EPS Foundation Year University of Birmingham LF Mathematics I
1 Exponentials and Logarithms
1.1 Numbers (Things you should know)
Below are several formalisations of things you should already be aware of from previous study.
Throughout this course, we will make extensive use of set notation. A set is a collection of
things which, in this course, will consist of numbers.
If we have a discrete set of numbers, we use curly braces, { and }, to contain the numbers
in the set. We will see some other brackets later on when we study the domain and ranges of
functions.
1.1.1 Natural Numbers, N
Natural numbers are used to count discrete objects:
N = {1, 2, 3, 4, 5, . . .} (1.1.1)
There is continued debate about whether zero is a natural number, although it doesn’t matter
for this course.
1.1.2 Integers, Z
Integers introduce us to the idea of negation and negative numbers. It is useful to be able to
count backwards from 1: If I have £10 in my bank account, and spend £12, then my balance
will be −£2 which means I owe my bank £2. Of course, negative numbers also appear in
nature; electronic charge can be positive or negative for example.
Z = {. . . , −2, −1, 0, 1, 2, . . .} (1.1.2)
The symbol, Z, is used because it is derived from the German word “zählen” which translates
into “to count”.
1.1.3 Rational Numbers, Q
Rational (from ratio) represent the division, or quotient (hence Q), of two integers. A rational
number can be expressed as the ratio of two integers a and b: a/b.
1 1 2 1 2 3
Q= , , , , , ,... (1.1.3)
1 2 1 3 3 1
All rational numbers can be expressed as either terminating or recurring decimals. For example:
1
= 0.5 (1.1.4)
2
1
= 0.111111 . . . = 0.1̇ (1.1.5)
9
1
= 0.142857142857 . . . = 0.1̇42857̇ (1.1.6)
7
Page 1
, EPS Foundation Year University of Birmingham LF Mathematics I
1.1.4 Irrational Numbers, Q′
Irrational numbers cannot be expressed as the ratio of two integers. The symbol Q′ is used to
represent all numbers that are not rational. The prime (′) symbol is used to denote a set that
contains only elements that are not in the original set.
When expressed in decimal notation, irrational numbers do not terminate. In other words,
the numbers after the decimal point go on forever and don’t contain any generally repeating
patterns. There are a few examples of irrational numbers that are useful to know about:
π ≈ 3.141592654 . . . (1.1.7)
e ≈ 2.718281828 . . . (1.1.8)
√
2 ≈ 1.414213562 . . . (1.1.9)
1.1.5 Real Numbers, R
All of the above sets of numbers are real numbers. Real numbers can be represented as a
number on a continuous number line.
1.1.6 Base 10 (Decimal) Notation
The modal number of digits on the human hand is 10. While difficult to prove this is the reason
why we generally count in base 10, it might be reasonable to expect it to have developed from
people counting on their fingers.
In general, if the base is n, then n different digits are needed to express numbers. In base
10, we have: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When we express a number in base 10, the
rightmost digit (next to the decimal point) is multiplied by 100 , then the next digit on the left
is multiplied by 101 and so on. All of these parts would then be added together. Numbers to
the right of the decimal point continue the decreasing trend in the exponent. For example:
342 = 3 × 102 + 4 × 101 + 2 × 100 = 300 + 40 + 2 (1.1.10)
27.06 = 2 × 101 + 7 × 100 + 0 × 10−1 + 6 × 10−2 = 20 + 7 + 0.06 (1.1.11)
1.1.7 Parity (Even/Odd)
Integers are either even or odd. This property is known as parity and the following statements
are true:
even ± even = even (1.1.12)
even ± odd = odd (1.1.13)
odd ± odd = even (1.1.14)
even × even = even (1.1.15)
even × odd = even (1.1.16)
odd × odd = odd (1.1.17)
Here, ± means “plus or minus”. It is a shorthand notation to mean that we could add or
subtract.
Even numbers are divisible by 2 (odd numbers are not). If a number, p, is divisible by another
number, q, then p/q will be an integer.
Page 2
1 Exponentials and Logarithms
1.1 Numbers (Things you should know)
Below are several formalisations of things you should already be aware of from previous study.
Throughout this course, we will make extensive use of set notation. A set is a collection of
things which, in this course, will consist of numbers.
If we have a discrete set of numbers, we use curly braces, { and }, to contain the numbers
in the set. We will see some other brackets later on when we study the domain and ranges of
functions.
1.1.1 Natural Numbers, N
Natural numbers are used to count discrete objects:
N = {1, 2, 3, 4, 5, . . .} (1.1.1)
There is continued debate about whether zero is a natural number, although it doesn’t matter
for this course.
1.1.2 Integers, Z
Integers introduce us to the idea of negation and negative numbers. It is useful to be able to
count backwards from 1: If I have £10 in my bank account, and spend £12, then my balance
will be −£2 which means I owe my bank £2. Of course, negative numbers also appear in
nature; electronic charge can be positive or negative for example.
Z = {. . . , −2, −1, 0, 1, 2, . . .} (1.1.2)
The symbol, Z, is used because it is derived from the German word “zählen” which translates
into “to count”.
1.1.3 Rational Numbers, Q
Rational (from ratio) represent the division, or quotient (hence Q), of two integers. A rational
number can be expressed as the ratio of two integers a and b: a/b.
1 1 2 1 2 3
Q= , , , , , ,... (1.1.3)
1 2 1 3 3 1
All rational numbers can be expressed as either terminating or recurring decimals. For example:
1
= 0.5 (1.1.4)
2
1
= 0.111111 . . . = 0.1̇ (1.1.5)
9
1
= 0.142857142857 . . . = 0.1̇42857̇ (1.1.6)
7
Page 1
, EPS Foundation Year University of Birmingham LF Mathematics I
1.1.4 Irrational Numbers, Q′
Irrational numbers cannot be expressed as the ratio of two integers. The symbol Q′ is used to
represent all numbers that are not rational. The prime (′) symbol is used to denote a set that
contains only elements that are not in the original set.
When expressed in decimal notation, irrational numbers do not terminate. In other words,
the numbers after the decimal point go on forever and don’t contain any generally repeating
patterns. There are a few examples of irrational numbers that are useful to know about:
π ≈ 3.141592654 . . . (1.1.7)
e ≈ 2.718281828 . . . (1.1.8)
√
2 ≈ 1.414213562 . . . (1.1.9)
1.1.5 Real Numbers, R
All of the above sets of numbers are real numbers. Real numbers can be represented as a
number on a continuous number line.
1.1.6 Base 10 (Decimal) Notation
The modal number of digits on the human hand is 10. While difficult to prove this is the reason
why we generally count in base 10, it might be reasonable to expect it to have developed from
people counting on their fingers.
In general, if the base is n, then n different digits are needed to express numbers. In base
10, we have: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When we express a number in base 10, the
rightmost digit (next to the decimal point) is multiplied by 100 , then the next digit on the left
is multiplied by 101 and so on. All of these parts would then be added together. Numbers to
the right of the decimal point continue the decreasing trend in the exponent. For example:
342 = 3 × 102 + 4 × 101 + 2 × 100 = 300 + 40 + 2 (1.1.10)
27.06 = 2 × 101 + 7 × 100 + 0 × 10−1 + 6 × 10−2 = 20 + 7 + 0.06 (1.1.11)
1.1.7 Parity (Even/Odd)
Integers are either even or odd. This property is known as parity and the following statements
are true:
even ± even = even (1.1.12)
even ± odd = odd (1.1.13)
odd ± odd = even (1.1.14)
even × even = even (1.1.15)
even × odd = even (1.1.16)
odd × odd = odd (1.1.17)
Here, ± means “plus or minus”. It is a shorthand notation to mean that we could add or
subtract.
Even numbers are divisible by 2 (odd numbers are not). If a number, p, is divisible by another
number, q, then p/q will be an integer.
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