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CONTENTS 1. Some elements of the history of the negative numbers 2. Use of the negative numbers in mathematics 3. Obstacles to the understanding the negative numbers 4. Particular problem of the rule of the signs for the product 5. Conclusion in the form of pedagogical reflection
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Argand, Essay on a manner of representing the imaginary quantities in the geometrical constructions, 1806. The conceptual introduction of the negative numbers was a process of a surprising slowness. There cannot be of course a negative number without the presence of a zero; however, in Europe, the mathematicians have the zero since the XIV century, and it will be necessary to await the end of the XV century to see appearing non-positive numerical beings, which therefore will not be accepted like numbers with whole share. Very quickly the rules of use will be laid down, and the mathematicians will handle the relative numbers, but they will have a very partial understanding of it, with astonishing gaps. Their existence as real quantities is refused. They will be for a long time a computational tool, facilitating the resolution of the equations, for which in addition one will adopt only the positive solutions. Several obstacles can explain this difficulty of recognition: one of the most obvious obstacles will be the absolute zero, below which there is nothing. This difficulty is particularly pointed, for example by the French mathematician Lazare Carnot (1753-1823), member of the Academy of Science and famous mathematician: " to really obtain an isolated negative quantity, it would be necessary to cut off an effective quantity from zero, to remove something of nothing: impossible operation. How thus to conceive an isolated negative quantity? " Geometry of position, 1803. An author of a handbook of mathematics, XIX century, (F Busset), will go as far as making carry the failure of the teaching of mathematics in France to the admission of the negative quantities. He is shocked that it is discussed if there exist "quantities smaller than nothing ". It is for him " the roof of the aberration of human reason ".There is a kind of prevention to handle the zero origin, besides the absolute zero. In the preceding texts, it could be noted that one does not speak about negative numbers, but of quantities. The numbers can be only positive; it is the quantities that can be negative or positive. A negative quantity is defined by an opposition to a positive quantity: a path in a direction, a path in the contrary direction; a profit, a debt... We propose to study here the slow birth of the negative quantities, and the obstacles which had to be overcome to reach the abstract concept of a negative number. 2. Use of the negative numbers in mathematics : It is usual to estimate that the concept of a negative number was born from countable needs (profits and debts). Chinese seems to have used since the first century of our era the " negative numbers ". On the calculation tables, generally, the black rods represent them; red rods represent the positive ones. However they seem only auxiliaries of calculation, there are no negative numbers in the statements of the problems, not either in the answers. They also appear with the Indian mathematicians (Hindu) of the VI and VII century; for example we find them in the writings of Bramagupta (VII century). He teaches the way of making additions, subtractions, etc... on the goods, the debts, nothingness. "a debt cut off from nothingness becomes a good, a good cut off from nothingness becomes a debt." The rules of calculation are given; but one is not worried to justify them. The " negative numbers " thus will appear in calculations, and the mathematicians throughout the history will concentrate better and better to practise operations on these "numbers ", even if the rules are not clearly laid down. In Occident they thus appear at the end of the XV century, at the time of the resolution of the equations, for example in the writings of the Italian mathematician Cardan (1501-1576). Cardan was the first to notice the multiplicity of the values of the unknown quantity in the equations, and the distinction into positives and negatives. This discovery which, with another of Viete, is the base of all those of Harriot and Descartes on the analysis of the equations, this discovery, I say, is clearly contained in its Ars magna. In the third article he observes that the root of a square is also more or less the side of this square, and in article 7 he proposes an equation which, reduced to our language, would be X2 + 4x = 21, and he notices that the value of X is also + 3 or - 7, and that by changing the sign of the second term, it becomes - 3 or + 7. He names these negative roots 4 pretended 4. Cardan will rectify in that the error of Pacioli, who not having mentioned any of these negative roots, seems not to have noticed them. J F Montucla, History of mathematics, 1758. At the same time, other mathematicians, like the French Viete (1540-1603), will only give the positive solutions of the equations.The rules of calculation are built in prolongation of the rules for the positives, and, throughout the history, the mathematicians will practise these calculations better and better, but with a certain embarrassment, because they are in fact generally rules of calculation concerning quantities or sizes which one adds or cuts off, and not positive or negative numbers. Cardan expresses thus his doubts: " It is a simple advise not to confuse the failing quantities with the abundant quantities. It is necessary to add between them the abundant quantities, to add between them also the failing quantities, and to cut off the failing quantities from the abundant quantities, but taking into account the species, i.e. to operate only on the similar ones; to combine the numbers between them, also the squares, in the same way cubes, etc... " Ars Magna, 1545. One imagines a book of accounts in which one writes in a column the expenditure, in the other the incomes, while especially taking care not to mix them.Clairaut (1713-1765), also, on this subject, give his rules, in his " Elements of algebra " of 1746: " It will be asked perhaps if one can add negatives with positives, or rather if one can say that one adds the negatives. With what I answer that this expression is exact when one does not confuse to add with increasing. That two person for example join their fortune, whatever they are, I say that that is to add their good, that one has some debts and some real goods, if the debt exceeds the goods, he will owe only negative, and the junction of his fortune with that of the first one decrease the goods of this one, so that the sum itself will be, or less than what the first had, or even entirely negative " This highlights the confusion between the sign of the operation and the sign of a number, and the differentiation between add and increase, difficulties which are real, as soon as one begins to teach the negative. The distinction will not be really made until the end of the XIX century, but the teaching problem will persist. From the time of Viete, at the beginning of the XVII century, the rules on literal calculation will be perfectly controlled, but the letters always represent positive quantities and never the negative ones. One cannot thus find, for example, X = - 3 as solution of an equation; it would be absurd. 3. Obstacles to the understanding the negative numbers: We already mentioned the problem of the absolute zero and the relative zero . There is for example in the "Dictionary of mathematics " of J Ozanam, of 1691, a score of kinds of numbers, the wholes, the broken (fractional), the incommensurable ones, the deaf ,..., and the negative ones are not mentioned. They appear in the resolution of equations, but they are then qualified as false roots, pretended, by the true one, which are the positive ones. The false root is the denied value of the unknown letter of the equation. Here how Descartes presents the various solutions of an equation: But often it happens, that some of these roots are false, or less than anything, as if one supposes that X indicates also the defect of a quantity, which is 5, one has X + 5 = 0, which being multiplied by x3 – 9xx+ 26x – 24 = 0 is x4- 4x3 -19xx +106x-120 = 0, for an equation in which there are four roots, namely three true ones which are 2,3,4, and a false one which is 5. Descartes, geometry, 1637 We will notice that in this text, Descartes speaks about a false root which is 5.The negative solutions of the equations pose problems with the mathematicians, because they should be interpreted. Here an example of what proposes De Morgan (1806-1871), in 1831, facing negative solutions of a problem: " imaginary expression An example: A father is 56 years old and his son is 29. In how many years the age of the father be will the double of that of his son? Let X be the number years; X checks: 56 + X = 2 (29 + X). We find X = - 2. This result is absurd but if we change X into - X and if we solve: 56 - X = 2 (29 - X) we find X = 2. The negative answer shows that we made an error in the first formulation of the equation. When the response to a problem is negative, by changing the sign of X in the equation which produced this result, we can discover that an error was made in the method used to form this equation or to show that the question raised by the problem is too much limited " One admits the negative quantities in calculations, like obligatory auxiliaries, even if they do not have any meaning by themselves. It is exactly the same position as that of imaginary (that we name the complex numbers nowadays). Faintness appears particularly in the pedagogical writings, because the authors do not manage to give satisfactory explanations. Let us notice that until XVIII century there are few opportunities to handle negative " numbers " which have a physical meaning. In 1730, Reaumur produces the first scientific thermometer and it will still be necessary a century to go so that general public get used to temperature values below zero. In 1713, Farenheit manages to avoid these kinds of temperatures. Some, despite everything, keep very careful feelings, even hostile to the use of the negative quantities, which are definitively not numbers. Here how expresses this Mac Laurin (1698-1746), in his Treaty of the fluxions, in 1742: " The use of the negative sign in algebra gives place to several consequences that one has difficulty to admit and that gave origin to ideas which appear not to have any real basis " Here some of these ideas: Pascal (1623-1662), in his thoughts: " Too much truth astonishes us; I know who can not understand, who of zero removes 4, remains zero " Arnauld (a friend of Pascal): (in connection with the equality And here a frankly hostile reaction, of Francis Maseres, an English mathematician, in his Essay on the use of the negative sign in algebra (1759): " They are useful only in so far as I am able to judge, to darken the very whole doctrines of the equations and to make dark of the things which are in their nature excessively obvious and simple. It would have been desirable in consequence that the negative roots were never allowed in algebra or that they were discarded" In front of such obstacles, avoiding strategies are born: In writing equations: for example, there will be several types of equations of the second degree, which we can quote with our contemporary algebraic writing: x2 + px = q x2 + q= px x2 = px + q (x2 = px is not really of the second degree); it will take a very long time to accept the zero (0) as a given solution that means "nothing ". p and q represent numbers, therefore they are essence positive. For the choice of the axis to locate the points: either one does not take into account the part of the curve corresponding to x or y negative (ex the curve which bears the name of Folium de Descartes, thus named because it represents the cubic of equation x3 + y3 = 3axy, with x and y positives), (see the figure) or one manages to choose axes for which the curve considered corresponds only to positive co-ordinates. It will have to be awaited the XVIII century, so that Mac Laurin, and especially Euler, explain how one can take negative co-ordinates; it is a timid approach of what will be called " the real line ".
For not to have to accept a negative solution of a problem, almost until XX century, if the resolution of an equation leads to a negative solution, one advises to rewrite the problem as we have seen in the text of De Morgan.. 4. Particular problem of the rule of the signs for the product: Here what wrote the French writer Stendhal, in its autobiographical novel: Life of Henri Brulard ", in 1835, to express his distress in respect to the rule of the signs: My great misfortune was this figure:
Let us suppose that RP is the line which separates the positives from the negatives, all that is above is positive, as negative all that is below; how, by taking the square B as many times as there are units in square A, can I manage to make change side the square C? And, while following an awkward comparison that the supremely slow accent from Grenoble of Mr. Chabert made still more awkward, let us suppose that the negative quantities are the debts of a man, how by multiplying 10 000 francs of debt by 500 francs, this man will have or manage to get a fortune of 5 000 000, five million francs? There is a kind of need to accept that negative x negative = positive, if one wants that the ensemble calculations on all the numbers is coherent. In fact, it is the matter, as we have noticed, more of an operation on the signs that on the numbers, given that "a negative number " , is a positive number preceded by the minus sign. Anyway, this need, handled formally without problem runs up against the common sense, even if some mathematicians, among the biggest ones, try to give justifications, often wobbly. To a certain point, the justification is perhaps not a major problem, insofar as everything works well, and one does not arrive to contradictions. It is necessary to arrive to a certain level of epistemological reflection, or to encounter cases where the properties do not work to have a need for non-attackable basis. Explanations:
That of Stevin (1625):
It is a question in fact to compare the surfaces of the rectangles by taking them globally, then by adding the different small parts, and to arrive to in some way while developing (a - b) (c - d) where a, b, c, d are positive reals to the need for writing that (- b) x (- d) = bd. Those of Mac Laurin, (1748) in advance of his time because being formal: One could from there deduce the rule from the signs such as one has habit to state it, which is that similar signs in the terms of the multiplier and the multiplicand give + to the product, and the different signs give -. We avoided this manner of presenting the rule, to save the begginners from the revolting expression - by - gives +, which is however a consequence necessary of the rule: one can, as we made, disguise it, but not contradict it or destroy it; the reader, without realizing it, has observed all its meaning in the preceding examples; familiarized with the thing, could he be still startled by words? If there remains to him/her some scruple on top, he should pay attention to the following demonstration which tackles the difficulty directly. + a - a = 0, thus by any quantity that one multiplies + a - a, the product must be 0: if I multiply it by n, I will have for the first term + na, therefore I will have for the second - na, since it is necessary that the two terms are destroyed. Then do the different signs give - to the product? If I multiply + a - a by - n, by the preceding case, I will have - na for the first term; therefore I will have + na for the second, since it is necessary always that the two terms are destroyed: therefore - multiplied by - gives + to the product. That of Euler, (1770), very naive and not very convincing. It remains still to solve the case where - is multiplied by - or, for example - a by - b. It is obvious initially that as for the letters, the product will be ab; but it is dubious still if it is the sign + or well the sign - that it is necessary to put in front of the product; all that one knows, it is that it will be one or the other of these signs. However I say that it cannot be the sign -; because - a by + b gives - ab and - a by - b cannot produce the same result that - a by + b; but it must result the opposite from it, i.e. + ab; consequently we have this rule: + multiplied by + made +, just as - multiplied by -. We understand well that up to now it is really about the rule of the signs, since there are in fact only negative quantities, indicated by a number positive, and preceded by the sign -. Really it is not a question of the product of two negative numbers. The explanation of Cauchy (1821) accentuates this consideration defining a rule operating on the symbols + and -, therefore not on the negative numbers. According to these conventions, if one represents by A either a number, or an unspecified quantity, and that one makes: a = + A, b = - A. One will have: +a = + A, + b = - A , -a = - A, -b = + A If in the four last equations one gives for A and B their values between brackets, one will obtain the formulas : + (+ A) = A + ( - A ) = - A - ( + A ) = - A - ( - A ) = + A In each one of these formulas the sign of the second member is what is called the product of the two signs of the first. To multiply two signs one by the other is to form their product. The inspection alone of the equations is enough to lay down the rule of the signs. There is a kind of confusion between the sign - which means the opposite; and Cauchy is based in fact on the fact that the opposite of the opposite is the number itself; there is no consideration on the product of negative numbers. Hankel (1867) tackles the problem in a completely different form, purely formal. The rules of the addition and the multiplication must be the same ones for all positive or negative reals. From this point of view the negatives have the status of a number, with whole share, and he distinguishes in a clear way the sign - from the opposite and the sign - from the subtraction. What is important is to be able to multiply opposites. His explanation can be summarized in the following way : 0 = a x 0 = a x (b + opp b) = ab + a x (opp b) 0 = 0 x (opp b) = (a + oppa) x (oppb) = a x (oppb) + (oppa x oppb) then (oppa) x (oppb) = ab. Other proposals were made at the beginning of the XIX century by Wessel, Argand..., giving a geometrical interpretation of the complex numbers, including the negatives. All these mathematicians were quite unknown, and their proposals were not going to be taken seriously until the "big ones ",like Gauss or Cauchy, took them into account themselves. In fact, upheaval brought by Hankel marks the ideological rupture of the mathematical thought of the end of the XIX century in connection with the relations between mathematics and physical reality. Until there, if new "numbers " were invented that shocked the generally accepted ideas, they were automatically qualified incomprehensible, inconceivable, absurd, deaf,irrational, false, imaginary ... Hankel rejects this ideology. He accepts that (-3)2 > (2)2, because this result is coherent with the formal deduction, and he is not concerned with that being shocking for the generally accepted ideas. There are no good models for the negatives, and Hankel refuses this search. The very significant step that it is possible to make at the time of Hankel, and which undoubtedly was not at the time of Mac Laurin, is to be able to consider the numbers as not related to a physical reality, but as mathematical entities which have certain relations between them. The number is not today any more a thing, a substance which would existindependently from the thinking subject or of the objects which occasionthem; it is not any more an independent principle as believed thepythagorians. The question of the existence of the numbers addresses either to the thinking subject, or to the thought objects for which the numbers present relations. The mathematician only holds for impossible, in a strict sense, that which is logically impossible, i.e. which implies a contradiction. It is not necessary to demonstrate that one can admitimpossible numbers in this sense. But if the numbers considered arelogically possible, if their concept is defined clearly and distinctly, if it is thus free of any contradiction, the question cannot be anymore to know if there is in the field of reality, in what is intuitive or iscurrently given, a substrate for this number; if there are objects which can give matter to the numbers as they are intellectual relations of acertain type. Theory of the system of the complex numbers, Hankel, 1867. Hamilton, in 1835, in its work: Theory of conjugate functions; on algebraas the science of Pure Time, will underline this difficulty: that tounderstand numbers and particularly a property as the rule on the sign of a product, it is necessary to remain in the purely formal field, and to be withdrawn from any reference to the physical world. On the contrary, he insists, in the field of the geometry, it is the reference to the physical world which makes it possible to admit, without discussion, for example the fifth postulate of Euclide on the parallels: The postulate of the parallels is admitted by everybody without discussion, because it can be "verified " everyday physically; the rule of the signs, on the contrary runs up against the common sense, therefore needs a solid justification. Let us notice that Hamilton, the inventor of the quaternions, built, at the time when he wrote what precedes, a theory of the couples which allowed a kind of algebraic justification of all the " numbers " and that he would be led to give up, for the quaternions precisely, a property which seemed related to the concept even of a number, namely commutation of the product. Let us remark also that at the same time, nonEuclidean geometries were put up, attacking the postulate of the parallels. Note finally that Hankel is one of those that will work on the ideas of Grassmann, which contributed largely to the construction of the vectors and vector spaces, on a mode rather different of the one of Hamilton. These new considerations on the numbers made their path very slowly, and at the beginning of the XX0 century it still persists a mistrust and a certain difficulty of explaining the negative numbers, in particular in the textbooks. Conclusion in the form of pedagogical reflection: Currently it is not so easy to teach negative numbers. The concrete model,in the form
" profit-debt " for example is a teaching help, in a certain way, but it is not
always possible, it can even become an obstacle. This history shows easily that it is
possible to acquire a certain facility, even an operational virtuosity, formally, without
having comprehension of what one handles. When the interrogations appear, the obstacle is
then created. Let us retain the reflections of Carnot, who posed fundamental problems: it
is not possible that except
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and
negative expression -b resemble in that each one, when they seem the solution of a
problem, they indicate that there is some inconsistency or nonsense. Concerning the
reality of their meaning, both are also imaginary since 0 - a is quite as inconceivable as
) " How a smaller number could be with
larger like larger with smaller? " Wallis (1616-1703): " a being a
positive number, the quotient a/0 is infinite; as a /-1 is larger, the denominator being
smaller, it is larger than infinite while being lower than zero, because the result is
negative " .
if one gives up some laid down rules. The " negatives "are not" numbers
" like the positives.It is perhaps necessary to get convinced that mathematics are
useful to solve theoretical or abstract problems, and not concrete problems. The
difficulty lies in the relations between physical reality and its mathematical modeling.
up