Download 3 When operations in the same order appear together, we operate

GLOSSARY
operations
addition
subtraction
multiplication
division
raise a number to a power
get a root of a number
the equals sign ‘ = ‘
(the things on both sides are of equal value; same value, different appearance)
the plus sign ‘ + ’
the minus sign ‘– ‘
the negative sign ‘– ‘
the times sign ‘·’
(if one of the factors is not a number, the multiplication sign can be left out)
the divided-by sign ‘÷’
the slash ‘/’
the fraction bar ‘––––‘
the radical bar ‘
’
the parentheses or round brackets ‘( )’ (but “parenthesis” in singular)
the square brackets ‘[ ]’
the operands
the result, the outcome
to evaluate
to perform operations on quantities
addends
the sum of
to add
minuend
substrahend
the difference between
to substract
factors or multiplicants
the product of
to multiply
the dividend
the divisor
the quotient
the remainder
to divide
power
base
exponent, index (exponents, indices)
‘a’ to the power of ‘b’
ab
‘a’ to the ‘b’
ab
‘a’ exponent ‘b’
ab
the square of ‘a’
a2
the cube of ‘a’
a3
to square
to cube
to raise to the power of four
to raise to the fourth power
radical
surd
the rooted
index (indices)
the square root of
larger than
smaller than
more than
less than
right
wrong
enter a figure on a calculator
scientific calculator
simple calculator
graphing calculator
fraction
numerator
denominator
reduced fraction
reciprocal or multiplicative inverse fraction
proper fraction
improper fraction
mixed fraction or number
three divided by five
3:5
3/5
three over five
3:5
3/5
three fifths
3:5
3/5
a half.
½
three fourths, three quarters
¾
one and three-fifths
1
3
5
1
3
5
3
5
3
5
1
2
3
4
8
5
3
5
LCM (lowest common multiple) of the denominators
to flip numerator and denominator
GCD (greater common divisor) of numerator and denominator
HCD (highest common divisor)
Overbar (to denote a repeating block of digits):
5
 0, 45
11
LESSON 1 Numbers
Revision exercises:
5·(–2)·( –8) – ( –4)·5=
42:2–1–82:2–1=
(–3)4 =
Note: this means (– 3)· (– 3)· (– 3)· (– 3)
4
–3 =
Note: this means – 3· 3 · 3 · 3
4–7– [–2–9+(–4–2) –5] =
[11+8: (7–5)]·2–7 =
2 + 6 – 8 + 1 – 3 = Note: add positives and add negatives separately
Order of operations
1
First operations that are inside round brackets, then operations inside square brackets, always inside out.
2
If there are no brackets or within a set of brackets:
3
2.1
Roots and powers
2.2
Multiplications and divisions
2.3
Additions and substractions
When operations in the same order appear together, we operate from left to right. (With sums and
differences it is better to add positive numbers and negative numbers separately)
mnemonic: PEMA ("Please Excuse My Aunt") (Parentheses, exponents, multiplications, additions)
note: numerator and denominator are evaluated as if they were inside a set of parentheses
note: expressions inside a radical bar is evaluated as if it was inside a set of parentheses
Mental calculation:
13 page 38
1
2
3

3:4=
4
1

1:4=
4
1

1:5=
5
2

2:5=
5
3

3:5=
5
Note: when the denominator is 5, just evaluate the numerator divided by ten and then
double it
15 page 38
16 page 38
1:2=

Revision exercises:
5 7 2
  
3 3 3
2 2 1
  
3 7 5
4
5 
5
7
2
3
21 8
 
4 12
2
5
:

10 1000
6 8 7 5
   
5 2 6 3
6 3 7
:  
5 2 12
3 4 6 2
   
5 9 5 3
35 1 4  3 1
     
8  3 2  11  4 5 
2
1
1
2   
4
2
5
26 
1 11
:
13 85
Mental calculation:
17 page 38
18 page 38
2
of 60
3
3
of 100
4
3
of 500
500
(two thirds of 60)
(three quarters of 1000)
(three over five hundreds of 500)
2
3
12
third part of
7
half the fifth part of – 6
half part of
NAME
Natural numbers
Integer numbers
Rational numbers
Real numbers
NUMBER SETS
SYMBOL
EXAMPLES
1,
2,
3,
4,
5,
6, 7, 2145, 10000…
N
…,– 100, – 5,– 2, – 1, 0, 3, 7,…
Z
Q
9
3
…– 20, – , – 1, 0, 2, , 4, …
7
5

9
3
…, – , –  , – 6, 0, 2, , 3,...
7
5
INTERNAL
OPERATIONS
+
+
+
·
·
·
–
–
:
+
·
–
:

Note: A real number is called “irrational” if it is not a rational number.
Note: in English, “whole numbers” means the set consisting of the natural numbers and the zero.
Decimal numbers are the same as real numbers:
-
A decimal number with no digits on the right of the decimal point is an integer.
A terminating decimal number or a repeating decimal number is a rational.
A non-terminating, non-repeating decimal number is not a rational; it is called an irrational number.
Note: In England, the symbol used to separate the integer part of a decimal number from its fractional part is a
decimal point. We use a decimal comma instead.
Location on the real line:
Integers are placed leaving always the same distance between each number and the following. Zero is in
the middle, positives are on the right and negatives are on the left. The numbers grow from left to right.
Examples:
– 5, – 3, 0, 2, 4
Proper fractions are pictured dividing the unit into as many parts as the denominator tells and moving to
the right (to the left, if negative) as many parts as the numerator tells.
3 3
Examples: – ,
5 7
Improper fractions are first changed into mixed numbers. Then the fractional part is pictured on the unit
segment following the integer.
12 8
Examples: –
,
5 7
A decimal number is pictured by repeatedly fractioning each unit into ten smaller-order units.
Examples:
– 1’207 , 2’13
Homework:
30 and 31 page 40
Improper and mixed fractions
An improper fraction is a fraction where the numerator is greater than the denominator. Examples:–
12 8
,
5 7
1
1
2
2

  2   , 1  1 
7
7
5
5

To get the mixed fraction that is equal to a given improper fraction, you divide numerator by denominator; the
quotient is the whole part, and the remainder is the numerator of the fractional part.
A mixed fraction is a fraction plus a whole number. Examples: – 2
A fraction may be changed into a decimal number
When we divide numerator by denominator we can obtain:
-
An integer number
A terminating decimal number
A repeating non-terminating decimal number (recurring decimal)
Examples: exercise 6, page 38
Note: Suppose the fraction is irreducible. If denominator is factorized in prime factors and these prime factors
are only numbers two and/or five, the outcome will be a whole number or a terminating decimal number.
Note: Suppose the fraction is irreducible. If denominator is factorized in prime factors and these prime factors
are numbers two and/or five as well as any other numbers, the outcome will be a mixed recursive decimal
number.
Note: Suppose the fraction is irreducible. If denominator is factorized in prime factors and these prime factors
are numbers different from two or five, the outcome will be a repeating non-terminating decimal number with
all the decimal part digits being recursive.
Some decimal numbers may be changed into fractions (rational numbers)
All rational numbers give decimal parts that either terminate or recur.
-A whole number
Write down the number divided by 1
-A terminating decimal number
Remove the decimal point and use the decimal number as the numerator.
The denominator is the number 1 followed by as many zeros as decimal places in the number.
Reduce the fraction.
-A repeating non-terminating decimal number (all the digits in the decimal part are recursive).
Count the number of digits that are repeating and write this power of ten
Multiply the decimal number by it and subtract the original number: it is the numerator
Denominator = the power of 10 minus 1
-A repeating non-terminating decimal number (there are no-repeating digits in the decimal part)
Count the number of decimal digits and write this power of ten
Count the number of digits that are not repeating and write this power of ten
Multiply the decimal number by both powers and subtract: it is the numerator
Denominator = the difference between both powers of 10
Examples: exercise 8, page 38
Some decimal numbers can not be changed into fractions (irrational numbers)
The numbers whose decimal part continues forever with no pattern are irrational. They can not be
changed into fractions.
-
Some irrational numbers are trascendental numbers like 
-
The surds that are not terminating numbers are irrational
Homework:
2, 5 and 25 page 38 (and following)
study page 37 on your textbook (powers laws)
41 page 40
5,
3
6,
2
3
Power rules
-
Exponents or 'powers' are a process of repeated multiplication.
Exponents are normally written in the form ab, where ‘a’ is the base and ‘b’ is the exponent.
We pronounce ab as ‘a to the power of b’ or ‘a to the b’ or ‘a exponent b’.
When calculating with ‘powers’ use the following rules:
a1  a
a0 1
282
Same base
a b  a c  a bc
ab
b
c
a : a  c  a b c
a
Same exponent
b
p b  q b   p  q
283
284
285
a 
b c
 a b c
286
Negative exponents
1
a b  b
a
287
 p
 
q
288
289
b
 p
pb
b
p : q  b   p : q     290
q
q
b
Exponent of an exponent
b
291
b
q
  
 p
b
Fractional exponents
1
c
a c a
b
a c  c ab
Exponential notation.
-
Exponential notation is a method of writing numbers that have many zeros.
The basis of exponential notation is the power of ten.
A number written in exponential notation has two parts.
The first part is a real number
The second part is a power of ten.
Examples:
15 600 000 = 156 × 105 = 156E5
0’0000000072 = 72 × 10–9 = 7,2E–10
Scientific notation.
-
Scientific notation is a method of writing very large and very small numbers.
The basis of scientific notation is the power of ten.
A number written in scientific notation has two parts.
The first part is a number between 1 and 10 called “mantissa”.
The second part is a power of ten.
Examples:
15 653 = 1’5653 × 104 = 15,653E4
0’0000000072 = 7’2 × 10–9 = 7,2E–9
-
Converting numbers into scientific notation
First move the decimal place so that only one integer is on the left side of the decimal.
Then, following the number, write a multiplication sign.
Then, the number 10 raised to the number of places you moved the decimal comma.
If the number is large, the exponent is positive, if the number is small the exponent is negative.
-
Entering scientific notation on your scientific calculator
Your calculator will have either an 'EE' key or an 'EXP' key.
To enter 2’53 × 102, you would punch 2.53EE2
To enter 3’95 × 10–3, you would punch 3.95EE3.
But to raise a number to some exponent use the 'yx' or 'xy' key depending on the calculator.
Exercises:
1) An electron weights 0’000 000 000 000 000 000 000 000 91 Kg more or less. Express it with
exponential notation.
2) Convert the following numbers into scientific notation:
a) 600 000
b) 7 000 000 000
c) 5 782 670 000
d) 7 334 971 650 000 000
e) 0’000 000 008 88
f) 750 000 000 000
g) 45 711 000 000 000 000 000
h) 0’000 000 72
i) 0’000 000 006 194 3
j) 0’000 000 000 0660529 8
k) 34 216 670 000 000
3) Andromeda constellation is two million light-years far from Earth. Calculate this distance in km and
write it in scientific notation (light speed = 299 792’458 km/s , 1 year = 365 days and 6 hours)
4) In your brain there are more than forteen thousand million neurons. Convert this number into
scientific notation.
5) Express these numbers as decimal numbers:
a) 3,479E13 b) 1,26E10 c) 4,81E2
f) 3,355E-2 g) 4,02E5
h) 9,96E-9
d) 5,70E-11 e) 3,92E-21
i) 5,0408E11
Accuracy
When expressing a number with respect to a fixed number of decimal places we are really deciding on
the accuracy to which we want to work. One often uses the shorthand ‘expressed to nD’ to emphasise that a
number has been expressed to n places of decimal.
Examples:
12’50913
3’142
194267’60992
2’718
0’00043
0’577
they are all expressed to 5D
they are all expressed to3D
Significant figures
Reading from the left the first non zero digit is the most significant and will be called the first significant
digit. The digit to the right of the first significant digit is the second significant digit and so on.
If we represent a number to n significant digits we say it is ‘displayed to nS’
Examples:
123’45
0’000305
0’000032004
123000
12’002
1’06
87’4
they are all displayed to 5S
they are all displayed to 3S
Approximation
Often we wish to rewrite a number with less significant digits. There are two methods to do this.
-
Chopping (one simply discards the superfluous digits)
Examples:
163’190385  163’1903
– 34’978992  – 3 4’9789
0’000299  0’0002
Rounding (we first chop the number at nD; if the digit following the last significant digit was 5 or
higher, we increase the last significant digit in one unit)
Examples:
163’190385  163’1904
– 34’978992  – 34’9790
0’000299  0’0003
26’288359  26’2884
26’288459  26’2885
Error
When one has to deal with numbers which have been rounded, we keep note of the error involved.
Actual absolute error
Given some number X and its approximation x the actual absolute error is:
ε = X – x
–n
The maximal absolute error for a number given to nD is 0’5 · 10 . It is the maximal error bound.
Example:
If we use 3’14159 as a decimal approximation of π, the actual absolute error is π – 3’14159 .
It is less than 0’000005 so this is the maximal error bound.
Actual relative error
Given some number X and its approximation x the actual relative error is:
The maximal relative error for a number given to nD is
0'5 10
x
r

X

X x
X
n
. It is the maximal relative error
bound
Examples:
1)
An absolute error of 0’5 is not very important if the number is X = 62 500
But it is very high if the number is X = –2
The relatives errors would be 0’000008 in the first approximation and 0’25 in
2)
If we use 3’14159 as a decimal approximation of π
0'000005
The relative error would be smaller than
, so smaller than 1’592·10–6 = 0’000001592
3'14159
Exercises on absolute and relative errors:
1) While measuring a length of 7m, four different outcomes were obtained:
a) 7'02 m.
b) 7'03 m.
c) 6'98 m.
d) 6'9 m.
Fill the table of actual errors calculations.
7'02 m.
7'03 m.
6'98 m.
6'9 m.
Absolute
error
Rounded up
or down?
Relative
error
2) Compare the two following relative errors and explain why do you think that having the same
absolute errors, relative errors are different:
a) something that lengths 7m. is measured and obtained 6'5m.
b) something that lengths 107m. is measured and obtained 106'5m.
3) A length of 108 m is measured so that the relative error is lower than 0'05. How much can the
absolute error be at most?
Homework:
34, 37 and 39 page 40 (and following)
read page 25 on your textbook (use of calculator)
Nth root of a number
(“nth root of a”)
n
The expression
a
th
raised to the n power equals “a”
n
means:
a number such that
a  R  Rn  a
“a” is called the rooted
“n” is called the index (when index is 2, it does not have to be written)
is the radical bar
5
Examples:
4
3
3
because
32  2
because
81  3
 125  5 because
64  4
100  10
2 5  32
34  81
 53  125
because
4 3  64
because
10 2  100
note: a 2-index root is called “square root”
a 3-index root is called “cube root”
a 4-index root is called “fourth root”, a 5-index root is called “fifth root”, and so on.
Surds
A surd is and expression involving roots.
Sometimes it is useful to work with them, rather than using an approximate decimal value. Surds can be
manipulated just like algebraic expressions.
When asked to give the exact value, approximate decimal answers will not do and you will have to
manipulate surds in order to give a final answer in simplified surd form.
Example:
4
3  4 27  1,31607·2,27951 = 2,99999
4
3  4 27  4 3  27  4 81  3
is not exact
is exact
Surds as indices
1
n
a a
A n root can be written as an exponent:
th
The laws of indices also apply to any n root. We can use these laws to manipulate surds.
th
note:
n
n
a a
p
Homework:
p
n
as well
42 and 43 page 40 (and following)
Rules of surds
p
a  a
Uses
np
1.

n
np
as
a a
p
1
np
 an  n a
9  4 32  3
Change surds to have the same index. For example to compare
4
Simplify radicals. For example:

3
p
586  586  6 343396 ,
2.

70  70  6 343000 therefore,
2
6
3
6
1
3
3
586 with
70 :
586  70
1
n
as
a b  n a  n b
a  b  a  b  n  a n  b n  n a  n b
Uses
Take a factor out of the radical. For example: 18  9  2  9  2  3 2
1
n

Write two radicals under the same radical bar. For example:
a

b
n
a
1
n
1
n
n
a a
a
a
   1  n
b b
b
bn
3.
n

Uses
Together with rules number 1 and 2 this rule allows to write surds products and quotients under the
n
as
15  20  15  20  300
b
n
same radical bar. For example:
4.
 a
n
p
 a
n
p
33 4
6
 a
n
as
24

6
33  6 4 2
6
23  3
33  2 4 6

 2  3 2  6 18
3
2 3
p
p
6
1
p
 1
  a n   a n  a p
 
 
1
n
 n ap
1
5.
m n
1
 1 m
a   a n   a nm  mn a
 
m n
as
a  mn a
6.
Two different surds can not be added unless you calculate their decimal approximations. Only alike
surds can be added.
Cases
3 2
7 3 7

For example:
or
can only be solved by approximation or they
should remain unsolved.
17 5
7 5  11 5  5 certainly can be simplified to


Sometimes the possibility that a surds addition can be simplified is hidden. For example:
8  18  4 2500  23  2  32  4 22  54  2 2  3 2  5 2  10 2
7.
Rationalising the denominator. Sometimes it is useful to get rid of surds from the bottom of a fraction.
To do so, you have to multiply both numerator and denominator by a suitable expresion.
Examples
3
3
1
1
5
5




3
3
3
2
2 3
5
25
5
5  5

Notes:
1
5 3
5 3


5 3
5 3 5 3
52  3

a a 
 a
2


 a2  a

5 3
22
 
a  b  a  b   a
2
2
b


a b 

a  b  a b
3
a  3 a2  a
Exercises:
12
1) Simplify: a)
x9
b)
12
x8
c)
5
y10
d)
6
e)
8
9
f)
64
2) Compare these two surds: a)
4
31 e 3 13
b)
3
51 e 9 132650
3) Multiply and simplify:
3
25 2
b)
3
96 3
a)
6) Solve:
5
9
3
b)
4
16
c)
a 3b 5 c
ab 3 c 3
a) 18  50  2  8
b)
3
5
7) Rationalise: a)
3
2
2
b)
5
7
c)
3
b)
0
a
2
3
1
 1 2
g)  2  
9 
 x  x
6
3
e)
4
g)
3 2
3
81a 3b 5 c c)

f) 


2

5
64
8
1
5
32
37  32  31  33 
c)
34 :31 
3
3
 1  1
e)   :   
 2  2
 2  2   3 1 
f)          
 3   2  
23  27  23  25
h) 4 6 2 9 
2 2 2 2
 9  11
i)  3 
 27 
5
0
 1
d) 2   3       0'07 
 5
0
3
a 4b 6
10
50a  18a
f)
2 3
Exercises on exponents:
a) 23  24  23  2 
0
d)
81
c)
4) Take as many factors as possible out of the radical: a) 3 32x 4 b)
5) Simplify: a)
8
3
1
1
 64 
j) 

 243

  12  2
 81 
k)  2  
 3 


1
3

4
 2  2
   
m)  3  3
2
3
1
l)
1
2 2  33  2
1

93  3

1
1
2

3
3
n) 16

1 1

2 4
1
Exercises on surds:
1) Simplify or take factors out of the root:
a) 6 125
b)

g) 5

 2  
m)
25  75a
3
6
h)

6
4
c) 4 1024
8
18
i)
j)
a)

3  2 5 6

3 2 
3
e)


j)
4
3
o)

3) Rationalise:
2
3 2
2 5
b)
c)
3
6
3

43 6
2 6
e)
l)
25 4 27 5
a
b
9
200
p) 4 144a 6 b 8 c 2

f)
3
2916
a  b3
c) 5 75  4 9  6 27
53 2  53 2
f)
75  3
2
3 2
8x 3 y 4
243
k)
4

f)
9317
1000
i)
x2 y2
27 x 6 y 3
e)
2ab  6 9a 2 b  3 5ab 4
k) 2 12  2 27 
d)
9
25200
2 xy 3
h)
4
3 2
3
b) 5 72  7 18  50  2 18
d) 12  15  18
g)
99
44
n) 4 16a 8 b12 ñ) 125a 2 b 3 c 5
2) Solve:
3  27  12
a)
d)
2 5
5 6
g)
9a 2 b 2
3
3ab
3 3
18  3 8
h)
2 3 6
2 54 6
Percentages
A percentage is a way of represent the relation between two quantities as a fraction of 100.
Example:
45
45
 20  9
45 percent (45%) means
45% of 20 is
100
100
Percentage increase and percentage decrease
If a quantity changes from one value to another then we calculate the percentage change from this
percentagechange differencebetweenvalues

equation:
100
oldvalue
Example:
In a sale the price of a bicycle is reduced from 120 € to 90 €
percentagedecrease 30
30 100

 25%
We write
and get percentagedecrease 
100
120
120
To calculate the outcome of a percentage increase we multiply the old value by 1 
Examples:
To increase 120 a 12%
To increase 120 a 5%
we add 100 + 12 = 112
we add 100 + 5 = 105
then multiply 120 · 1,12 = 134,4
then multiply 120 · 1,05 = 126
To calculate the outcome of a percentage decrease we multiply the old value by 1 
Examples:
To decrease 120 a 12%
To decrease 120 a 5%
Class work:
Homework:
we substract 100 – 12 = 88
we substract 100 – 5 = 95
percentage
100
percentage
100
then multiply 120 · 0,88 = 105,6
then multiply 120 · 0,95 = 114
34, 38 and 43 page 62
33 and 35 page 62
Read page 58
Interest
Interest is the money paid by a borrower to a lender in return for the loan. It is usually expressed as a
percentage by year.
Examples:
If I borrow 100 € from a company and the interest of the loan is “5% per annum” I will have to
give back the company 105 € if I keep the money for one year.
If I lend 200 € to a bank for one year and they promise to pay an interest of “3% per annum”,
they must pay 6 € to me after that year, when I get my 200 € back. These 6 € are the interest.
Simple interest
Simple interest is interest calculated as a straight percentage of the amount loaned or invested by the
A p t
formula I 
where I is the interest, A is the original amount, p is the percentage and t is the time
100
(years, usually)
Example:
If I deposit 200 € into a bank over seven years with a simple interest of “3% per annum”, I get
A  p  t 200  3  7
I

 42€
100
100
Compound interest
When simple interest is calculated, only the interest on the original capital is added.
In compound interest, the interest earned over a period of time is added to the investment, so that at the
end of the next period interest is paid on that total.
t
p 

We can use the formula: f A o A  1 
 to calculate the final amount of money. oA is the original
 100 
amount, p is the percentage and t is the time (years, usually).
Example:
2000 € to a 10% compound interest over 3 years convert into: 2000 · 1,13 = 2662 €
Class work:
Homework:
48 and 50 page 62
8 and 13 page 60
Ratios. Direct proportionality
The relative size of two quantities expressed as a quotient is called ratio. The ratio of a to b is written
as a:b or a/b. An equality between two ratios is a proportion.
To solve a direct proportionality problem it is possible to use the “Formula Triangle” or a proportion.
It is also possible to reduce one of the ratios to the form 1:n and draw a conclusion.
Ratios. Inverse proportionality
It can be managed as direct proportionality by changing one of the fractions of the proportion into its
reciprocal one. It is also possible to reduce one of the ratios to the form 1:n and to reason out from this.
Class work:
10 and 15 page 60
18 page 60
Homework:
19 page 60
Read page 51
Proportional division
In a proportional division question, a total amount is to be split in a certain ratio. You have to follow
these steps:
Add up the parts
The sum is propotional to the total amount as each amounts is proportional to each part.
Example:
Homework:
1800 € is to be split in the ratio 2:3:1
We add:
2+3+1=6
We solve ratio questions:
6
1800
2
x
Read page 51
20, 21 and 23 page 60
x=(1800·2)/6=600 €
6
3
1800
x
x=(1800·3)/6=900 €
6
1
1800
x
x=(1800·1)/6=300 €
Speed, distance and time
These three magnitudes follow the rule speeed 
Examples:
dis tan ce
when an object moves with constant speed.
time
A car that moves with a constant speed of 100 Km/h can run 200 Km in 2 hours.
If a walker moves 32 Km in 8 hours, his/her average speed is 4 Km/h
“Speed” and “distance” are in direct proportion.
“Time” and “distance” are in direct proportion.
“Speed” and “time” are in inverse proportion.
Two moving objects.
Two moving objects with constant speeds and opposite directions approach with a speed equal to the
addition of their speeds.
Two moving objects with constant speeds and same direction approach with a speed equal to the
difference between their speeds.
Examples:
Two cars with the same speed 50 Km/h head-on crash. It is the same as if one of them ran
into a wall at a speed of 100 Km/h
One car with a speed of 80 Km/h follows a truck that moves at a speed of 50 Km/h and
finally crash. It is the same as if one of them crashed at a speed of 30 Km/h.
Homework:
Read page 53
28 and 30 page 61