htp math arithmetic theory 1 (numbers)

8/8/2019 HTP Math Arithmetic Theory 1 (Numbers)

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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



ARITHMETIC THEORY

NUMBERS

NUMBER LINE

-3 -2 -1

0 +1 +2 +3

SUM DIFFERENCE

PRODUCT (DIVIDEND & DIVISOR) QUOTIENT



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



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



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



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



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 (-)



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 (-)



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



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.



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



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



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



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



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



ARITHMETIC THEORY

NUMBERS

FRACTIONS

ADDITION

Only fractions with the same denominator can be added or subtracted

Example

+ + + = 16 = 2

8



ARITHMETIC THEORY

NUMBERS

FRACTIONS

ADDITION

LOWEST COMMON DENOMINATOR

Example

+ + = 3 + 16 + 20 = 39 = 1 15

24 24 24 24 24



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



ARITHMETIC THEORY

NUMBERS

FRACTIONS

MULTIPLICATION

Multiply the numer atorsthen

Multiply the denominators

Example.

¾ x = 2132

Convert mixed numbers or simplify as necessary.



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.



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.



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



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



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



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



ARITHMETIC THEORY

WEIGHTS and MEASURES

Common Systems

1. Systeme Internationale (SI) ± metres, kilogr ams, litres, seconds «

2. Imperial Systems ± feet, pounds, gallons «



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.



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.



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



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



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.



ARITHMETIC THEORY


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



ARITHMETIC THEORY


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



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.



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



ARITHMETIC THEORY

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



ARITHMETIC THEORY

POWERS


Second law: the law of division

Example 2

6a6¹ 3a2

= 6a6

3a2

= 2a6-2

= 2a4



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



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



ARITHMETIC THEORY


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



ARITHMETIC THEORY


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



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:



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



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.



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



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



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.



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 ...



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



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



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



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.

htp math arithmetic theory 1 (numbers)

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