Introduction

However, one of the great achievements of the 19th century was the full realization that the true domain ofnumber is not one-, but rather is two-dimensional. The plane of the complex numbers is the natural arena ofdiscourse for much ofmathematics. This has been brought home to mathematicians and scientists through problem solving–to be able to carry out the investigations required to solve real-world problems, many ofwhich seem to be only about ordinary counting numbers, it becomes necessary to expand your number horizon. The explanation as to how this extra dimension emerges will come towards the end ofthis chapter and be explored further in Chapter 8.

7. Central portion of the numberline near 0

Pluses and minuses

The integers is the name applied to the set of all whole numbers,positive negative, and zero. This set, often symbolized by the letter Z, is therefore infinite in both directions:

{…-4,-3,-2,-0,1,2,3,4,…}.

The integers are often pictured as lying at equally spaced points along a horizontal number line, in the order indicated. The additional rules that we need to know in order to do arithmetic with the integers can be summarized as follows:

(a)to add or subtract a negative integer,-m, we move mspaces to the left in the case ofaddition, and mspaces to the right for subtraction;

(b)to multiply an integer by -m, we multiply the integer by m, and then change sign.

In other words, the direction ofaddition and subtraction of negative numbers is the opposite to that ofpositive numbers, while multiplying a number by-1 swaps its sign for the alternative. For example,8+(-11)=-3,3×(-8)=-24, and(-1)×(-1)=1.

You should not be troubled by this last sum. First, it is reasonable that multiplying a negative number by a positive one yields a negative answer:when a debt(a negative amount)is subject to interest(a positive multiplier greater than 1)the outcome is greater debt, that is to say a larger negative number. We are all well aware of this. That multiplication of a negative number by another negative number should have the opposite outcome, that is a positive result, would then appear consistent. The fact that the product of two negative numbers is positive can readily be given formal proof. The proof is based on the assumptions that we want our expanded number system of the integers to subsume the original one of the natural numbers, and that the augmented system should continue to obey all the normal rules of algebra.Indeed, the result on the product of two negatives follows from the fact that any number multiplied by zero equals zero.(This too is not an assumption but rather is also a consequence of the laws of algebra.)For we now have:

-1×(-1+1)=-1×0=0;

if we then multiply out the brackets, we see that in order that the left-hand side equal zero,(-1)×(-1)must take the opposite sign to(-1)×1=-1;in otherwords(-1)×(-1)=1.

Fractions and rationals

and so we recover the Egyptian decomposition:

This kind of trick is often used to simplify an expression that involves an infinite repeating process. For example, consider the following little monster:

By squaring, and then squaring again, the left-hand side becomes a4, while the expression on the right gives:

Since what follows the 5 is another copy of the expression for a, we infer that a4=20a so that a3=20 or, ifyou prefer, a is the cube root of 20. We will call on this technique again in Chapter 7 when we introduce so-called continued fractions.

Does the class of fractions provide us with all the numbers we could ever need? As mentioned earlier, the collection of all fractions, together with their negatives, form the set of numbers known as the rationals, that is all numbers that result from whole numbers and the ratios between them. They are adequate for arithmetic in that any sum involving the four basic arithmetic operations of addition, subtraction, multiplication, and division will never take you outside the world of rational numbers. Ifwe are happy with that, this set of numbers is all we require. However,we explain in the next section how numbers such as a above are not rational.

Irrationals

Arguments along these lines allow us to show that quite generally,when we take the square root(or indeed the cube or a higher root)of a number, the answer, if not a whole number, is always irrational, thus explaining why the decimal displays on your calculator never show a recurring pattern when asked to calculate such a root.

This problem remained untouchable in classical times. That the answer is cube root of 2 lies outside the range of the euclidean tools was only settled in 1837 by Pierre Wantzel(1814–38),as it requires a precise algebraic description of what is possible using the classical tools in The reason why the smallest solution is in the bill order to see that the cube root of 2 is a number of a fundamentally not hard to see. Any solution nhas to have the for different type. It does indeed come down to showing that you can some positive powers r and s and where the rema never manufacture a cube root out of square roots and rationals.factors are collected together into a single integer When put that way, the impossibility sounds more plausible.divisible by 3 or 5. If we first focus on the possible However, that in no way constitutes a proof.

Transcendentals

Within the class of irrationals lies the mysterious family of transcendental numbers. These numbers do not arise through the ordinary calculations of arithmetic and the extraction of roots. For the precise definition, we first introduce the complementary collection of algebraic numbers, which are those that solve some polynomial equation with integer coefficients:for example x5-3x+1=0 is such an equation. The transcendentals are then defined to be the class of non-algebraic numbers.

It is not at all clear that there are any such numbers. However,they do exist and they form a very secretive society, with those in it not readily divulging their membership of the club. For example,the number π is an instance of a transcendental but this is not a fact that it openly reveals. It will be explained in the next chapter when we explore the nature of infinite sets why it is that‘most’numbers are transcendental, in a sense that will be made precise.

Another way in which the mysterious e arises is through the sum of the reciprocals of the factorials, and this gives a way of calculating e to a high degree of accuracy as this series converges rapidly because its terms approach zero very quickly indeed:

The real and the imaginary

Therstve chapters of this Very Short Introduction dealt mainly with positive integers. We emphasized factorization properties of integers, which led us to consider numbers that have no proper factorizations, which are the primes, a set that occupies a pivotal position in modern cryptography. We also looked at particular types of numbers, such as the Mersenne primes, which are intimately connected with perfect numbers and took time to introduce some special classes of integers that are important in counting certain naturally occurring collections. Throughout all this, the backdrop was the system of integers, which are the counting numbers, positive, negative, and zero.

In this chapter we have gone beyond integers,rst to the rationals(the fractions, positive and otherwise), then to the irrationals,and within the class of irrationals we have identied the transcendental numbers. The underlying system in which all this is taking place is the system of the real numbers, which can be thought of as the collection of all possible decimal expansions. Any positive real number can be represented in the form r=n.a1a2…,where n is a non-negative integer and the decimal point is followed by an innite trail of digits. If this trail eventually falls into a recurring pattern, then r is in fact rational and we have shown how to convert this representation into an ordinary fraction. If not, then r is irrational, so the real numbers come in those two distinctavours, the rational and the irrational.

In our mathematical imaginations, we often picture the real numbers as corresponding to all the points along the number line as we look out from zero, to the right for the positive reals, and to the left for the negative reals. This leaves us with a symmetrical picture with the negative real numbers being a mirror image of the positive reals, and this symmetry is preserved when dealing with addition and subtraction–but not with multiplication. Once we pass to multiplication, the positive and negative numbers no longer have equal status as the number 1 is endowed with a property that no other number possesses, for it is the multiplicative identity, meaning that 1×r=r×1=r for any real number r. Multiplication by 1 fixes the position of any number, but in contrast multiplication by-1 swaps a number for its mirror image on the far side of 0. Once multiplication enters the scene,the fundamental differences in the nature of positive and negative numbers are revealed. In particular, negative numbers lack square roots within the real number system because the square of any real number is always greater than or equal to zero.

This is the cue for imaginary numbers to make their entrance. This topic is one that we shall take up again in the final chapter;for the time being, we willjust make some introductory comments.

This first struck home in the 16th century when Italian mathematicians learnt how to solve cubic and fourth-degree polynomial equations in a fashion that extended that used to solve quadratic equations. The Cardano method, as it came to be known, would of ten involve square roots of negatives even though the solutions to the equations eventually turned out to be positive integers. By stages from this point, the use of complex numbers,which are those of the form a+bi, where a and b are ordinary real numbers, was shown to facilitate a variety of mathematical calculations. For example, in the 18th century Euler revealed and exploited the stunning little equation eiπ=-1, which cannot fail to surprise anyone on their first encounter.

Around the beginning of the 19th century, the geometric interpretation of complex numbers as points in the coordinate plane(the standard system of xy-coordinates), was investigated by Wessell and Argand, from which point the use of the‘imaginary’became accepted as normal mathematics. Identifying the complex number x+iywith the point with coordinates(x, y)allows examination of the behaviour of complex numbers in terms of the behaviour of points in the plane, and this proves to be very illuminating. The theory of so-called complex variables, whose subject matter is represented by functions of complex numbers,rather thanjust real numbers,flourished spectacularly in the hands of Augustin Cauchy(1789–1857). It is now a cornerstone of mathematics, underpins much of electrical signal theory, and the entire field of X-ray diffraction is built on complex numbers.These numbers have proved to have real meaning, and moreover the system is complete in that every polynomial equation has its full complement of solutions within the system of complex numbers. We shall return to these matters in the final chapter.Before doing that, however, we shall in the next chapter look more closely at the infinite nature of the real number line.