3. Number
"There are more things in heaven and earth, Horatio
Than are dreamt of in your philosophy."
W Shakespeare, ‘Hamlet’
3.1 Number theory is one of the more esoteric areas of mathematics and deals basically with the nature and properties of numbers as such. Arithmetic, implicitly included in the National Curriculum 'Number' Programme of Study, deals with operations which combine numbers and is, strictly speaking, a separate topic. Number representation (base and place value) is also fundamentally distinct from both these - 'number' and 'arithmetic facts' are independent of the number representation used and confusion between them is best avoided where possible.
3.2 The corresponding Attainment Target has, with scant regard for consistency, been called 'Number and Algebra'. This echoes the modern approach which treats number and arithmetic as part of Abstract Algebra, notably Group Theory - a much wider topic dealing with the properties of sets of mathematical objects and associated combining operations. Although too abstract for Key Stage 2 pupils in itself the basic properties listed below appear so widely at all levels of mathematics that a cursory encounter with the basic ideas in passing could provide a valuable foundation for future studies. Depending on these properties systems can be divided into very general classes which show common behaviour for many different types of object and operation eg in the context of KS2 both arithmetic and rigid transformations form groups.
| 3.3 Group theory | |
3.6 Integers | |
| 3.4 Natural numbers | |
3.7 Rational numbers | |
| 3.5 Positive integers | |
3.8 Real numbers | |
The significant properties of a group are
a) Completeness - the combination of every pair of members of the set is itself a member of the set:
b) Identity - there is one member of the set (the 'identity') which when combined with each member leaves it unchanged
c) Inverse - for every member there is some member which when combined with it produces the identity element ie combination with it produces the 'identity' object.
d) Associativity - In sequences of operations the order of application is irrelevant
e) Commutivity - the order of operands in combination is irrelevant
The first four of these are necessary conditions for the system to constitute a group; the fifth is not necessary but simplifies the behaviour considerably.
The concept of natural numbers has been proposed as the fundamental foundation for the development of a consistent formulation of all mathematics (as in the now largely abandoned 'New Mathematics' and SMP) but it suffers from a dual personality whose origins are obscured in the mists of time. As ordinal numbers the natural numbers represent the names associated with counting while as cardinal numbers they carry a quantitative significance amenable to arithmetic processes (addition and multiplication). In this sense the reference to the 'base and place value of our counting system' in the KS2 is incorrect - the names of numbers in the context of counting are merely symbols representing their position in the ordered sequence and the '2' in 'twenty one' cannot be interpreted strictly as implying two tens (or the 'one' one unit for that matter) in this context. Coming third in the sack race is not equivalent to 2/3 of coming second. Evidence of this primitive distinction is found in many languages in the verbal forms "first, second, third, fourth, etc" and "one, two, three, four etc" which show a separate origin for the two concepts.
The basic operation in counting is 'succession' ie s.n is to be interpreted as 'the successor of n' as in, for example, 4 = s.3 = s.s.2 = s.s.s.1 etc.
Here the operator s is unary (ie has one operand rather than two). The set is complete, since the successor of every member is also a member, but it has no identity element - there is no element such that s.n = n In general a group which lacks an identity element cannot have inverses but in this case a rank m can be found which satisfies n = s.m except for n = 1. The distributive and commutative properties are inapplicable to unary operators so that the (ordinal) set satisfies all the relevant conditions for a group except for the anomalies of lacking an identity element and inverse of 1.
As cardinals the natural numbers express basically the total size of a set of objects and are associated with two binary operators viz addition ( + ) and multiplication ( . ). In this context they also satisfy the conditions for a group, including associativity and commutivity, but this time with the exceptions of lacking an identity element, an inverse of 1 for addition, and an inverse for multiplication.
The distinction between ordinal and cardinal natural numbers is sometimes regarded as somewhat pedantic but it is often convenient, even essential, to maintain it. To avoid the ambiguity cardinal numbers can be classified as a separate entity - the positive integers.It is a moot point whether or not zero should be included in the natural numbers in the cardinal number context, but it is usually omitted.
Practical instances of the use of ordinal numbers can be found in team running races, for example, where in cross-country racing a team rank is found by aggregating individual ranks, with the winner being the team having the lowest aggregate. In track athletics individuals are awarded 'points' according to their finishing position with the team having the greatest total winning; this appears superficially as a cardinal system but it can equally well be interpreted as ranking individuals in order of slowness, thus using ordinal numbers. Both of these systems give the same outcome but in Association Football Leagues the old system (0 for a loss, 1 for a draw, and 2 for a win - an ordinal system in disguise) and the new system (0 for a loss, 1 for a draw, and 3 for a win - unequivocally a cardinal system) may give different outcomes.
As noted above the positive integers are closely related to the natural numbers, being their cardinal number incarnation ie they have a definite quantitative significance. It remains a moot point whether or not zero is included and this should be (but is not always) stated specifically; a precise convention is to call the set with zero 'non-negative integers' and the set without zero 'positive integers' but this is not always observed.
The set is closed under both addition and multiplication - the sum or product of any two members is itself a member of the set, and the set of non-negative integers has an identity element for both addition and multiplication (0 and 1 respectively) but the set of positive integers has no identity element for addition. Neither have inverses for addition or multiplication - the result may be negative or fractional, neither of which are members of the set. The properties of associativity and commutivity are satisfied in both cases for both addition and multiplication however.
The non-negative integers are appropriate for situations such as monitoring the egg production of a flock of chickens or the number of people in a room for example, where the values cannot be negative or fractional.
The integer number system is derived from the positive integers by extension to include zero and negative integers. This removes the difficulty of not having an identity and inverse operation for addition but perpetuates the lack of a multiplicative inverse.
The concept of negative numbers raised philosophical objections at the time of its introduction (around 1500 AD in Europe) viz that it is not possible to display a negative number of any concrete object. The objection possibly derives from the fact that arithmetic techniques in Europe were then largely based on counting processes, which require that some concrete objects exist (at least in the form of a 'tally') to be counted. The objection disappears when the distinction between ordinal and cardinal numbers is recognised and the modern view is that numbers are abstract constructions and any mathematical object that can be defined precisely with self-consistent rules for its manipulation is a proper subject for study whatever practical significance might be appropriate to it. It is noted however that proof of the self-consistency of arithmetic is not only notably lacking but now known to be fundamentally impossible to achieve.
It is worthy of note that the issue of negative numbers has not entirely disappeared since in computer languages the question of whether -5, say, represents a different number from 5 or is the number 5 acted on by a unary operator which reverses the sign has some practical significance. Multiple use of symbols such as this (called symbol overloading - the minus sign may be a binary operator 'subtract' or a unary operator 'reverse sign') influences the design of programming language compilers significantly.
In the practical applications of negative numbers there are two slightly different situations. In measuring physical quantities such as temperature, for example, the measuring scale may have an arbitrary zero (as in the Celsius but not the Kelvin scale) and there is no inherent difficulty in conceiving or displaying an object with a temperature less than zero - positive and negative temperatures have intrinsically the same nature. In accounting and banking however it is conventional to assign positive numbers to credits and negative numbers to debits - entities which have intrinsically different natures.
The rational number system is derived from the integer system by including as numbers the ratio of any two integers except zero as denominator. This removes the problem of an inverse for multiplication except for division by zero, which is forbidden.
It is noted however that the inverses of both addition and multiplication, regarded as operations, are non-commutative ie (a - b) <> (b -a) and a/b <> b/a.
Rational numbers are found in one form or another in many of the most ancient systems we know of, probably due to the practical importance of calculations involving sharing resources between a number of objectives. Most however have preferred the ratio form to the expansion form for non-integers. This preference may stem from the fact that in general the expansion form is less compact than the fraction form - no small matter when paper is a scarce commodity and calculations are all by hand eg decimal expansions (of rational numbers) terminate when the prime factors of the denominator contain only 2's and 5's but otherwise they fall into a pattern of repeating digits which extends to infinity.
In contrast to the integer number systems, rational numbers provide a continuous scale ie a rational number arbitrarily close to any given number can always be found. The number system used by digital computers is basically rational, sometimes slightly disguised, but practical limitations (in computer word length) mean that arbitrarily fine graduations cannot be achieved in practice, with consequences which are not always given the attention they merit.
The most fundamental limitation is that many of the operations of classical analysis, such as differentiation and integration, are undefined for discrete variables and have to be replaced by approximations which do not always satisfy exactly the familiar relationships applicable to continuous functions. In fact functions of a continuous variable cannot be defined uniquely in the framework of discrete mathematics appropriate to a computer.
A significant practical limitation is that computer calculations other than those involving integers exclusively are always approximate. For example an error is almost always involved in converting a decimal fraction to the internal binary fraction system used by computers: These errors are generally small but their accumulation through the large number of manipulations common in computer operations may become significant. A more serious problem is the large magnification of errors that can occur in many situations of practical importance eg testing for the occurence of zero or equality of two calculated numbers becomes problematic when error is present, however small. A common cause of error magnification in computations is taking the difference of two nearly equal large numbers - the relative error in the difference can increase enormously. These considerations are fairly subtle and likely to cause confusion in some pupils at Key Stage 2 but perhaps care should be taken to avoid contradicting the salient features as far as possible viz
a) Calculations using rational fractions are exact
b) Decimal expansions are almost always approximations
c) In verifying results an estimate of any approximating errors and their effect is as important as the answer.
d) Uncritical belief in the correctness of the results of computer and desk-calculator computations is unjustified.
In contrast to all the previous systems real numbers are unrelated to counting. They are in fact uncountable - the 'infinite' class of real numbers is 'bigger' than the 'infinite' class of integers or of rationals (somewhat surprisingly, the same size as the class of integers). The need for extension of the number system stems from the discovery by the Pythagoreans (between 600 BC and 500 BC) that the square root of 2 cannot be expressed as a rational number. In fact their proof (which was kept secret and persuaded them to abandon arithmetic for geometry permanently) can be extended to cover the square root of any prime number, demonstrating that an infinite class of numbers which cannot be expressed as rationals exists. This class can be generalised as 'algebraic numbers' which result from solving polynomial equations with integer coefficients.
The need for further extension, to include transcendental numbers, was established in the late 19th century with proof that e is not an algebraic number (1873), followed by pi (1882). Many numbers suspected to be transcendental have not yet been proved to be so.The real numbers satisfy all the conditions to qualify as a group provided, again, that division by zero is forbidden, and include all the other classes as special cases.
Perhaps the most relevant point about real numbers from the Key Stage 2 teaching point of view is that the magnitude of any irrational number can never be known exactly (it would take an infinite time to enumerate it, let alone calculate it). They have to be assigned special symbols which are not part of any systematic number representation scheme.
The inner nature of real numbers is difficult to visualise - a common model is to identify them with points on a line (the Greek refuge!) but this itself gives little intuitive feel for the situation. It can be shown that any real number can be approximated arbitrarily closely by a rational number but, nevertheless, an infinite number of irrationals exist between any two distinct rationals (however close) while similarly an infinite number of rationals exist between any two distinct irrationals. It is a mistake however to think of rational and irrational numbers as 'alternating' in some fashion. The pathological, but perfectly valid, function defined by
f(x) = 0 if x is rational, f(x) = 1 if x is irrational
spread havoc through many fields of mathematics in the 20th century.
Necessary and sufficient conditions determining the class of a real number are known viz
Rational: The decimal expansion either terminates or ultimately produces a repeating pattern of digits
Irrational: The decimal expansion does not terminate nor produce a repeating pattern of digits
Alegbraic: A continued fraction expansion of the number ultimately produces a repeating pattern of coefficients
Transcendental: The continued fraction expansion does not produce a repeating sequence of coefficients
Despite the fact that these are necessary and sufficient conditions they do not provide a constructive method of determining the type of any real number since the period of the repeating pattern and the position of its onset are not known in advance in any particular case.
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