Department of Mathematics and Physics
Mathematics 1
Core Notes
,1. Sets A set is a collection of objects. We use curly brackets { } to show sets. The
members of a set are called its elements, symbol . eg. The set P = {2,4,6}
Definition
has three elements. We write this as n(P) = 3 Also, 2 P but 3 P.
Subset If every element of P is also in Q we say P is a subset of Q. This is written
P Q.
Intersection The intersection, symbol , of two sets has elements common to both sets
Union The Union , , of sets is the set we get when all the members of the sets are
put together.
Universal Set The universal set , U, contains all elements relevant to the discussion.
Empty Set The empty or null set ,denoted, or { } has no members in it
Venn Diagrams A Venn diagram is useful to show the relation between two sets. We draw a
circle to represent each set.
Example Draw a Venn Diagram to show the relation between the sets: P = {2,4,6,8}
and Q = { 1,2,3,4,5}. What is n(P Q)? What is n(PQ)?
P and Q have common members, 2 and 4. We shows this in the overlapping
Solution region
Venn Diagram P 1 Q
6 3 P Q ={2, 4} , (PQ) = {1,2,3,4,5, 6,8}
2
8 5 , n(P Q) = 2 , n(PQ) = 7
4
Sets of numbers The table below shows important sets of numbers
Set name Description Members
N Natural numbers {1, 2, 3…..}
W Whole numbers {0,1, 2, 3…}
Z Integers; {3, 2, 1, 0, 1, 2…}
Q Rationals includes integers, {3/2 , 7/9, 4/1 …}
Irrationals: Non terminating, 2 , 5 , etc
Non- repeating
Real numbers Rationals + irrationals
Number line The number line is useful to show sets: Example {x | –1< x 4}
Practice
1. Given A = { 1, 2, 3, 4} B = { 1, 4, 6, 8} C = {2, 4, 5, 8, 10} and
U = {1,2,3,….11} find (a) A B (b) B C (c ) A (BC} (d) (AB) (e) n(AC)
1
,2. Solve each inequality over . and draw the graph:
(a) 4 – 10x 8 (b) 3x2 + 10x – 8 < 0 (c ) x2 > 2x + 3 (d) x2 – 5x 0
3. A survey of companies was carried out to determine which companies had stock control and
payroll systems. 32 companies had both stock control and payroll systems, 65 had just one of these
systems and 40 firms had a payroll system but no stock control system. If 22 companies had
neither of these systems how many companies were surveyed? Ans 119
4. In a survey of 39 Internet users, it was noted that 10 use Google (G) , 18 use MXit (M) and 19
use Facebook (F), 3 use all three, , 4 use MXit and Facebook, 12 use MXit but not Google or
Facebook, and 4 use Google and Facebook but not MXit. Draw a Venn diagram and determine
how many a) use none of the 3 search engines b) use Google and MXit c) use MXit or Facebook
d) use exactly one search engine
2. Approximation
In practical calculations most real numbers must be rounded. Approximate numbers are derived from
measurements or calculations where rounding has been applied. The answers you get by multiplying, or
dividing measurements are also approximate. For example,
Length of hypotenuse
Area of circle radius
1 1 cm = R2= (1)2=
= 12 2 2 5 is often rounded to 3,1416
c 2
5 is 2,23606…. m
Other examples of rounded numbers are: The number of atoms in one mole is 6,022 1023 ( means
approximately, given to 4 s.f), atomic diameters are 1 – 3 Å (1 Å = 1010 m), and the speed of light is
3108 m. s1.
Rounding Numbers . Decimal places.
In general when rounding look at the digit after the one of interest and use the rule: Round down if the digit
is < 5; Round up if it is 5 or more.
Significant figures (s.f)
To determine which digits in a measurement are significant use these rules:
1. Every nonzero digit in a recorded measurement is significant. 12,7 m, 0,143 m and 201 m all have three
significant figures.
2. Zeroes between nonzero digits are significant. The measurements 1003 m, 40,23 m, and 2,503 m all
have four significant figures.
3. Zeroes in front of all nonzero digits are merely placeholders; they are not significant. 0,0000054 has two
significant figures.
1. If the digit is 0, 1, 2, 3 or 4 round down
2. If the digit is 5, 6, 7, 8 or 9 round up
For example: 23,453 rounds to 23,5 correct to 3 s.f
5,0746 rounds to 5,08 correct to 2 d.p
0,00271 rounds to 0,003 correct to 3 decimal places (d.p) 0,00517 rounds to 0,0052
correct to 2 s.f 2
, Standard Scientific notation
The mass of the Earth is 5 979 000 000 000 000 000 000 000 000 kg. This can be expressed in scientific
notation as 5,979 1027 kg. In standard scientific form numbers are expressed as A 10 n where 1,0 A <
10,0 and the exponent on 10 represents the number of places the decimal should be moved. It is much easier
to tell at a glance what the order of magnitude is. The diameter of an ammonia molecule is 0,000 000 029 7
cm which is more easily written as 2,97 108. The significant figures, in a number in scientific notation,
are the number of digits in A. The number
4 ×105 has only one digit in A, so it has one significant figure. 9,304×105 has 4 significant figures.
Scientific notation is a convenient way of expressing very large or very small numbers. (rather than
counting zeros). The scientific notation for a length of 0,06051 m is 6,051102 m.
Order of Operations
In science and maths, the solution of many problems involve a series of calculations and we need to follow a
set of rules regarding which calculations to do first. For example, what is:
5×3 + 2 6 (23−5)2 = ?
The correct order of operations is: Parentheses, Exponentiation, Multiplication or Division, and finally
Addition or Subtraction. (mnemonic BODMAS). So the answer is 18.
Prefixes
The following metric prefixes are common. Remember them.
prefix exa peta tera giga mega kilo deka deci centi milli micro nano pico femto
1018 1015 1012 109 106 103 101 101 102 103 106 109 1012 1015
Notes:
1. Your answer should not have more significant figures than the number of significant digits in the least
precise measurement. For example, If R = 3,47 m , the area of the circle = R2 = (3,47)2 = 37,827603 (full
calculator display) should be given as 3,78 103 m2 to 2 d.p as the original radius was given to 2 d.p.
2. In calculations involving several stages, do not round off intermediate results. Round only the final
answer.
3. Be careful when the base is a unit of measure. Illogical results such as 2,3 L2 or 5,4 cm4 implies you have
failed to properly cancel terms.
Practice.
1. Write the following in standard form rounded to 2 significant figures:
(a) (7,21 106)(4,27103) (b) (0,003046 )1/3
2. Write the following in expanded notation (without the prefixes) giving the final answer in standard form
to 2 s.f:
(a) 2,76 MW (b) 12,75 mg (c) 2,57 m
(d) 12,75 pF (e) 2,97 104 TW (f) 12,765 kPa
3
Mathematics 1
Core Notes
,1. Sets A set is a collection of objects. We use curly brackets { } to show sets. The
members of a set are called its elements, symbol . eg. The set P = {2,4,6}
Definition
has three elements. We write this as n(P) = 3 Also, 2 P but 3 P.
Subset If every element of P is also in Q we say P is a subset of Q. This is written
P Q.
Intersection The intersection, symbol , of two sets has elements common to both sets
Union The Union , , of sets is the set we get when all the members of the sets are
put together.
Universal Set The universal set , U, contains all elements relevant to the discussion.
Empty Set The empty or null set ,denoted, or { } has no members in it
Venn Diagrams A Venn diagram is useful to show the relation between two sets. We draw a
circle to represent each set.
Example Draw a Venn Diagram to show the relation between the sets: P = {2,4,6,8}
and Q = { 1,2,3,4,5}. What is n(P Q)? What is n(PQ)?
P and Q have common members, 2 and 4. We shows this in the overlapping
Solution region
Venn Diagram P 1 Q
6 3 P Q ={2, 4} , (PQ) = {1,2,3,4,5, 6,8}
2
8 5 , n(P Q) = 2 , n(PQ) = 7
4
Sets of numbers The table below shows important sets of numbers
Set name Description Members
N Natural numbers {1, 2, 3…..}
W Whole numbers {0,1, 2, 3…}
Z Integers; {3, 2, 1, 0, 1, 2…}
Q Rationals includes integers, {3/2 , 7/9, 4/1 …}
Irrationals: Non terminating, 2 , 5 , etc
Non- repeating
Real numbers Rationals + irrationals
Number line The number line is useful to show sets: Example {x | –1< x 4}
Practice
1. Given A = { 1, 2, 3, 4} B = { 1, 4, 6, 8} C = {2, 4, 5, 8, 10} and
U = {1,2,3,….11} find (a) A B (b) B C (c ) A (BC} (d) (AB) (e) n(AC)
1
,2. Solve each inequality over . and draw the graph:
(a) 4 – 10x 8 (b) 3x2 + 10x – 8 < 0 (c ) x2 > 2x + 3 (d) x2 – 5x 0
3. A survey of companies was carried out to determine which companies had stock control and
payroll systems. 32 companies had both stock control and payroll systems, 65 had just one of these
systems and 40 firms had a payroll system but no stock control system. If 22 companies had
neither of these systems how many companies were surveyed? Ans 119
4. In a survey of 39 Internet users, it was noted that 10 use Google (G) , 18 use MXit (M) and 19
use Facebook (F), 3 use all three, , 4 use MXit and Facebook, 12 use MXit but not Google or
Facebook, and 4 use Google and Facebook but not MXit. Draw a Venn diagram and determine
how many a) use none of the 3 search engines b) use Google and MXit c) use MXit or Facebook
d) use exactly one search engine
2. Approximation
In practical calculations most real numbers must be rounded. Approximate numbers are derived from
measurements or calculations where rounding has been applied. The answers you get by multiplying, or
dividing measurements are also approximate. For example,
Length of hypotenuse
Area of circle radius
1 1 cm = R2= (1)2=
= 12 2 2 5 is often rounded to 3,1416
c 2
5 is 2,23606…. m
Other examples of rounded numbers are: The number of atoms in one mole is 6,022 1023 ( means
approximately, given to 4 s.f), atomic diameters are 1 – 3 Å (1 Å = 1010 m), and the speed of light is
3108 m. s1.
Rounding Numbers . Decimal places.
In general when rounding look at the digit after the one of interest and use the rule: Round down if the digit
is < 5; Round up if it is 5 or more.
Significant figures (s.f)
To determine which digits in a measurement are significant use these rules:
1. Every nonzero digit in a recorded measurement is significant. 12,7 m, 0,143 m and 201 m all have three
significant figures.
2. Zeroes between nonzero digits are significant. The measurements 1003 m, 40,23 m, and 2,503 m all
have four significant figures.
3. Zeroes in front of all nonzero digits are merely placeholders; they are not significant. 0,0000054 has two
significant figures.
1. If the digit is 0, 1, 2, 3 or 4 round down
2. If the digit is 5, 6, 7, 8 or 9 round up
For example: 23,453 rounds to 23,5 correct to 3 s.f
5,0746 rounds to 5,08 correct to 2 d.p
0,00271 rounds to 0,003 correct to 3 decimal places (d.p) 0,00517 rounds to 0,0052
correct to 2 s.f 2
, Standard Scientific notation
The mass of the Earth is 5 979 000 000 000 000 000 000 000 000 kg. This can be expressed in scientific
notation as 5,979 1027 kg. In standard scientific form numbers are expressed as A 10 n where 1,0 A <
10,0 and the exponent on 10 represents the number of places the decimal should be moved. It is much easier
to tell at a glance what the order of magnitude is. The diameter of an ammonia molecule is 0,000 000 029 7
cm which is more easily written as 2,97 108. The significant figures, in a number in scientific notation,
are the number of digits in A. The number
4 ×105 has only one digit in A, so it has one significant figure. 9,304×105 has 4 significant figures.
Scientific notation is a convenient way of expressing very large or very small numbers. (rather than
counting zeros). The scientific notation for a length of 0,06051 m is 6,051102 m.
Order of Operations
In science and maths, the solution of many problems involve a series of calculations and we need to follow a
set of rules regarding which calculations to do first. For example, what is:
5×3 + 2 6 (23−5)2 = ?
The correct order of operations is: Parentheses, Exponentiation, Multiplication or Division, and finally
Addition or Subtraction. (mnemonic BODMAS). So the answer is 18.
Prefixes
The following metric prefixes are common. Remember them.
prefix exa peta tera giga mega kilo deka deci centi milli micro nano pico femto
1018 1015 1012 109 106 103 101 101 102 103 106 109 1012 1015
Notes:
1. Your answer should not have more significant figures than the number of significant digits in the least
precise measurement. For example, If R = 3,47 m , the area of the circle = R2 = (3,47)2 = 37,827603 (full
calculator display) should be given as 3,78 103 m2 to 2 d.p as the original radius was given to 2 d.p.
2. In calculations involving several stages, do not round off intermediate results. Round only the final
answer.
3. Be careful when the base is a unit of measure. Illogical results such as 2,3 L2 or 5,4 cm4 implies you have
failed to properly cancel terms.
Practice.
1. Write the following in standard form rounded to 2 significant figures:
(a) (7,21 106)(4,27103) (b) (0,003046 )1/3
2. Write the following in expanded notation (without the prefixes) giving the final answer in standard form
to 2 s.f:
(a) 2,76 MW (b) 12,75 mg (c) 2,57 m
(d) 12,75 pF (e) 2,97 104 TW (f) 12,765 kPa
3
