Chapter 1

Numbers and algebra

The backstory of algebra

In our schooldays, the arrival of x and y on the scene represented the point where mathematics went beyond mere arithmetic and, by acquiring a language all of its own, entered a higher realm. By passing through the portal of algebra, the subject develops surprising power, showing us things that could not be discovered in any other way. Modern science is based on mathematics, which comes about through algebraic manipulation of symbols representing quantities of interest. Algebra is the tool through which exact physical relationships are revealed, including that most famous of equations, E = mc², along with a host of others. Equations like this one, which arises in Einstein’s theory of special relativity, are consequences of physical models based on experiment. Nonetheless, the relationship itself is arrived at through algebra, and it is the undoubted soundness of the underlying algebra that gives authority to the momentous conclusion that energy and mass are one and the same thing. Algebra underpins all modern systematic research. Although its contribution may be embedded in scientific software, without algebra, progress would be impossible.

The word ‘algebra’ is derived from the Arabic word al-gebr, meaning reunion of broken parts. During the 11th century, it was perhaps the Islamic world that represented the most mathematically sophisticated civilization. However, there was no algebraic manipulation of the kind seen in modern texts, and medieval mathematical writing throughout the world of Marco Polo was rhetorical, with everything being described in words. Algebra of a kind that we might recognize did not appear until the 17th century. The scarcity of paper may have held back the spontaneous development of mathematical symbology, but it should be appreciated also that ancient scholars faced obstacles that obscured the underlying mathematical landscape of arithmetic.

When we carry out algebraic manipulation we introduce arbitrary symbols, x and y being the most common, which stand for fixed but unspecified numbers, and these symbols are manipulated according to the laws of arithmetic. The argument underpinning all we do is that, no matter what the numbers x and y may be, the relationships that emerge from our manipulations are true as they are consequences of our initial assumptions and of rules that apply to all numbers, independently of their particular values. The use of algebraic symbols to stand for unknown quantities is a convenient abbreviation, and although that brevity certainly facilitates reasoning, the real power of algebra stems from the universality of interpretation that the symbols afford, which allows them be wielded in a powerful manner that cannot be matched by words alone.

In order to realize the potential of algebra, we need to be able to move our symbols about in an uninhibited fashion, making free use of the operations of arithmetic, particularly the fundamental pairs of operations: addition and subtraction, multiplication and division. For that we need a number system fit for purpose. If, for example, we reject negative quantities as meaningless or, more fundamentally still, we fail to treat zero as a number, we will be handicapped and deny ourselves the freedom that algebra offers to explore the world of unknown quantities. A cloud of confusion needed to be dissipated before the existence of the algebraic world that we take for granted could even be glimpsed, never mind properly understood and developed.

Great minds of the past would have been stunned at the ease with which a modern student can use algebra to completely solve problems that they found impossible and perhaps even had difficulty formulating with any clarity. For example, school algebra is enough to prove that the square root of a whole number, for example or , either is another whole number or is not a fraction at all. The ancient Greek scholars put great effort into this question and used their geometric methods to show that some particular square roots up to were not fractions. The general problem, however, defeated them, yet this and many others beyond the reach of the ancients can be completely understood by the reader of an OUP Very Short Introduction, as you shall see.

The number system

In order to harness the power of algebra we need a number system that meets its demands. Part of those demands is the freedom to perform the four basic arithmetic operations with symbols standing for arbitrary or unknown numbers. However, the collection of ordinary counting numbers has shortcomings in this regard. The numbers that arise through counting, 1, 2, 3, … are known as the natural numbers because they emerge more or less of their own accord once we begin to tally things up. This set of numbers is denoted by ℕ, and ℕ is closed under the operations of addition and multiplication, meaning that if we begin with two natural numbers we may add or multiply them together and the answer is always another natural number. Subtraction, however, is a different story. Subtraction is the taking away of one number from another and is the reverse or, as mathematicians prefer to say, the inverse operation to addition. Applying subtraction in sums such as 3 − 5 where the second number is larger that the first takes us out of ℕ and into the realm of the negative integers, as they are called. When this kind of difficulty arises, we do not give up but rather adopt the attitude that our number system is currently inadequate and should be extended to allow continuation of our calculations.

The standard model of numbers that pervades all of advanced mathematics and engineering is the field of so-called complex numbers, denoted by ℂ. The journey from ℕ all the way to ℂ was long and was not truly completed until the 19th century. Prior to that there was much philosophical agonizing as to the reality, meaning, and validity of numbers other than the natural numbers. We shall, however, introduce the required number types without hesitation.

Having said this, we begin by adjoining to ℕ the number zero, denoted by 0, which may be added to or subtracted from any number without changing its value. It must be conceded that 0 is not a member of the positive integers, as the natural numbers are sometimes called, but 0 is a number nonetheless and will need to find its place in our system of arithmetic. We next introduce a negative mirror image for each positive number; for instance, −6 is the negative partner for 6.

Although not necessary for the development of the subject, it is often easiest to picture and explain the behaviour of numbers by imagining them sitting along the number line. This is a horizontal line with the integers placed at equally spaced points along its length. We place 0 in the middle, the positive integers marching off to the right in their natural ascending order and the negatives occupying the mirror positions to the left of zero.

The collection of all integers, as this set is called, positive, negative, and zero, is represented by the symbol ℤ, while ℚ stands for the collection of rational numbers, which comprise all fractions together with their negatives. The set ℤ lies within ℚ, as an integer n is equal to the rational number n/1. (We say that ℤ is a subset of ℚ, and, in like manner, ℕ is a subset of ℤ.) However, two rational numbers such as 3/9 and 7/21 are considered to be equal as they both cancel down to the same fraction, in this case 1/3. Any positive rational number has a unique representation as a fraction cancelled to lowest terms, a/b, where a and b have no common factor other than 1. The rational numbers too may be pictured as lying along the number line in their natural order, densely and uniformly spread throughout its length.

To add a positive number n to another number m (positive or not), we begin at m and move n places to the right on the number line, while to subtract n we move n places to the left. In the set ℤ, each number n has an opposite, −n, and we now use this feature to define addition of negatives in terms of subtraction. We declare that subtraction of any number n is to mean the same thing as the adding of its opposite, −n, so that adding a negative number −n moves us n places to the left on the number line. It follows that to subtract a negative number −n, we add its opposite, n. In other words, to subtract the negative number −n we move n places to the right on the number line.

This way of looking at things leads to familiar sums such as

(−1) + 4 = 3, 6 + (−11) = −5, (−8) + 6 = −2, 1 − (−9) = 1 + 9 = 10,

as pictured in Figure 1.

The brackets around −1 and other negatives here are not strictly necessary but are introduced to avoid either beginning a string with an operation symbol or a clash of two operation symbols, for instance + and −. The need to do this comes about because we have loaded the minus symbol with two slightly different meanings: the minus sign is used both to indicate the taking of the opposite of an integer, which is an operation on a single number, and also to stand for subtraction, which is an operation on two numbers taken in a particular order.

1. Addition and subtraction on the number line.

Up to this point, we have not invoked anything that you might call a Law of Algebra to explain how our arithmetic works. The justification for our rules depends, rather, on extending the idea of subtraction to the entire collection of integers, which has been ordered in a natural linear fashion. In Chapter 2, we explore the laws that govern arithmetic operations and explain how these rules are extended so that they continue to be respected as we pass from one number system to a greater one that subsumes the former.

Number factorization

Although division of one integer by another generally leads to a fractional answer that lies outside the integers, division of one whole number by another may have an integer outcome, and the nature of how and when this happens is important and finds analogues in other algebraic systems we shall meet, such as polynomials. For that reason, we now record the main features of integer division. We will begin to use both power notation (for instance, writing 2 × 2 × 2 as 2³) and the ‘less than’ and ‘less than or equal to’ signs, < and ≤, respectively (for example, 4 < 7 and −3 < 2, as in each instance the first number lies to the left of the second on the number line). When it is understood that we are dealing with multiplication of numbers represented by letters such as a and b, we normally take the multiplication sign as given and so write ab or sometimes a ⋅ b instead of a × b. We tend to avoid the cumbersome × sign and sometimes write arithmetic expressions like 2 × (−3) × 4 as (2)(−3)(4).

An integer a is a factor or divisor of another integer b if b can be written as b = ac, where c is itself an integer (equally, of course, c is then also a factor of b). A prime is a positive integer such as 71 that has just two positive factors, those necessarily being 1 and the number itself. An integer exceeding 1 that is not prime is called composite, as it is composed of smaller factors. For example 72 = 8 × 9. We say that 8 is a factor of 72 or that 8 divides 72 or that 72 is a multiple of 8: we sometimes denote this relationship by 8|72, which is simply shorthand for ‘8 is a factor of 72’. Successively factorizing the divisors of a given number as far as possible will eventually yield the prime factorization of the number. In our example, 72 = 8 × 9 = 2³ × 3². We could have found the prime factorization of 72 by another route by writing 72 = 6 × 12 = (2 × 3) × (4 × 3) = (2 × 3) × (2 × 2 × 3), but rearranging the prime factors from lowest to highest yields the same result as before, and we say that 2³ × 3² is the prime factorization of 72. The Fundamental Theorem of Arithmetic says that the prime factorization of any natural number n (with the prime factors written in ascending order) is unique. This uniqueness can be deduced from an even more basic property of numbers, Euclid’s Lemma, which says that if a prime number p divides a product ab, so that p|ab, then p is a factor of a or a factor of b (or perhaps a factor of both). An equivalent formulation of Euclid’s Lemma is that if neither a nor b is a multiple of the prime p, then nor is their product ab. Although plausible, this property is not self-evident, and we do not prove Euclid’s Lemma here. We shall, however, explain more about why it holds later in this section. (My VSI Numbers explains in detail all the properties of integers that are taken for granted here.)

The general pattern that arises when one natural number, a, is divided by another, b, is as follows. To divide b into a, we subtract as many b’s as we can from a, q say, until the remainder r < b. In this way, we get a = bq + r. This expression is unique: there is only one value for q and one for r that make this equation true, remembering that we are insisting that 0 ≤ r ≤ b − 1. There are special cases, for instance q = 0 exactly when a < b, in which case r = a. More interestingly, r = 0 exactly when b|a, in which case a/b = q.

As a representative example, if a = 72 and b = 13 then 72 = 13 × 5 + 7, so here we have q = 5 and r = 7. This process of producing the equation a = bq + r for given a and b is known as the Division Algorithm.

One fundamental algebraic idea that we first meet in arithmetic is that of the greatest common divisor (gcd), also known as the highest common factor, of two positive integers a and b. As the name seeks to convey, the gcd of a and b is the largest number d that is a factor of both a and b; since a and b always have at least one common factor, that being the number 1, the gcd certainly exists. We call two numbers a and b relatively prime to each other if their gcd is 1. For example, 15 = 3 × 5 and 28 = 2² × 7 are relatively prime (although neither number is itself prime). The question remains, however, as to how we may compute the gcd of two given numbers.

The gcd, d, can be found through comparison of the prime factorizations of a and b, for the prime factors of d are just those common to a and b. There is, however, a better way of finding it, known as the Euclidean algorithm, which not only is quicker but also reveals other useful relationships. We shall explain the algorithm shortly, but first we draw attention to certain basic properties of common factors.

Suppose that c is any common factor of a and b, so that a = ct and b = cs, say. Then c is also a factor of any number r of the form r = ax + by, where x and y are themselves integers (which may be negative or zero). To show this we locate and ‘take out’ the common factor of c in the expression ax + by, as follows:

Since tx + sy is another integer, we have that c is indeed a factor of r.

An immediate consequence of (1) is that it applies to our Division Algorithm equation written in the form r = a − bq, for it tells us that any common factor of a and b is also a factor of r. By the same token, it follows from a = bq + r that any common factor of b and r is also a factor of a. Hence the set of all common factors of a and b is the same as the set of all common factors of b and r and, in particular, the gcd of a and b is likewise the gcd of b and r. This allows us to work with the pair b and r instead of b and a and, since r < b, this simplifies our problem of finding the gcd as we can now apply the Division Algorithm to the pair (b, r) and repeat the process until the gcd of a and b emerges. This process is known as the Euclidean Algorithm.

Let us act with the algorithm on the pair a = 189 and b = 105. We underline the two numbers in hand at each stage and divide the smaller into the larger, discarding the larger as we proceed from one line to the next. We halt the procedure when the remainder becomes 0, indicating that the remainder on the previous line is the required gcd:

and so the gcd of 189 and 105 is 21 ( 189 = 9 × 21 and 105 = 5 × 21).

These equations themselves have uses as they can be reversed to express the gcd, d, in terms of the original numbers, a and b. We begin with the second last equation and make d the subject, giving in this case 21 = 105 − 84. Then we use each equation in turn to eliminate an intermediate remainder: in our example, the first equation gives 84 = 189 − 105 and so overall we have

There are interesting theoretical consequences as well, which we will call upon in Chapter 6. We proved earlier in this section that any common factor c of a and b is also a factor of any number of the form ax + by, and since the gcd d of a and b has this form, which may be found by reversing the steps of the Euclidean Algorithm, it follows that any common factor c of a and b divides their gcd d. Moreover, a′ = a/d and b′ = b/d have a gcd of 1, for suppose that t is a common factor of a′ and b′ so that a′ = ta″ and b′ = tb″, say. We shall verify that t = 1. (The use of dashes is a way of reminding ourselves that a′ and a″ are factors of a: of course, any new symbol could be used.) The previous equations imply that a = da′ = dta″ and b = db′ = dtb″, whence dt is a common factor of a and b. Since, however, d is the gcd of a and b, it follows that t = 1 and a′ and b′ are indeed relatively prime. In our example, a = 189, b = 105, and d = 21; dividing through by the gcd gives 189/21 = 9 and 105/21 = 5, and 9 and 5 have no common factor (apart from 1).

The fact that the gcd of two numbers a and b may be written in the form ax + by lies at the heart of a host of algebraic proofs about numbers, Euclid’s Lemma being just one of many examples.