Real Numbers

Real numbers (or just reals) can be thought of as a set of values that represent every possible position on a line that starts at some arbitrary point of origin (which we will call zero) and stretches to infinity in two opposite directions. This number line is typically represented in text books as a horizontal line, as illustrated below. The origin (zero) is shown as a point at the centre of the line. Evenly-spaced points are marked on the line to either side of the origin. The points to the right of the origin represent the positive integers, while the points to the left of the origin represent the negative integers. An arrowhead is appended to each end of the line to show that the line extends indefinitely in both directions.


Real numbers are represented as points on an infinitely long number line

Real numbers are represented as points on an infinitely long number line


The set of real numbers is usually signified using the symbol ℝ (a double-struck R). You have probably already realised that since the positive and negative integers lie on the real number line, then they too must be real numbers. The same is true of zero. By definition, the set of real numbers includes the natural numbers (or counting numbers) and the whole numbers (the set of natural numbers plus zero). It also includes what we call rational numbers and irrational numbers (more about these later). You may well have some questions at this point. For example, why are they called "real" numbers, and what kind of number is not real? The simple answer is that there are such things as imaginary numbers, the subject of which will be dealt with elsewhere. Real numbers only started to be called real numbers because mathematicians needed to differentiate between them and these so-called imaginary numbers.


This Venn diagram shows the hierarchy of real numbers

This Venn diagram shows the hierarchy of real numbers


Although we sometimes talk about the set of real numbers, mathematicians more often refer to it in terms of a field or continuum. The word set implies that we are dealing with something that can be counted. When we talk about the set of natural numbers, for example, we know that we can start at one (or zero, depending on how you choose to define the natural numbers) and count to . . . well, infinity. It is of course true that we would never actually finish counting, no matter how long we kept going. Nevertheless, we could continue to count. For that reason, mathematicians tend to describe the set of natural numbers as being countably infinite. So why can't we simply do the same thing with the reals? Well, the key word here is continuum. If you take any two points on the number line, how many points are there between those two points? How many points are there, for example, between zero and one?

Once you really start to think about this question, you begin to realise why we can't count the real numbers. Imagine, for example, a point halfway between zero and one on the number line that represents the value zero-point-five (0.5). Halfway between zero-point-five and one, imagine another point that will represent the value zero-point-seven-five (0.75). Then imagine yet another point halfway between zero-point-seven-five and one that represents zero-point-eight-seven-five (0.875). In fact, we could keep halving the difference like this forever. No matter how small the distance is between two points on the number line, we can always find another point half-way between them. From this, then, we can conclude that there are infinitely many real numbers between zero and one. In fact, we can take any two points on the number line, no matter how close they are together, and there will be an infinite number of points between them. It's a bit like being able to zoom in on a picture indefinitely.

This leads us to some rather strange conclusions. From the above, it is fairly easy to see that there must be infinitely many real numbers on the number line. At the same time, however, we can see that there must be infinitely many real numbers between any two points on the number line, no matter how close together those points may be. Obviously, then, not all infinities are equal! All we can say at this point is that there is a paradox here, but it is not one that we need to worry too much about unless, and until, we undertake the study of higher mathematics. For now, it is enough to understand that we cannot count the real numbers in the same way that we can count the natural numbers. They are said to be uncountably infinite. We can therefore also say that the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers, despite both sets having an infinite number of members.

Real numbers are very useful when we want to measure things in the physical world. Some examples that spring to mind include temperature, voltage, the pressure of a gas, or the density of a liquid. Of course, we often talk about these things in everyday life using integer values. For example, it is sufficient for the purposes of a public weather forecast to give the maximum and minimum temperatures expected on a particular day to the nearest degree. And, let's face it, these predictions are often wrong anyway. In many areas of science and technology, however, we often need to be far more precise. When talking about global temperature variations over a number of years or decades, for example, even small fractions of a degree may be significant. And when we get into areas such as particle physics, the ability to measure unimaginably small differences in things like mass is absolutely crucial. We very much depend on real numbers to enable us to express quantities that constantly vary, especially when the amount by which they vary may be infinitesimally small.

As you can see from the Venn diagram above, all real numbers are either rational (including integers, whole numbers and the natural numbers) or irrational. Let's look at rational numbers first. Significantly, the word rational contains the word ratio. When we talk about rational numbers, we are talking about numbers that can be expressed as the ratio of two integer values. One of the ways in which we can express rational numbers is as a fraction. So, any number that we can express as a fraction is a rational number. Fairly obviously, it follows that any number we can't express as a fraction is not a rational number. Many students at this point ask how this can be, when clearly the set of rational numbers includes the integers (and therefore by definition the set of natural numbers, which is a subset of the integers). These numbers, after all, are not fractions. Nevertheless, they can be expressed as the ratio of two integer values. In fact, they can even be expressed as fractions. We just need to use a denominator of one (1). The following table presents some examples of rational numbers, together with their representation as fractions:


NumberAs a fraction
66/1
√93/1
0.3753/8
1.717/10
3.141592392,699/125,000
0.333r1/3
0.142857142857142857...1/7

The first number in our table is six (6), and demonstrates how we can represent an integer value as a fraction simply by using the number itself as the numerator, with a denominator of one. The second number is the square root of nine (√9). Since nine is a square number (i.e. the result of multiplying an integer value by itself), we can easily simplify √9 to its integer value (in this case three) and then represent it as a fraction in the same way we did previously with six. The decimal fraction zero-point-three-seven-five (0.375) represents the fraction three hundred and seventy-five over one thousand (375/1,000), but since both three hundred and seventy-five and one thousand can be divided by one hundred and twenty-five (125), we can simplify this fraction to three-eighths (3/8). In similar fashion, we can express the number one-point-seven (1.7) as seventeen-tenths (17/10).

The next example may have given you some pause for thought. Surely this is Pi (π)? Well, yes and no. It is an approximation of Pi. Had we written the actual symbol for Pi, rather than a decimal number that approximates Pi, then it would definitely be out of place in our table of rational numbers. Pi is not a rational number, because it cannot be expressed as the ratio of two integers. At the time of writing, a decimal representation of Pi has been calculated to a precision of over ten trillion (10,000,000,000,000) digits, but this is still only an approximation. Having said that, a value of Pi calculated to just forty significant digits is usually more than adequate, even for the kind of calculations typically carried out by astrophysicists and cosmologists.

Remember that any decimal number that terminates (has a finite number of digits after the decimal point) is rational by definition, since it can be represented as the quotient of two integers. This is true even if the number is being used to represent a non-rational value like Pi. A decimal representation that has either a single digit after the decimal point that repeats forever, or a whole sequence of decimal digits that repeats forever, is also a rational number. The penultimate example in our table, zero-point-three recurring (0.333r), is the decimal representation of one-third (1/3), in which the digit three after the decimal point repeats forever. The letter r here stands for recurring, and simply indicates that the digit which it follows will be repeated forever. In the final example, we see the decimal representation of one-seventh (1/7), in which the sequence of digits one-four-two-eight-five-seven (142857) repeats continuously after the decimal point (the repetition is indicated using ellipsis (...). If, as an exercise, you try dividing one by seven (1 ÷ 7) using traditional long division techniques, you will be able to see how this repeating pattern of digits emerges.

We can formally define a rational number as one that can be represented as the quotient of two integers. In other words, we can say that a rational number is a number that can be expressed in the form p/q, where p and q are integer values, and q is not equal to zero (the result of dividing any number by zero is said to be undefined). In the fifth century BCE, The Greek philosopher and mathematician Pythagoras, together with his followers, believed that everything in the universe could be quantified using rational numbers, i.e. either integers or numbers that could be expressed as the quotient of two integers. This belief is understandable up to a point, since even today we are only able to represent quantities that are not rational using either an abstract symbol, or a rational number that is an approximation of the actual value. Admittedly, these approximations are often calculated with an extremely high degree of precision, but they are still approximations.

For the Pythagoreans, everything in the physical world could be described by, or measured using, rational numbers. This concept was central to their philosophy and religion. Indeed, there is certainly no shortage of rational numbers. There are an infinite number of rational numbers, just as there are an infinite number of natural numbers. Indeed, between any two rational numbers, no matter how close together they are on the number line, we can always find another rational number. In fact, we can find an infinite number of rational numbers between any two rational numbers. We can therefore say that the density of rational numbers on the number line is infinite. At the same time, since we know that there are numbers on the number line that are not rational, there must be gaps in between the rational numbers. We can also say that, like the set of natural numbers, the set of rational numbers (usually denoted using the symbol ℚ - a double-struck Q, for quotient) is countably infinite.

This last concept is somewhat more difficult to grasp than the concept that the set of natural numbers and the set of integers are both countably infinite, but it follows from the fact that we can express any rational number as the quotient of two integer values. A graphical representation may help to clarify matters. Consider the table below, in which we show every possible rational number that can be expressed using pairs of positive integers taken from the number range one to ten, expressed as a fraction. Now consider that we could extend the table to have an infinite number of rows and columns, in both the positive and negative directions. Even in our extended table, both the number of rows and the number of columns can be said to be countably infinite, because we can actually count them (even if the task would be a never ending one). Now, since every rational number that exists must be represented by at least one of the cells in our extended table, then the set of rational numbers must itself be countably infinite. In fact, if you think about it for long enough, you will realise that every rational number is represented by an infinite number of cells in our table!


The rational numbers are countably infinite

The rational numbers are countably infinite


Any real numbers that cannot be expressed as the quotient of two integers are said to be irrational numbers. We have already established that the set of real numbers is uncountably infinite, and that the set of rational numbers is countably infinite. Since all real numbers are either rational or irrational, the conclusion must be that there are far more irrational numbers than rational numbers. In fact, it turns out that almost all real numbers are irrational. According to some sources, the Pythagoreans became aware of the existence of irrational numbers thanks to the efforts of one of Pythagoras' students, Hippasus. Although we don't really know that much about Hippasus, one story has it that he discovered the existence of irrational numbers whilst trying to find the length of the diagonal of a unit square (i.e. a square with side lengths of one unit). Far from being celebrated for this discovery, he was allegedly thrown overboard and drowned by some of his fellow Pythagoreans whilst on a sea voyage. They were presumably less than happy that one of their most sacred beliefs - that everything in the universe could be described or measured in terms of rational numbers - had been proved false.

Whether or not there is any truth to the stories about Hippasus, the example of the unit square is as good as any to demonstrate the existence of a number that cannot be expressed as the ratio of two integers - in other words, an irrational number. Consider the diagram below, which shows two adjacent sides of a unit square and one of its diagonals. From Pythagoras' theorem, we know that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. In this case, the two legs of the right-angled triangle are two adjacent sides of a unit square, and the hypotenuse is one diagonal of the same unit square. Since the legs of the right-angled triangle are both one unit in length, their squares added together are equal to two (2). The hypotenuse (i.e. the diagonal of the unit square) must therefore be equal to the square root of two (√2). The decimal representation of this value shown in the diagram was obtained using a calculator, which can obviously only show a limited number of digits after the decimal point (in this case, thirty-one digits). The ellipsis after the last digit simply indicates that the sequence of digits continues, but cannot be displayed.


The length of the diagonal of a unit square is an irrational number

The length of the diagonal of a unit square is an irrational number


One of the defining characteristics of an irrational number is that its decimal representation does not terminate or repeat. In other words, the sequence of digits after the decimal point goes on forever, and contains no regularly repeating pattern. Looking at the decimal representation of √2 displayed by our calculator, there would certainly appear to be no repeating pattern of digits. Of course it is not really possible to say, just from looking at the calculator output, whether or not the sequence of digits would eventually terminate or start to repeat itself. Admittedly it doesn't seem particularly likely, but we need to be certain before we can claim that this number is indeed irrational. As it turns out, we can prove that √2 is not a rational number relatively easily using a little bit of algebra. Let's start with the premise that √2 can be expressed in the form p/q, where p and q are integer values, and q is not equal to zero. We can state this formally as:

√2  =  p
q

The proof we are looking for relies on the assumption that the fraction p/q is reduced to its lowest form, i.e. it cannot be further simplified. This being the case, then either p or q (or both) must be odd, since if both p and q were even, both the numerator and the denominator would have two (2) as a factor, and the fraction could be simplified further. Let's modify our original equation by squaring it to eliminate the square root:

2  =  p2
q2

Rearranging this equation gives us the following:

p2  =  2q2

This means that p2 must be even, since it is equal to the square of some integer (q) multiplied by two. At this point we should point out, for those of you who did not already know, that squaring an even number always gives us an even number, while squaring an odd number always gives us an odd number. It therefore follows that if p2 is an even number then p must also be an even number. Furthermore, if p is an even number, it must be equal to two multiplied by some other number. For argument's sake, let's call this other number n, so that p = 2n. Let's modify our equation once more to include this new term:

(2n) 2  =  2q2

Expanding the term in brackets gives us:

4n2  =  2q2

Simplifying and re-arranging gives us:

q2  =  2n2

So, p2 must also be even, since it is equal to the square of some integer (n) multiplied by two. This in turn means that p must also be even, because we have already established that only an even number can be squared to produce another even number.

What have we proved here? We said above that, for a fraction to be in its lowest form, either the numerator or the denominator (or both) must be odd numbers. We have also stated that the fraction p/q (which is the hypothetical representation of √2 as a rational number) is already in its simplest form. Therefore, if √2 really were a rational number, either p or q (or both) would have to be an odd number. However, we have now shown that both p and q in our original equation must be even numbers. That being the case, then the fraction p/q cannot be in its simplest form, since p and q have a common factor (two). The only conclusion we can reach, given this obvious contradiction, is that √2 cannot possibly be represented as the quotient of two integers, and is therefore irrational.

When you think about it, the square root of any number n can only be rational if n is a perfect square, i.e. a number that results from squaring a rational number. This may be stating the blindingly obvious, but saying it serves to underline the fact that any number that is not a perfect square cannot have a rational root.

There are a couple of other things about real numbers - both rational and irrational - that are worth mentioning. First of all, it should be noted that all rational numbers are also algebraic numbers. An algebraic number is any number that can be a root (i.e. a solution) of a non-zero polynomial equation with rational coefficients. To clarify, consider the following polynomial equation:

x2  -  4x  -  5  =  0

This equation satisfies our requirements in terms of having rational coefficients. In this case, the coefficients of the first two terms are one and four respectively (the last term is a constant, and therefore by implication has a coefficient of one). Obviously, since the coefficients in the polynomial are not all equal to zero, it also fulfills the requirement of being a non-zero polynomial. The root (or roots) of the equation will be any value of x that satisfies the equation (i.e. makes it true). In this case, there are two values of x that satisfy the equation - minus one (-1) and five (5). Both of these values are rational numbers. The graph of the polynomial's corresponding function is shown below, and clearly shows the roots of the equation (i.e. the x coordinates of the points at which the graph intercepts the x-axis).


The graph of y = f(x) = x^2 - 4x - 5

The graph of y = ƒ(x) = x2 - 4x - 5


For every rational number n, there will exist at least one non-zero polynomial equation with rational coefficients that has n as one of its roots. This means that, by definition, every rational number must be an algebraic number. The same cannot be said for irrational numbers. In fact, most irrational numbers are not algebraic numbers. An irrational number that is not an algebraic number is called a transcendental number. This doesn't mean that there is anything mystical about it, just that it cannot be a root of a non-zero polynomial equation with rational coefficients. An example commonly given of an irrational number that is also transcendental is the mathematical constant Pi (π), which represents the ratio of a circle's circumference to its diameter. In this context, however, there is nothing particularly special about Pi, since the majority of irrational numbers - indeed the majority of real numbers - are transcendental. For an example of an irrational number that is not transcendental, we need look no further than the square root of two (√2), which is an algebraic number because it satisfies the simple polynomial equation x2 - 2 = 0. In fact this equation has two roots, √2 and -√2, as you can see below from the graph of the polynomial equation's corresponding function.


The graph of y = f(x) = x^2 - 2

The graph of y = ƒ(x) = x2 - 2