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Foundation Mathematics
Dr Dana Mackey
School of Mathematical Sciences
Room A305 A
Email: [email protected]
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Integers
The integer (whole) numbers are: 1,2,3,4, ...(positive integers) and also
-1,-2,-3,... (negative integers).
When two or more integers are multiplied together to form a product, the
numbers are called factors of that product.
For example, 2 × 7 = 14. The numbers 2 and 7 are the factors while 14 is
the product.
Any integer has at least two factors (namely 1 and itself). Some integers
have only 1 and itself as factors. These are called prime numbers. For
example: 13 = 1 × 13.
Other integers have more than two factors. These are called composite
numbers. For example: 24 = 1 × 24 = 4 × 6 = 3 × 8
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Divisibility
A number is said to be divisible by another number if the second number is
a factor of the rst.
Example
26 is divisible by 13 because 26 = 13 × 2.
Integers divisible by 2 are called even numbers (for example, 2,4,6,8, etc).
Integers which are not divisible by 2 are called odd numbers (1,3,5,etc.)
A prime number is not divisible by anything, except 1 and itself.
How do we check whether a larger number is divisible by 2,3,5, etc.?
For example, is 572 divisible by 11?
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It is useful to learn the following divisibility tricks:
A number is divisible by 2 if and only if its last digit is divisible by 2.
A number is divisible by 10 if and only if its last digit is 0.
A number is divisible by 5 if and only its last digit is either 0 or 5.
A number is divisible by 3 (or 9) if and only if the sum of its digits is
divisible by 3 (or 9).
A number is divisible by 4 if and only if the number formed by the last
two digits is divisible by 4.
A number is divisible by 11 if and only if the alternating sum of its
digits is equal to zero.
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Factorizing integers
To factorize an integer means to write it as the product of prime factors
(that is, factors that cannot be factorized further). This factorisation is
then unique.
Example
231 = 3 × 7 × 11;
Z
200 = 2 × 2 × 3 × 5 × 5
(Fundamental theorem of arithmetic)
Every integer can be uniquely written as the
product of prime factors
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The highest common factor (HCF)
If two integers have a factor in common then this is called a common
factor of the numbers. The largest of all common factors is called the
highest common factor, or HCF, of the two integers.
Example
The number 24 has factors 1,2,3,4,6,8,12,24. The number 80 has factors
1,2,4,5,8,10,16,20,40 and 80. The common factors are: 1,2,4,8. The HCF
is 8.
An easy way to nd the HCF of two numbers is to write their prime
factorisation.
Example
Find the HCF of 210 and 720.
210 = 2 × 3 × 5 × 7;
720 = 2 × 2 × 2 × 2 × 3 × 3 × 5
Hence the HCF is 2 × 3 × 5 = 30.
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The lowest common multiple (LCM)
The lowest common multiple (LCM) of two integers is the smallest number
which is divisible by both integers.
Example
The LCM of 2 and 5 is 10. The LCM of 10 and 6 is 30.
To nd the LCM of two numbers we use again prime factorisation.
Example
Find the LCM of 24 and 36.
24 = 2 × 2 × 2 × 3;
36 = 2 × 2 × 3 × 3
Hence the LCM is 2 × 2 × 2 × 3 × 3 = 72.
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Interesting fact
The product of any two integers is equal to the product of their HCF and
LCM.
Example
The HCF of 24 and 36 is 12. Their LCM is 72. Also,
24 × 36 = 864 = 12 × 72
The highest common factor is found by multiplying all the factors which
appear in both factorisations.
The lowest common multiple is found by multiplying all the factors which
appear in either factorisation.
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Powers
Repeated multiplication of a number by itself is called raising to a power.
For example, 3 × 3 × 3 × 3 = 34 (3 raised to the power of 4). the number 4
is called the index and 3 is called the base.
In general, an stands for a × a × a · · · a (n times).
With this notation, we can write prime factorisation in a more concise form:
Example
1350 = 3 × 3 × 3 × 5 × 5 × 2 = 33 × 52 × 2
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Power Rules
1
2
3
Any integer raised to the power 1 is equal to itself, a1 = a.
Any integer raised to the power 0 is equal to 1, a0 = 1.
When multiplying two powers of the same base, we add the indices:
am · an = am+n
4
5
When dividing two powers of the same base, we subtract the indices:
am
m−n
an = a
The power of a power is achieved by muliplying the indices,
(am )n = am·n
6
7
A negative power is the reciprocal of the corresponding positive power,
a−n = a1n .
The product of two powers with the same index is the power of the
product, am · bm = (a · b)m .
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Review of fractions
Recall the basic operations with fractions:
To multiply two fractions, we multiply both the numerators (top) and
the denominators (bottom):
a·c
a c
· =
b d
b·d
12 2
8
(For example, 29
· 3 = 24
87 = 29 .)
To divide two fractions, we invert and then multiply:
a
a c
a d
÷ = b
= ·
c
b d
b c
d
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To add and substract, ensure that the denominators are the same
and then add (or substract) the numerators. For example, to nd
1
3
6 + 4 we make both denominators 12. Note that 12 is the LCM of 6
and 4! Then we get
1 3
2
3
2+3
5
+ =
+
=
=
6 4 12 12
12
12
To simplify a fraction, factorise both top and bottom and then
eliminate (cancel) common factors. For example,
28 4 · 7 7
=
=
12 4 · 3 3
Example: Simplify
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105 .
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Fractional powers
What does an mean when n is not an integer?
Example
35 = 343. We say that 3 is the fth root of 343 and write
√
5
343 = 3
Fractional power rules:
√
1
an = n a
m √
a n = n am
1
√
1
In particular, a 2 = a (square root) and a 3 =
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√
3
a (cube root).
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Recall that we are not allowed to take the square root of a negative
number. More√generally, we cannot take even roots of negative numbers.
For example, 4 −7 does not exist.
Example
Simplify the following expressions:
2
(−8) 3
1
16− 4
4
27− 3
√
−2
3
a
a √
a5
√
3
(−6) 8 · 5 9
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Algebraic manipulation
The order in which arithmetic operations must be performed is the following
Z
First multiplication/division
then addition/subtraction
So, 6 + 2 × 3 + 4 = 16.
To change the order in which operations are performed we use brackets:
Example
(6 + 2) × 3 + 4 = 28
(6 + 2) × (3 + 4) = 56
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We can get rid of brackets (this is called expanding) by using the
distributive law:
3(x + y ) = 3x + 3y ;
a(x + y ) = ax + ay
Example
Expand (x + y )2 .
(x + y )2 = (x + y )(x + y ) = x (x + y ) + y (x + y ) = xx + xy + yx + yy
= x 2 + 2xy + y 2
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Example
Calculate the following
1
x
−
1
1
=
−
1
x +1
x −1
x +1
=
1 1
−
=
9 11
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Equations
An equation is a statement that two quantitites are equal, taking the form
Left hand side = Right hand side
The equation can be true, for example 2 · 3 = 6 or false such as 0=1.
Usually an equation contains an unknown and to solve the equation
means to nd the value of the unknown for which the equation is true.
Example
Solve the equation
2x + 3 = 13 − 3x
This equation can be written
2x + 3x = 13 − 3
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so 5x = 10
MATH
which means x = 2.
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Note:
An equation can have
Exactly one solution, for example 2x + 1 = 13 has one solution: x = 6;
No solution at all, for example 2x + 1 = 4 + 2x or x 2 = −1;
More than one solution; for example x 2 = 1 has two solutions (1 and
−1, while x 3 = x has three solutions (1,0 and −1).
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