unit 1 fractions. rational numbers. - Over-blog

3º ESO
1
UNIT 1
FRACTIONS. RATIONAL NUMBERS.
1. Vocabulary
A fraction is a number like this:
a
b
Numerator
Fraction line
Denominator b ≠ 0
The numerator and the denominator are integer 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 decimal number, but in mathematics 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 division 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
Example:
10 6
= 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
=
7 ×4 28
← Amplification
40 ÷20 2
=
60 ÷20 3
← Simplification
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 ⋅ 2 3 ⋅ 3 down 2 2 4
22 ⋅ 25 ⋅ 35 ⋅ 36 2 ⋅ 11 ⋅ 5 2 ⋅ 5 ⋅ 7 ⋅ 2 2 ⋅ 32
7 7
= 3 3
= 3 =
3
125 ⋅ 24 ⋅ 33 ⋅ 8
8
5 ⋅ 2 ⋅ 3 ⋅ 11 ⋅ 3 ⋅ 2
2
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
⋅ =
b d b⋅d
Multiply numerators together, multiply denominators together.
Remember: of means multiply.
Examples:
4
4 30 120
4
of 30 is equivalent to ⋅ 30 = ⋅
=
= 24 .
5
5
5 1
5
1
9
1 9
of of 1500 = ⋅ ⋅ 1500 .
8
3
3 8
4.2. Division of fractions
a c a d
: = ⋅
b d b c
Change the division into a multiplication by turning the second fraction
upside down and multiply both fractions together, that is, multiply by the
reciprocal.
a
a c a⋅d
b = a⋅d
: =
The important is:
c
b⋅c
b d b⋅c
d
4.3. Addition and subtraction of fractions
We have two possibilities
A) Same denominator:
a b a+b
+ =
c c
c
Examples
1
5 1 3 5 1+ 3 + 5 9
+1+ = + + =
= =3
3
3 3 3 3
3
3
18 3 18 + (−3) 15 simplify 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
a)
Integer numbers: the numerator is a multiple of the denominator. E.g.
63
= 7.
9
− 15
−3
.
= −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.
We always write them in the whole way. Example:
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
5
17
2
5
3
17
2
2
= 3+ = 3
5
5
5
The quotient 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.)
Exercise: Convert into mixed numbers:
5 26
52
,
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 repeating decimal: decimal periódico.
Recurring cycle: periodo.
Classification of decimal numbers
1. Terminating decimals


i) pure


a) recurring decimals
ii) mixed
2. Non - terminating decimals
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
,
(simplify always!)
1.5 =
= ,
0.36 =
=
2.512 =
=
100 25
1000 125
10 2
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
are rational numbers.
6
a)
Fractions like
b)
3
−5
Every integer number is also a rational number: 3 = , − 5 =
1
1
c)
Terminating decimals are a rational numbers, as we know: 2.5 =
d)
25 5
= .
10 2
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.
The set of integer numbers is denoted by Z:
The set of rational numbers is denoted by Q.
N = {1, 2, 3, …}
Z= {…−3, −2, −1, 0, 1, 2, 3, …}
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
Z
−1, −2, ...
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:
a)
81
27
d)
5 2 ⋅ 35 ⋅ 2 3
5 ⋅ 33 ⋅ 2 2
b)
42
189
e)
212
7 ⋅ 34
c)
7980
6118
f)
25 ⋅ 35 ⋅ 75
15 ⋅ 21 ⋅ 10
g)
34 ⋅ 5 −7 ⋅ 2
2 5 ⋅ 3 −6 ⋅ 5 4
2. Cancel these algebraic fractions down to their simplest from:
a)
2a
4a
b)
8b
12 b
3. Calculate: a)
d)
2
of 18
3
2
3
of
of 665.
5
7
c)
3c
12cd
e)
d)
ef 2
ef
f)
b)
22
of 891
33
e)
3
3
2
of 24 + of 28 + of 36
4
2
3
c)
(ef )2
ef
2g 2h
6 gh
12
of 585
13
4. Write down as improper fractions:
a)
5
1
3
c)
1
b)
6
2
5
d)
−2
1
4
1
3
5. Calculate the following and simplify (A.S.A.P.- as soon as possible):
a)
5 3 3
− :
2 4 5
d)
− 3 6 5 −1 2
− : +
⋅
5 8 3 6 7
b)
5 9 7 −5
⋅ − :
8 10 4 3
e)
−3 6
2 3 2
− :5+ 6⋅ − ⋅
5 8
7 2 3
c)
1 1 3
+ :
3 7 7
f) 
1
 12 1 
+ :3−
2
 5 3
Susana López
3º ESO


g)  9 −
9
1 7 2
⋅ +
4 3 5
h) 2 ⋅
3 4 3
− :
5 7 4
i)
1  2  1 1 
+ − − 
2  3  2 6 
j)
−8 5 −7 − 2
−
+
−
+ (−9)
3 − 2 −1 2
1
2
k)
1
1−
2
1+
4 1 7
− ⋅
5
4 3 +1
l)
4 1 7
 − ⋅
5 4 3
Susana López