the denominators. For instance, 3 2 3·2
6
· =
= 2 5 2 · 5 10
Exercises on web page: http://recursostic.educacion.es/secundaria/edad/3esomatematicas/3quincena1/
index_3quincena1.htm •
Rule for dividing fractions To get the reciprocal of a fraction, to turn it upside down. For instance, IES GONZALO NAZARENO Author: José Antonio Santos-‐Mora Murillo 7 MATHEMATICS 3rd ESO !
!
The reciprocal of ! is ! To divide fractions, multiply the first one by the reciprocal of the second one. For instance, 3 2 3 · 5 15
: =
= 2 5 2·2
4
Exercises on web page: http://recursostic.educacion.es/secundaria/edad/3esomatematicas/3quincena1/
index_3quincena1.htm IES GONZALO NAZARENO Author: José Antonio Santos-‐Mora Murillo 8 MATHEMATICS 3rd ESO Reference web pages: http://en.wikipedia.org/wiki/Fraction_(mathematics)#Common.2C_vulgar.2C_or_
simple_fractions http://recursostic.educacion.es/secundaria/edad/2esomatematicas/2quincena2/
index2_2.htm http://recursostic.educacion.es/secundaria/edad/3esomatematicas/3quincena1/
index3_1.htm IES GONZALO NAZARENO Author: José Antonio Santos-‐Mora Murillo 9
