3º ESO 1

UNIT 1 . RATIONAL NUMBERS. 1. Vocabulary A is a number like this: a Numerator Fraction line b Denominator b ≠ 0

The numerator and the denominator are numbers.

The fraction line is read “over” 5 Is a fraction? No, because in a fraction, the denominator must be a non-zero number. 0 a a Negative fractions. The opposite of a fraction is the fraction − , and it is negative. b b But there are different notations: a − a a − = = b b − b Examples: − 4 4 3 3 = − , = − are negative fractions. 5 5 − 7 7 − 2 2 1 1 − 5 5  7  7 And notice: = , − = , − = , − −  = − 3 3 − 3 3 6 6  9  9

Remark 1. A fraction means a part of a whole: 4 The fraction means “four parts out of five” 5

Remark 2. A fraction can be seen as a number, but in context we only use the decimal form when we need to figure out how big or small is the number 4 can be considered as a 4:5 to get its corresponding decimal, 0.8. 5

Never use decimal numbers when working with fractions

Susana López 3º ESO 2

3. Equivalent fractions

Equivalent fractions are fractions that are equal in value, even if they look different. a c Two fractions = are said to be equivalent if a ⋅ d = b ⋅ c b d

10 6 Example: = because 10 ⋅9 = 15⋅ 6 15 9

The way to get equivalent fractions to a given one is to multiply or divide its numerator and its denominator by the same number. Verbs: to amplify and to simplify or to cancel down.

Examples: 5 ×4 20 40 ÷20 2 = ← Amplification = ← Simplification 7 ×4 28 60 ÷20 3

This is used to change the denominator of a fraction, getting a new fraction with the same value.

a) To simplify you can divide top and bottom by the same number until they won’t go any further. Example: 16 divide 2 divide 1 = = 48 by 8 6 by 2 3

b) Or you can do it at once, by factorizing firstly and cancelling down.

Examples: 66 11⋅ 2 ⋅3 11⋅ 2 ⋅3 cancel 1 1 = = = = 264 11⋅8⋅3 11⋅ 23 ⋅3 down 22 4

22 ⋅ 25 ⋅35 ⋅36 2⋅11⋅52 ⋅5⋅7 ⋅ 22 ⋅32 7 7 = = = 125⋅ 24 ⋅33 ⋅8 53 ⋅ 23 ⋅3⋅11⋅3⋅ 23 23 8

Irreducible fraction: (or fraction in lowest terms) is a fraction in which the numerator and denominator are smaller than those in any other equivalent fraction.

Susana López 3º ESO 3

4. OPERATIONS WITH FRACTIONS

When using integer numbers and fractions together, remember that an integer has 7 always denominator 1 and it is also a fraction. E. g.:7 = . 1 Always simplify before operating.

4.1. Multiplication of fractions a c a ⋅ c ⋅ = Multiply numerators together, multiply denominators together. b d b ⋅ d Remember: of means multiply. 4 4 4 30 120 Examples: of 30 is equivalent to ⋅ 30 = ⋅ = = 24. 5 5 5 1 5 1 9 1 9 of of 1500 = ⋅ ⋅1500. 3 8 3 8

4.2. Division of fractions a c a d : = ⋅ Change the division into a multiplication by turning the second fraction b d b c upside down and multiply both fractions together, that is, multiply by the reciprocal. a a c a ⋅ d b a ⋅ d The important is: : = = b d b ⋅ c c b ⋅ c d

4.3. Addition and subtraction of fractions

We have two possibilities a b a + b A) Same denominator: + = c c c Examples 1 5 1 3 5 1+ 3 + 5 9 18 3 18 + (−3) 15 simplify 3 +1+ = + + = = = 3 − = = = 3 3 3 3 3 3 3 25 25 25 25 5

B) Different denominators: first get the same denominator (the lcm of the denominators) and after, do the same as in A. 3 2 15 8 15 + 8 23 Example: l.c.m.(4, 5) = 20 + = + = = 4 5 20 20 20 20

WHEN MORE THAN TWO FRACTIONS ARE INVOLVED, WORK CAREFULLY WITH THE SIGNS AND BRACKETS.

Susana López 3º ESO 4

To know more:

5. Classification of fractions

4 1. Proper fractions: the numerator is less than the denominator. E.g. . 5 9 2. Improper fractions: the numerator is greater than the denominator. E.g. . 7 63 a) Integer numbers: the numerator is a multiple of the denominator. E.g. = 7. 9 −15 − 3 We always write them in the whole way. Example: = −3, not . 5 1 1 1 b) Mixed numbers: formed by an integer plus a proper fraction. 2 means 2 + . 2 2 The integer number is called its whole part.

Take into account:  Proper fractions: their absolute value is less than one. This means that these fractions are between − 1 and 1. (absolute value is the distance to zero)

 Improper fractions: their absolute value is greater than one. This means that these fractions are numbers whose distance to zero is greater than one.

 Mixed numbers can always be expressed as improper fractions.

Examples: 1 1 15 +1 16 2 2 35 + 2 37 5 = 5 + = = 7 = 7 + = = 3 3 3 3 5 5 5 5 Check these examples using the fraction button on your calculator.

 Conversely, an improper fraction may be expressed as a mixed number:

17 17 5 17 2 2 = 3 + = 3 5 2 3 5 5 5

The is the whole part. The remainder is the numerator of the proper fraction.

Check these results on your calculator.

(Quotient: cociente. Remainder: lo que queda, resto. to remain: seguir, continuar; quedarse, permanecer, (= to be left) quedar.)

5 26 52 Exercise: Convert into mixed numbers: , and . 4 9 3

Susana López 3º ESO 5

6. Decimal numbers

Some vocabulary: Digit: dígito. Whole part: parte entera. Decimal part: parte decimal Decimal places: lugares o cifras decimales. Decimal point: la coma del decimal. Terminating decimal: decimal exacto. Recurring or : decimal periódico. Recurring cycle: periodo.

Classification of decimal numbers

1. Terminating    i) pure  a) recurring decimals 2.Non - terminating decimals ii) mixed    b) non - terminating non - recurring decimals

• From fractions to decimals: Just divide numerator over denominator and get some decimal digits.

• From decimals to fractions: (Fracción generatriz de un número decimal) 1) Terminating decimals: decimal numbers with a fixed number of decimal places. To get the fraction form, put in the numerator all the digits of the number −without the decimal point−, and the denominator is 1 followed by as many zeros as decimal places has the number. Simplify if possible. Examples: 15 3 36 9 2512 314 1.5 = = , 0.36 = = , 2.512 = = (simplify always!) 10 2 100 25 1000 125

2) Decimal numbers with infinitely many decimal digits: a) Recurring decimals: Those having a recurring cycle in decimal part. i) Pure recurring decimal numbers: recurring cycle after the decimal point. Conversion into fraction: Examples: Convert into a fraction Convert into a fraction: N = 2.565656…  x = 1.3.

Susana López 3º ESO 6

ii) Mixed recurring decimal numbers: recurring cycle after non-recurring digits. Conversion into fraction:

Examples: a) Convert into a fraction m = 2.35151515…

b) Convert into a fraction de number: x = 3.2655555…

c) Convert into a fraction de number: a = 1.2777777…

b) Non-terminating non-recurring decimals. Their decimals go on forever without repeating. These numbers will be studied on next unit. E.g.: Pi, 2 0.41441444144441…

Susana López 3º ESO 7

7. Rational numbers.

Definition. Rational numbers are numbers that can be expressed as fractions.

Examples: 5 a) Fractions like are rational numbers. 6 3 − 5 b) Every integer number is also a : 3 = , − 5 = 1 1 25 5 c) Terminating decimals are a rational numbers, as we know: 2.5 = = . 10 2 d) 16 is a rational number because 16 = 4.

Since all recurring decimals can be written as fractions: terminating and recurring decimals are rational numbers. (Since= As: como…- causa)

Note: (set = conjunto) The set of natural numbers is denoted by N. N = {1, 2, 3, …} The set of integer numbers is denoted by Z: Z= {…−3, −2, −1, 0, 1, 2, 3, …} The set of rational numbers is denoted by Q.

Any natural number or integer is also a rational number, i. e. these sets are included in the set of rational numbers. Here we have a layout for the relationship between the different sets of numbers:

Q −1, −2, ... Z N 0, 1, 2 ... − 5 1 1 , , ... 9 2 3

In set notation we write this as N⊂ Z ⊂ Q. The symbol ⊂ means “included in”.

Examples: a) 25 is rational , it equals 5 which is an integer. 4 b) is a rational number because it is a fraction. 9 4 2 c) 0.4 is rational, because it equals = . 10 5 d) 10 is not a rational number.

Susana López 3º ESO 8

EXERCISES 1. Simplify:

81 52 ⋅35 ⋅ 23 34 ⋅5−7 ⋅ 2 a) d) g) 27 5⋅33 ⋅ 22 25 ⋅3−6 ⋅54 42 b) 212 189 e) 7 ⋅34 7980 c) 25 ⋅35 ⋅ 75 6118 f) 15 ⋅ 21⋅10

2. Cancel these algebraic fractions down to their simplest from:

2a 3c ef 2 a) c) e) ( ) 4a 12cd ef

2 8b ef 2 b) d) 2g h 12b f) ef 6gh

2 22 12 3. Calculate: a) of 18 b) of 891 c) of 585 3 33 13

2 3 3 3 2 d) of of 665. e) of 24 + of 28 + of 36 5 7 4 2 3 4. Write down as improper fractions: 1 1 a) 5 c) 1 3 4 2 1 b) 6 d) − 2 5 3

5. Calculate the following and simplify (A.S.A.P.- as soon as possible):

5 3 3 − 3 6 5 −1 2 a) − : d) − : + ⋅ 2 4 5 5 8 3 6 7

5 9 7 − 5 − 3 6 2 3 2 b) ⋅ − : e) − : 5 + 6 ⋅ − ⋅ 8 10 4 3 5 8 7 2 3

1 1 3 12 1  1 c) + : f)  +  : 3 − 3 7 7  5 3  2

Susana López 3º ESO 9

 1  7 2 3 4 3 g) 9 −  ⋅ + h) 2 ⋅ − :  4  3 5 5 7 4

1 2  1 1  i) +  −  −  2 3  2 6 

− 8 5 − 7 − 2 j) − + − + (−9) 3 − 2 −1 2

1 1+ k) 2 1 1− 2

4 1 7 − ⋅ 5 4 3 l) +1  4 1  7  −  ⋅  5 4  3

Susana López