Preliminary Mathematics for online MSc programmes in Data AnalyticsUnit 1: Mathematical notation, sets, functions, exponents and logarithms
Mathematical notation and symbols
Introduction
This introductory block is here to remind you of some important notations and conventions used in Mathematics and Statistics.
Numbers and common notations
The numbers are called natural numbers. These are denoted by (whereas the set denotes all natural numbers including the number 0).
Integers are denoted by and include negative numbers too:
Numbers that can be expressed as a ratio of two integers (that is, of the form where
and are integers, and ) are said to be rational.
Numbers such as cannot be expressed as a ratio of integers; thus they are called irrational.
The set of real numbers includes both rational and irrational numbers and is denoted by .
The reciprocal of any number is found if we divide 1 by that number. For example, the reciprocal of is and the reciprocal of is 3. Note that the old denominator has become the new numerator, and the old numerator has become the new denominator.
The absolute value of a number can be thought of as its distance from zero. This is denoted by vertical lines around the number. For example, (read "the absolute value of 6") is , and is again.
The factorial of a non-negative integer number is denoted by (read " factorial") and is the product of all positive integers less than or equal to . For example . We also define to be equal to .
Using symbols
Mathematics provides a very rich language for the communication of different concepts and ideas. In order to use this language it is of high importance to appreciate how symbols are used to represent physical quantities, and to understand the rules and conventions that have been developed to manipulate them.
The choice of which letters or symbols to use is up to the user, although it is helpful to choose letters that have some meaning in any particular context. For example, if we wish to choose a symbol to represent the temperature in a room we might choose the capital letter . Usually the lowercase letter is used to represent time. Since both time and temperature can vary we refer to and as variables. In a particular calculation some symbols represent fixed and unchanging quantities and we call these constants.
We often reserve the letters , and to stand for variables and use the earlier letters of the alphabet, such as , and , to represent constants. The Greek letter is used to represent the constant which appears in the formula for the area of the circle. Other Greek letters are frequently used, and for reference the Greek alphabet is given below.
Letter
Upper case
Lower case
Letter
Upper case
Lower case
Alpha
Nu
Beta
Xi
Gamma
Omicron
Delta
Pi
Epsilon
or
Rho
Zeta
Sigma
Eta
Tau
Theta
or
Upsilon
Iota
Phi
or
Kappa
Chi
Lambda
Psi
Mu
Omega
Mathematics is a very precise language and care must be taken to note the exact position of any symbol in relation to any other. If and are two symbols, then the quantities , and can all mean different things. In the expression , is called a superscript while in the expression it is called a subscript.
If the letters and represent two numbers, then their sum is written as .
Subtracting from yields . This quantity is also called the difference of and .
The instruction to multiply and is written as where usually the multiplication sign is omitted and we simply write . This quantity is called the product of and .
Note that is the same as . Because of this we say that multiplication is commutative.
Multiplication is also associative. When we multiply three quantities together, such as , it doesn't matter whether we evaluate first and then multiply the result by , or evaluate first and then multiply the result by . In other words, .
The quantity (or x/y) means that is divided by . In the expression the top line is called the numerator and the bottom line is called the denominator. Division by leaves any number unchanged (i.e. is simply ) while division by is never allowed.
The equals sign, , is used in several different ways:
It can be used in equations. The left-hand side and right hand side of an equation are equal only when the variable involved takes specific values known as solutions of the equation. For example, in the equation , the variable is and the left-hand side and right-hand side are equal when has the value . If has any other value the two sides are not equal.
It can be used in formulae. Physical quantities are often related through a formula. For example, the formula of the length, , of the circumference of a circle expresses the relationship between the circumference of the circle and its radius . It specifically states that . When used in this way the equals sign expresses the fact that the quantity on the left is found by evaluating the expression on the right.
It can also be used in identities. At first sight an identity looks like an equation, except that is true for all values of the variable. For example, is true for all values of the variable .
The sign is read "is not equal to". For example it is correct to write .
The notation (read "Sigma notation") provides a convenient way of writing long sums. The sum is written using the capital Greek letter sigma, , as .
The notation (read "product notation") provides a convenient way of writing long products. The product is written using the capital Greek letter Pi, , as .
Inequalities
Given any two real numbers and , there are three mutually exclusive possibilities:
( is greater than ),
( is less than ), or
( is equal to ).
The inequality in the first two cases is said to be strict.
The case where " is greater than or equal to " is denoted by . Similarly, we have that .
In these cases, the inequalities are said to be weak.
Some useful relations are:
If and ; then .
If ; then for any .
If ; then for any positive .
If ; then for any negative .
Laws of indices
Indices or powers provide a convenient notation when we need to multiply a number by itself several times. the number is written as and read "5 raised to the power of 3". Similarly we could have
More generally, in the expression , is called the base and is called the index or power.
There are a number of rules that enable us to manipulate expressions involving indices. These rules are known as the laws of indices and they occur so commonly that it is worthwhile to memorise them.
The laws of indices state:
(when multiplying two numbers that have the same base we just add their indices)
(when dividing two numbers that have the same base we subtract their indices)
(if a number is raised to a power and the result itself is raised to a power, the two powers are multiplied together)
Note that in all the previous rules the base was the same throughout.
Two important results that can be derived from these laws are that:
(any number raised to the power of is ), and
(any number raised to the power of is itself).
A generalisation of the third law states:
(when two numbers, and , are multiplied together and they are raised to the same power, each number is raised to that power and they can then be multiplied together).
Negative indices
A number can be raised to a negative power. This is interpreted as raising the reciprocal number to the positive power. For example, .
Generally, we have that and .
Fractional indices
Let's now consider the expression . Using the third law of indices we can write it as
So is a number that when it is raised to the power of equals . That means that it could be or . In other words is a square root of , that is . There are always two square roots of a non-zero number, and we write .
Similarly, we have that
so that is a number that when it is raised to the power of equals . Thus is the cubic root of , that is which is equal to . Each number has only one cubic root.
Generally, we have that is the -th root of , that is defined as .
The generalisation of the third law of indices states that . By taking and we have that .
Polynomial expressions
An important group of mathematical expressions that use indices are known as polynomial expressions. Examples of polynomials are
Notice that they are all constructed using non-negative whole-number powers of the variable. Recall that and so the number appearing in the first expression can be thought of .
A polynomial expression takes the form
where are all constants called the coefficients of the polynomial. The number is also called the constant term. The highest power in a polynomial is called the degree of the polynomial. Polynomials with degree , , and are known as cubic, quadratic, linear and constant respectively.
Tasks
Task 1
Write out explicitly what is meant by the following:
A set is a well-defined, unordered collection of objects. We typically use curly brackets to denote sets, for example .
The objects that make up the set are also known as elements of the set.
If is an element of , we can say that belongs to and write (the symbol reads "belongs to" or "in"). If, on the other hand, an element does not belong to we can write . To give an example, for , , but .
A set may contain finitely many or infinitely many elements.
A set with no elements is called the empty set and is denoted by the symbol .
The number of elements within a set is called the cardinality of the set and is denoted by or .
Given sets and , we say that is a subset of if every element of is also an element of . We can then write . In that case, we can also say that is a superset of ; and write it as . The diagram below (which is known as a Venn diagram) illustrates the definition.
Given sets and , their union is the set of elements that are either in or (or in both).
Given sets and , their intersection is the set of elements that are both in and .
A set is called the complement of if it contains all the elements that do not belong to it. The complement of is written as (or or upper T prime).
Given two sets, upper S and upper T, the differenceupper S minus upper T contains all elements of upper S that are not contained in upper T. The set difference can be, more formally defined as the intersection of upper S and the complement of upper T, upper S minus upper T equals upper S intersection upper T Superscript complement.
Example 1
Let's assume that I asked 43 people if they like dogs or cats. 23 of them said they like dogs, 14 of them told me they like cats while there were 6 people who like both dogs and cats.
If we denote as upper C and upper D the sets referring to the people who like cats and dogs respectively; then we are given the following information:
StartAbsoluteValue upper D EndAbsoluteValue equals 23, StartAbsoluteValue upper C EndAbsoluteValue equals 14 and StartAbsoluteValue upper D intersection upper C EndAbsoluteValue equals 6. Also, there are are 43 people in total. This information is shown on the diagram below (which is known as a Venn diagram).