TYPES OF NUMBERS
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DIRECTED NUMBERS
What are directed numbers?
Directed numbers include positive and negative numbers.
!
Positive numbers include " , 2.5 or 4. They can be written with a positive sign but are
usually written without it and are understood to be positive.
Negative numbers are written as -3.5 , -2 or -435.4. These are always written with the
minus sign.
Subtraction means move to the left
Addition sign means move to the right
E.g. Compute the following.
-3 + 2 =
5+3=
-3 – 2 =
Compute the following.
1.
-6 – 3 =
2.
-11 + 12 =
3.
7 – 10 =
4.
10 – 17 =
5.
11 – 12 =
6.
10 – 15 =
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DIRECTED NUMBERS
If 2 different signs are next to each other then we combine the signs.
The signs are combined in the following ways.
+ and - = - and - = +
- and + = + and + = +
E.g. Compute the following.
2 – (-3) =
7 + (-5) =
10 + (-7) =
Compute the following.
1.
-6 – (- 3) =
2.
11 - 12 =
3.
7 – (-10) =
4.
-10 – 17 =
5.
11 – (-12) =
6.
-10 – ( -15) =
7.
(-42) - 2 =
8.
150 – 200 =
9.
30 – (-30) =
10.
(-35) – 5 =
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DIRECTED NUMBERS
Similar to addition and subtraction, there are some rules we follow when multiplying
and dividing.
+ and - = - and - = +
- and + = + and + = +
When multiplying and dividing like signs give a positive answer unlike signs give a
negative answer
E.g. Compute the following.
2 x (-3) =
-20 x -2 =
-10 ÷ -5 =
40 ÷ 2 =
Compute the following.
1.
-(6) x (– 3) =
2.
(-11) x 12 =
3.
7 x (– 10) =
4.
(-10) ÷ (–2) =
5.
(-121) ÷ 11 =
6.
(-20) ÷ (-5) =
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MULTIPLYING AND DIVIDING DECIMALS
Using your calculator find the exact value of the questions below.
1.
#.%&%.'"!
(Jan20,Q1)
(.(("
2.
5.47 −
(.%(%#
(Jan17, Q1)
%."
3.
3.6 +
51.84 ÷ 3.75 (Jan16, Q1)
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FACTORS AND MULTIPLES
If a number is divided by another and there is no remainder that number is a factor of
the original number. (Division)
Example: 10 ÷ 5 = 2
Therefore 5 is a factor of 10.
A multiple is the product of an integer and any number. (Multiplication)
Example: 10 x 2 = 20
Therefore 20 is a multiple of 10.
List the factors of the following numbers.
1.
25 =
2.
49 =
3.
50 =
4.
100 =
5.
7=
List the first 5 multiples of the following numbers.
1.
2=
2.
7=
3.
15 =
4.
100 =
5.
20 =
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HIGHEST COMMON FACTOR
The HCF of a group of numbers is the largest number that is a factor of all the
numbers in the group.
To Find The Highest Common Factor:
1.
List the factors for each number
2.
Identify the common factors
3.
Select the highest among the factors
Find the highest common factor of the following numbers.
1.
8 and 12
2.
15, 25, 40
3.
24,36, 60
4.
9, 12
5.
15, 45, 75
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LOWEST COMMON MULTIPLE
The lowest common multiple of a set of numbers is the lowest multiple that is common
to all of the numbers.
Find the lowest common multiple of the following numbers.
1.
8 and 3
2.
9, 12, 18
3.
4, 5, 6
4.
14, 21
5.
5, 8, 20
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ADDITION AND SUBTRACTION OF
FRACTIONS
When adding and subtracting fractions we need to convert the fractions to the same
denominator in order to add or subtract.
The lowest common multiple can be used to find a common denominator when
simplifying fractions.
E.g.,1. Add the fractions below.
2
3
+
5 10
7
1
−
12 16
2
3
−
5 10
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ADDITION AND SUBTRACTION OF
FRACTIONS
3
1
5 +7
7
2
3
6
6 −3
4
7
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MULTIPLYING FRACTIONS
When multiplying fractions we;
1.
Convert to an improper fraction if needed.
2.
Simplify the fractions
3.
Multiply the numerator by the numerator and the denominator by the
denominator.
E.g.,1. Multiply the fractions below.
5x
3 1
x
4 5
3
5
3
1
2
x4
2
3
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DIVIDING FRACTIONS
When dividing fractions we;
1.
Convert to an improper fraction if needed.
2.
Change the division sign to multiplication and flip the second fraction.
3.
Multiply the numerator by the numerator and the denominator by the
denominator.
E.g.,1.Divide the fractions below.
36 ÷
2
1
÷
3
6
4
7
1
11
5 ÷2
4
12
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FRACTIONS PRACTICE
2 marks
2 marks
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DECIMAL PLACES AND SIGNIFICANT FIGURES
To correct 0.07822 to two decimal places (d.p.) we look at the third decimal place.
If it is 5 or more we add 1 to the figure in the second place.
If it is less than 5, we do not alter the figure in the second decimal place.
0.07. 822
8 is 5 or greater therefore, we add one to the figure in the second place.
Our answer = 0.08
Give each of the following numbers correct to the number of decimal places in brackets.
1.
2.7846 (2 d.p.)
2.
0.1572 (3 d.p.)
3.
0.009638 (4 d.p.)
4.
0.456 (2 d.p.)
Another way of approximating numbers is to express them correct to a number of
significant figures.
The first non-zero digit in a number reading from left to right is the first significant
figure.
Example: 0.0406754
0.0406754, rounded to 4 significant figures is 0.04068
0.0406754, rounded to 3 significant figures is 0.0407
0.0406754, rounded to 2 significant figures is 0.041
0.0406754, rounded to 1 significant figure is 0.04
Express the following to 3 significant figures;
1.
72.35
2.
6.735
3.
5.008
4.
0.0000005768
5.
34567
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SCIENTIFIC NOTATION / STANDARD FORM
We’ve seen that we can use smaller and larger versions of the same unit to represent a
measurement that would have otherwise been tedious to write.
We can also keep the same unit but write the answer in standard form/ scientific
notation.
What is scientific notation?
Scientific notation is a way of expressing numbers that are too large or small to
conveniently be written in decimal form (the way that we usually write numbers).
When a number is written in scientific notation it is written as:
𝐴×10)
Where,
A= coefficient
10 = base
n = exponent/index/power
Where A must be a number between 1 and 10 n is the number of places to move the
decimal point.
Why use scientific notation?
Diameter of an atom in decimal form in metres = 0.0000000001m
Diameter of an atom in scientific notation in metres = 1 x 10-10 m
How to express a number in scientific notation?
Place the point in the coefficient to make the
number between 1 and 10
(the coefficient must be less than 10 and it can be a
mixed number. i.e. a decimal)
Place a mark (*) at the original decimal point.
To get the power, we move the new decimal point
to the original point
(where we placed the mark).
The number of ‘jumps’ made is the
power/exponent.
If the point moves to the right the power is positive.
If the point moves to the left the power is negative.
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SCIENTIFIC NOTATION / STANDARD FORM
1. Convert 67 to standard form.
2. Convert 458 to standard form.
1. Convert 0.00567 to standard form.
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SCIENTIFIC NOTATION / STANDARD FORM
How to reverse scientific notation?
Look at the power of 10 to determine how to move the point. The number tells us how
many places to move the point.
Positive means move to the right.
Negative means move to the left.
Move the point in the direction determined and the number of placed determined.
1. Convert 6.5×10#.to decimal (normal)
form.
2. Convert 8.9×10&!.to decimal (normal)
form.
3. Convert 1.45×10&%.to decimal (normal) form.
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