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.

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

Sets

Introduction to sets

• 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.

Subset

• 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 ).

• Given two sets, and , the difference contains all elements of that are not contained in . The set difference can be, more formally defined as the intersection of and the complement of , .

Let's now look at some rule for manipulating expressions involving sets.

• For any two sets, and , the intersection of and is the same as the intersection of and , . Similarly, the union of and is the same as the union of and , . This property (which also holds for addition and multiplication of real numbers) is called commutativity. Note that the set difference is not commutative: is not the same as (you can illustrate this on a Venn diagram).
• For any three sets, , and , i.e. the order in which we take intersections does not matter. This property is called associativity in Mathematics. We can use Venn diagrams to illustrate why this property holds. The left column below identifies , whereas the right column identifies . We can see both are the same.

• Similarly, for any three sets, , and , i.e. the order in which we take unions does not matter either. This property is called associativity in Mathematics.

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• Furthermore, for any three sets, , and , This property is called distributivity in Mathematics. We can again try to understand this rule by identifying both the left-hand sided and the right-hand side in a Venn diagram.

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• Similarly, for any three sets, , and ,

You might at first be slightly puzzled by these rules, but you have already been familiar with most of them. You know them from doing arithmetic with numbers. Just think of unions as additions and intersections as multiplications.

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The final two rules are important rules, but have no equivalent in terms of addition or multiplication. They are called De Morgan's laws.

• The complement of the union of two sets equals the intersection of their complements, i.e.

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• De Morgan's laws also state that the complement of the intersection of two sets equals the union of their complements, i.e.

Again, you have already been familiar with these rules. This time not from arithmetic with numbers, but from logical statements involving the English words "not" (complement), "or" (union) as well as "and" (intersection).

Actually, all the rules we have seen so far, not just De Morgan's laws (including commutativity, associativity and distributivity) hold for logical statements involving "and" and "or".

So far we have defined all sets by listing their entries. We can also define sets by stating a property that lets us determine what is an element of the set. For example, the set S consisting of all number which are at least 1 can be written as .

Bounded sets

A set of real numbers is bounded above if there exists a real number that is greater than or equal to every element of the set. That is, for some we have for all . The number , if it exists, is called the upper bound of the set .

A set of real numbers is bounded below if there exists a real number that is less than or equal to every element of the set. That is, for all . The number , if it exists, is called the lower bound of the set .

A set that is bounded below and bounded above is called a bounded set.

Maximum and minimum

If a set :

• has a largest element , we call the maximum element of the set.

• has a smallest element , we call the minimum element of the set.

Can you see a relationship between the existence of bounds and of a minimum or maximum?

The general rules are:

• Finite sets have a maximum if and only if they have an upper bound. Similarly, they have a minimum if and only if they have a lower bound.

• Infinite sets are more complicated. If an infinite set has a minimum (maximum) then it also has a lower bound (upper bound), but the converse is not true. We have seen infinite sets that were bounded below but that did not have a minimum. Similarly, there can be infinite sets that are bounded above but do not have a maximum.

Intervals

If and are two real numbers, the set of all numbers that lie in between and is called an interval. An open interval does not contain its boundary points. The following is an open interval

A closed interval contains its boundary points. The following is a closed interval

Finally, we may have a half-open interval like

The intervals listed previously are all bounded. The following intervals are all unbounded.

The figure below illustrates the definition of these intervals.

Self help

Below is a list of resources you can access for additional reading.

Tasks

Functions

Introduction to functions

• A function is a rule that associates a unique value with any element of a set. A function, , from a set to a set defines a rule that assigns for each a unique element .
• It can also be thought of as a rule that operates on an input , sometimes called the argument of the function, and produces an output .
• In order for a rule to be a function it must produce a single output for any given input. It is however possible that two (or more) inputs are mapped to the same output.
• The set of all values that it "maps" from is called the domain.
• The set of values it maps to is called the range. The mapping can be denoted as where and are the domain and the range of the function respectively.

Standard classes of functions

• Algebraic functions: are functions that can be expressed as the solution of a polynomial equation with integer coefficients. Some examples of polynomial functions are:

  • Constant function ,
  • Identity function ,
  • Linear function ,
  • Quadratic function ,
  • Cubic function .

• Transcendental functions are functions that are not algebraic. Some examples are:

  • Exponential function ,
  • Logarithmic function ,
  • Trigonometric functions like .

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Inverse of a function

We have seen that a function can be regarded as taking an input , and processing it in some way to produce a single output . A natural question is whether we can find a function that will reverse the process. If we can find such a function it is called an inverse function to and is given the symbol . Do not confuse the with an index or a power. Here, the superscript is used purely as the notation for the inverse function.

To find the inverse of a function you can:

1. Replace with (this will make the rest of the process easier).

2. Replace every with a and replace every with an .

3. Solve the equation from Step 2 for , and finally

4. replace with .

Can you find the inverse of the function where is the Celsius measurement and the associated value in Fahrenheit?

(Hint: You will have to end up with a function where the input is the Fahrenheit measurement and is the associated value in Celsius.)

A function , defined on domain , is one-to-one if never has the same value for two distinct points in . This means that if we choose two different values and such that ; then we will have . Functions such as do not possess an inverse since there are two values of associated with each (e.g. if we use and we have that ). Note that every one-to-one function has an inverse.

Tasks

Self help

Exponents and logarithms

Introduction to exponents

An exponent is another name for a power (or an index). Expressions involving exponents are called exponential expressions. For example, consider any positive real number . We define the exponential function to base as .

Sometimes the irrational number is used as the base for the exponential function (called as the natural exponential function). In that case we have . The domain of the function is all the real numbers (this can be also written as while the range of the function is only the positive real numbers .

The figure below shows the graph of the natural exponential function for .

Since the exponential function is monotonic it has an inverse. Its inverse function is the natural logarithm, but more on that later.

Some properties of the exponential function can be seen from the figure:

1. As becomes large and positive, increases without bound. We express this mathematically as as (the symbol reads "goes to" and the symbol refers to infinity).

2. As becomes large and negative, approaches . We write as .

3. The function is never negative.

The property that increases as increases is referred to as exponential growth.

Laws of exponents

The laws of indices and the rules of algebra apply to exponential expressions.

Introduction to logarithms

Logarithms are an alternative way of writing expressions that involve powers, or indices.

Consider the expression . Remember that is the base and is the power. Another way to write this expression is and is stated as "log to base of is ". We see that the logarithm, , is the same as the power in the original expression. The base in the original expression is the same as the base of the logarithm.

The two statements are equivalent. If we write one of them, we are automatically implying the other.

In general, if is a positive constant and then . The number is called the base of the logarithm. In practice most logarithms are to the base or . The latter logarithms, to base , are called natural logarithms and are usually denoted by or (without a subscript, though some use to refer to the logarithm with base , which we will denote by ).

We define the logarithmic function to base as . If we use as the base we then have the natural logarithmic function that we write as . We can see the results of plotting the graphs of the logarithmic (with base ) and the natural logarithmic function for in the figure below.

Some properties of the logarithmic function can be seen from the figure:

1. As increases, both and increase indefinitely. We write this mathematically as as , as .

2. As approaches both and approach minus infinity. We express this as as , as .

3. The value for both functions is when the argument takes the value .

4. Both functions are not defined when is negative or zero. The domain of the function is all the positive real numbers, , while the range of the function is all real numbers .

Laws of logarithms

Just as expressions involving indices can be simplified using appropriate laws, so expressions involving logarithms can be simplified using the laws of logarithms. These laws hold true for any base. However it is essential that the same base is used throughout an expression before the laws can be applied.

Tasks

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Self help

Quadratic Equations

Quadratic Functions

We will take a closer look at quadratics, which are polynomials of degree 2. You may come across quadratics in different forms, such as:

• an expression
• a function
• an equation
• etc.

where , and are constants and

Solving Quadratic Equations

These can be solved by factorising, using the quadratic formula or by completing the square. We will focus on the first two methods only.

Solving quadratic equations by factorising:

To solve a quadratic equation , let's first start by factorising . To do this we need to find two numbers, and , that multiply to give us (i.e., factors of ) and add to give us so that we end up with:

It's easier to illustrate this using an example.

Let's try a few more examples.

Using the Quadratic Formula

Many quadratic equations cannot be solved by factorisation easily, sometimes because they do not have simple factors. The way round this is to use the quadratic formula. The solution of an equation is given by:

The symbol means that the square root has a positive and a negative value, both of which must be used in solving for x.

Let's try out an example that cannot be easily factorised.

For a quadratic equation , is called the discriminant.

• If , the roots are real and distinct;
• If , the roots are real and repeated;
• If , the roots are complex;

This last option takes us into the realm of Complex Numbers. For example, consider the quadratic

This means that .

At this point, one would say that as we cannot find the square root of a negative number, this problem cannot be solved. However, a way to overcome this issue is that we define an imaginary number as

and this allows us to now find . Bearing in mind that (or that , we can say that

Numbers formed by combining the imaginary number with real numbers are called complex numbers. These numbers take the form where is the real part and is the imaginary part; and are real numbers and is the imaginary number.

Self help

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