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ARITHMETIC THEORY
NUMBERS
Natural numbers: 1, 2, 3, 4, 5, 6, 7 .....
Intergers: -3, -2, -1, 0, +1 +2 only +1 & + 2 are natural
numbers
Fractions: ½, -2 or decimal fractions: 0.25, 0.38
Rational numbers: an integer or fraction. A decimal number
that terminates: 0.3876 or -1.125
Irrational numbers: decimals that do not terminate: 2
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ARITHMETIC THEORY
NUMBERS
NUMBER LINE
-3 -2 -1
0 +1 +2 +3
SUM DIFFERENCE
PRODUCT (DIVIDEND & DIVISOR) QUOTIENT
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ARITHMETIC THEORY
NUMBERS
1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10
2 2 4 6 8 10 12 14 16 18 203 3 6 9 12 15 18 21 24 27 30
4 4 8 12 16 20 24 28 32 36 40
5 5 10 15 20 25 30 35 40 45 50
6 6 12 18 24 30 36 42 48 54 60
7 7 14 21 28 35 42 49 56 63 70
8 8 16 24 32 40 48 56 64 72 80
9 9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100
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ARITHMETIC THEORY
NUMBERS
SIMPLE TESTS FOR DIVISIBILITY
A NUMBER IS DIVISIBLE BY:
2 if it is an even number
3 if the sum of its digits is divisible by 3
4 if the last 2 digits are divisible by 4
5 if the last digit is 0 or 5
10 if the last digit is 0
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ARITHMETIC THEORY
NUMBERS
DIRECTED NUMBERS
A number that has a + or ± sign attached to it.
ADDITION: To add numbers of likesigns: + 12 + 6 = +18
To add numbers of different (unlike) signs:
subtract the smaller from the larger number
-12 + 6 = (6 ± 12) = - 6
If there are more than 2 numbers carry out the operation 2 numbers at a time:
-15 ± 8 + 13 ± 19 + 6
= (-15 ± 8)
= -23 + 13
= - 10 ± 19
= - 29 + 6
= - 23
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ARITHMETIC THEORY
NUMBERS
DIRECTED NUMBERS
SUBTRACTION: To subtract, change the sign of the number to be
subtracted and add the numbers.
-10 ± (- 6)= - 10 + 6 = - 4
or 7 ± (+ 18) = 7 ± 18 = -11
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ARITHMETIC THEORY
NUMBERS
DIRECTED NUMBERS
MULTIPLICATION:
The product of two numbers with like signs is positive (+)
The product of two numbers with unlike signs is negative (-)
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ARITHMETIC THEORY
NUMBERS
DIRECTED NUMBERS
DIVISION:
The quotient of two numbers with like signs is positive (+)
The quotient of two numbers with unlike signs is negative (-)
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ARITHMETIC THEORY
NUMBERS
FACTORS
2 X 6 = 12
2 and 6 are FACTORS of 12
W
hat are the factors of 60 ?
Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 & 60
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ARITHMETIC THEORY
NUMBERS
FACTORS
PRIME NUMBERS
A PRIME NUMBER is a number that has no other factors than itself and 1
The prime numbers between 1 and 30 are:
2, 3, 5, 7, 11, 13, 17, 19, 23 & 29
Notes:
1. 2 is the only even prime number, as all other even numbers have 2 as a factor.
2. 1 is a special case and not considered a prime number.
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ARITHMETIC THEORY
NUMBERS
HIGHEST COMMON FACTOR (HCF)
The highest common factor (HCF) of two or more numbers is the HIGHEST number
that is a FACTOR of BOTH numbers.
Example:
Find the HCF of 40 & 48?
Factors of 40 = 1, 2, 4, 5, 8, 10, 20 & 40
Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Therefore () HCF of 40 & 48 = 8
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ARITHMETIC THEORY
NUMBERS
LOWEST COMMON MULTIPLE (LCM)
THE LOWEST NUMBER THAT IS A MULTIPLE OF TWO OR MORE NUMBERS
Example
Find the lowest common multiple (LCM) of 6 & 8
Method
Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72 .........
Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96 .......
Answer: LCM of 6 & 8 = 24
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ARITHMETIC THEORY
NUMBERS
ARITHMETIC PRECEDENCE
The order in which we carry out arithmetic functions
2+3x4 = ?
BIDMAS
B = Br acketsI = indices (powers)D = DivisionM = MultiplicationA = AdditionS = Subtr action
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ARITHMETIC THEORY
NUMBERS
ARITHMETIC PRECEDENCE
BIDMAS EXAMPLE
Find the value of 64 ¹ (-16) + (-7 -12) ± (-29 + 36)(-2 +9)
64 ¹ (-16) + (-19) ± (7)(7) B
= (-4) + (-19) ±(7)(7) D
= (-4) + (-19) ± 49 M
= -23 -49 A
= -72 S
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ARITHMETIC THEORY
NUMBERS
FRACTIONS
11 is an example of a PROPER FRACTION16
NUMERATOR = the number abovethe line.
DENOMINATOR = the number below the line.
IMPROPER FRACTION
5 ¾ MIXED NUMBER
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ARITHMETIC THEORY
NUMBERS
FRACTIONS
ADDITION
Only fractions with the same denominator can be added or subtracted
Example
+ + + = 16 = 2
8
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ARITHMETIC THEORY
NUMBERS
FRACTIONS
ADDITION
LOWEST COMMON DENOMINATOR
Example
+ + = 3 + 16 + 20 = 39 = 1 15
24 24 24 24 24
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ARITHMETIC THEORY
NUMBERS
FRACTIONS
SUBTRACTION
The same as for addition except subtr act the numer ators
Example.
3 - 1 7 = 10 19 = 40 19 = 21 = 19 = 1¾12 3 12 12 12 12 12
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ARITHMETIC THEORY
NUMBERS
FRACTIONS
MULTIPLICATION
Multiply the numer atorsthen
Multiply the denominators
Example.
¾ x = 2132
Convert mixed numbers or simplify as necessary.
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ARITHMETIC THEORY
NUMBERS
FRACTIONS
DIVISION
INVERT the divisor (the fr action we are dividing by) and MULTIPLY
Example.
¾ ¹ = ¾ 8 = 24 = 12 = 6
7 28 14 7
Convert mixed numbers or simplify as necessary.
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ARITHMETIC THEORY
NUMBERS
DECIMAL FRACTIONS
Decimal fr actions are fr actions where the denominator is equal to somepower of 10 e.g. 10, 100, 1000
Example:125
1000
Decimal fractions are usually re-written as decimals by using the DECIMAL POINT.
Example
125 = 0.125
1000
Note: Any fraction can be expressed as a decimal by dividing the Numerator by the
Denominator.
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ARITHMETIC THEORY
NUMBERS
DECIMAL FRACTIONS
ADDITION & SUBTRACTION
Example:
2 . 683 + 34 . 41
2 . 6 8 3
+ 3 4 . 4 1 0
3 7 . 0 9 3
34 . 41 2 . 683
3 4 . 4 1 0
- 2 . 6 8 3
3 1 . 7 2 7
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ARITHMETIC THEORY
NUMBERS
DECIMAL FRACTIONS
MULTIPLICATION
The same as normal ³long´ multiplication. Except the number of decimalplaces in the answer must equal the SUM of the decimal places in the
numbers being multiplied.
Example.
6 . 24 x 3 . 1 2 1 = 19 . 47504
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ARITHMETIC THEORY
NUMBERS
DECIMAL FRACTIONS
DIVISION
The same as normal ³long´ division. Except convert both numbers to adecimal by multiplying both the divisor and the dividend by a power of ten
until the divisor becomes a WHOLENUMBER.
Example.
3650 ¹ 45.56becomes 365000 ¹ 4565
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ARITHMETIC THEORY
NUMBERS
APROXIMATING DECIMALS
Also known as ³Rounding Off´ .
Example 1.
3. 1427694
rounded to 3 decimal places = 3. 143
Example 2.
3. 1427964
rounded to 2 decimal places = 3.14
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ARITHMETIC THEORY
WEIGHTS and MEASURES
Common Systems
1. Systeme Internationale (SI) ± metres, kilogr ams, litres, seconds «
2. Imperial Systems ± feet, pounds, gallons «
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ARITHMETIC THEORY
WEIGHTS and MEASURES
Conversion f actors
Changing a quantity from one unit into a different unit.
ExampleConvert 25 gallons into litres
25 x 4.546 = 113.65 litres
Note: You will normally be given the conversion factor.
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ARITHMETIC THEORY
RATIO and PROPORTION
Ratio.
Two or more quantities that are linked together.
Example. A 3 to 1 mixture of sand and cement can be written as 3:1
The ratio of sand to cement is 3 parts sand and 1 part cement.
Note: The mixture above has a total of 4 parts.
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ARITHMETIC THEORY
RATIO and PROPORTIONExample 1.
An engine turns at 4000 r pm and its propeller turns at 2400 r pmThe r atio of the two speeds can be written as 4000:2400
Simplified this r atio = 5:3
Example 2.
The volume of a cylinder at the bottom of its stroke is 240 cm3.At the top of the stroke the cylinder¶s volume is 30 cm3.
It¶s compression r ation = 240:30 = 8:1
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ARITHMETIC THEORY
RATIO and PROPORTIONExample 3 (Proportion).
Divide $240 between 4 men in the r atio of 9:11:13:15
Procedure
A. Add all the individual proportions to find the total number of parts (9+11+13+15 = 48)
B. Divide the total amount ($240) by the total number of parts (48) togive the value of one part. (240 ¹ 48 = $5)
C. Multiply each r atio by the number of parts to find its value9 x 5 = $4511 x 5 = $5513 x 5 = $6515 x 5 = $75 Total = 45+55+65+75= $240
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ARITHMETIC THEORY
AVERAGES and PERCENTAGES
When working with a series of numbers sometimes it is useful to know the
average of those numbers.
For example.
The average speed of a car or the average fuel used by a car.
In the above average we calculate the average by dividing the TOTAL by the
DISTANCEor TIME.
In these cases the average we calculate is know as the MEAN average.
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ARITHMETIC THEORY
AVERAGES and PERCENTAGES
Example.
A car travels a total distance of 200 km in 4 hours. What is the average
speed in km/hr (Kph)?
Average speed = 200 = 50 Km/hr (Kph)
4
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ARITHMETIC THEORY
AVERAGES and PERCENTAGES
Example.
The weight of 6 items are as follows:
9.5, 10.3, 8.9, 9.4, 11.2, 10.1 (kg)
What is the average mean weight?
Average mean weight = 9.5 + 10.3 + 8.9 + 9.4 + 11.2 + 10.1 = 59.4 = 9.9 kg6
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ARITHMETIC THEORY
POWERS
5 x 5 can be written as 52 = 25
5 is said to have been r aised to the power 2 (or squared )
5 x 5 x 5 can be written as 53 = 125
5 is said to have been r aised to the power 3 (or cubed )
Note:
52 Is NOT the same as 5 x 2.
In the expression 52
5 is called the BASE number and 2 is called the INDEX or POWER.
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ARITHMETIC THEORY
POWERS
L AWSOF INDICES (OR POWERS)
First law of indices: the law of multiplication
When two powers of the same value are multiplied, the index of theproduct is the sum of the indices of the factors.
Examples
54 x 52 = 56
a
2
x a
3
= a
5
2p2 x 5p4 = 10p6
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ARITHMETIC THEORY
POWERS
L AWSOF INDICES (OR POWERS)
Second law: the law of division
When dividing a power of a number by another power of the same value,subtract the index of the divisor from the index of the dividend.
Example1
a5¹ a2 = a5
a2
= a x a x a x a x a
a x a
= a5±2 = a3
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ARITHMETIC THEORY
POWERS
L AWSOF INDICES (OR POWERS)
Second law: the law of division
Example 2
6a6¹ 3a2
= 6a6
3a2
= 2a6-2
= 2a4
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ARITHMETIC THEORY
POWERS AND ROOTSL AWSOF INDICES (OR POWERS)
Third law of indices: the law of powers.
Raising a power term to a further power is carried out by multiplying the
powers together.
Examples
(a2)3 = a2 x a2 x a2 = a6
(b4)5 = b4x5 =b20
(a-2)4 = a-2x4 = a-8 = 1
a8
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ARITHMETIC THEORY
POWERSL AWSOF INDICES (OR POWERS)
The power of 0
Any number (other than zero) that is raised to the power 0 is equal to 1
Example
90 = 1 ; 10 = 1 ; 12456760 = 1
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ARITHMETIC THEORY
POWERSL AWSOF INDICES (OR POWERS)
The power of 1
Any number (other than zero) that is raised to the power 1 is equal to that
number
Example
91 = 9 ; 21 = 2 ; 12456761 = 1245676
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ARITHMETIC THEORY
POWERSL AWSOF INDICES (OR POWERS)
Negative indices
If the index is a negative number, the minus sign indicates the inverse (or
reciprocal) of that number, with a postive index
Example
2-3 is the same as 1
23
= 1 ___
2 x 2 x 2
= _1__
8
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ARITHMETIC THEORY
ROOTSThe root of a number is the value which when multiplied by itself a certain number of times produces that number.
Example:
4 is a root of 16 because 4 x 4 = 16.
4 is also a root of 64 because 4 x 4 x 4 = 64
The symbol used to indicate a root is called the radical () sign. This sign is
placed over the number. On its own it indicates the square root of the number.
Any number placed outside of the radical sign indcates the number of the root:
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ARITHMETIC THEORY
ROOTSFRACTIONAL INDICES
Another way of expressing a root is to show the root as an index.
However, with a root the index must be shown as a fraction.
Examples
can be expressed as a½ or can be expressed as a
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ARITHMETIC THEORY
STANDARD FORM
If the number 8. 347 is multiplied by 10,000 then the product is 83470.
This calculation can also be written as:
8.347 x 104 This know as STANDARD FORM.
Any number written in standard form has two parts.
The first part is any number between 1 and 10 (but does not equal 10). This
part is called the MANTISSA
The second part is 10 raised to a power. This part is called the EXPONENT.
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ARITHMETIC THEORY
STANDARD FORM
To express a number in standard form move the decimal place either to the
right or to the left to create the mantissa, then create the exponent.
Moving the decimal to the left increases the power positively.
Moving the decimal place to the left decreases the power negatively.
Examples:
67.9 in standard form = 6.79 x 101
679 in standard form = 6.79 x 102
0.00679 in standard form = 6.79 x 10-3
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ARITHMETIC THEORY
LOGARITHMS
Logarithms are a mathematical tool developed to simplify multiplicationand division of large numbers by enabling those calculations to be doneusing addition and subtr action.
Example: 1000 = 103 then 3 = log10 1000
y = 32 = 25
then 5 = log2 32
If a postive number y is expressed in index form with a base a:
y = ax
then the index x is known as the logarithm of y to the base a:
y = ax, then x = loga y
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ARITHMETIC THEORY
LOGARITHMS
Logarithms can be calculated to any base. But they are nor mally onlybase 10 or base e are used.
Logarithms to base 10 are called COMMON LOGARITHMS and abreviated
to log10or lg.
Logarithms to base e are called NATURAL or NAPERIAN logarithms. These
are abbreviated to logn or ln.
Logarithm values to any base can be determined using the log function on a
scientific calculator.
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ARITHMETIC THEORY
LOGARITHMS
Examples:
Using a calculator determine the following:
a. Log10 17.9 = 1.2528 ...
b. Log10 462.7 = 2.6652 ...
c. Log10 0.0173 = -1.7619 . .
d. Loge 3.15 = 1.147 ...
e. Loge
0.156 = - 1.8578 ...
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ARITHMETIC THEORY
LOGARITHMS
Rules for using logarithms:
There are 3 rules for using logarithms. These apply to logarithms of any
base.
1. To multiply two numbers:
log (A x B) = log A + log B
Example:
log10
10 = 1
log10
5 + log10
2 = 0.69897 + 0.301039 = 1
Therefore: log10
(5 x 2) = log10
10 = log10
5 + log10
2
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ARITHMETIC THEORY
LOGARITHMS
1. To divide two numbers:
log (A) = log A ± log B(B )
Example:
log10
5 = log10
2.5 = 0. 39794
2
log10 5 - log10 2 = 0.69897 - 0.301039 = 0.39794 = log102.5
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ARITHMETIC THEORY
LOGARITHMS
1. To r aise a number to a power :
log An = nlogA
Example:
log10
52 = log10
25 = 1.39794
Also2
log105
=2
x0
.69897
=1
.39794
= log1025
Therefore: log10 5
2 = 2 log10
5
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ARITHMETIC THEORY
LOGARITHMS
Working with logarithmes with different bases.
Example:
log2
3 = log10
3 = 0.04771 = 1.5850
log10
2 0.3010
Use the following formula:
logbh = log10 hlog10 b
You can check the above answer using the lognx function on your calculator.