Unit (mathematics)

“ We must apply the Law of Unity, and must reduce all identical alternatives before we can count with certainty and use the processes of numerical calculation. Identical alternatives are harmless in logic, but are wholly inadmissible in number. Thus logical science ascertains the nature of the mathematical unit, and the definition may be given in these terms—A unit is any object of thought which can be discriminated from every other object treated as a unit in the same problem. ”

“ Therefore, in future, on this part of the subject I shall cease to talk of any particular units, such as pebbles or acres, and speak of numbers only. A number, considered without intending to allude to any particular things, is called an abstract number: and it then merely signifies repetitions of a unit, or the number of times a unit is repeated. ”

“ The third, fourth, and fifth figures are in the places of the hundreds, thousands, and tens of thousands, for a similar reason. [Pg 8]
16. If the quantity measured had been acres of land, an acre of land would have been called the unit, for the unit is one of the things which are measured. Quantities are of two sorts; those which contain an exact number of units, as 47 yards, and those which do not, as 47 yards and a half. Of these, for the present, we only consider the first.
17. In most parts of arithmetic, all quantities must have the same unit. ”

“ Again, let none be alarmed at vast numbers and fractions, for in calculation it is as easy to set down or to reflect upon a thou­sand as a unit, or the thousandth part of an integer as an integer itself. ”

“ Shew that the nth number in the first is the number of ways in which n can be made up of numbers which, divided by 3, leave a remainder 1; and that the nth number in the second is the number of ways in which n can be made up of numbers which, divided by 3, leave a remainder 2.
It is very easy to shew in how many ways a number can be made up of a given number of numbers, if different orders count as different ways. Suppose, for instance, we would know in how many ways 12 can be thus made of 7 numbers. If we write down 12 units, there are 11 intervals between unit and unit. ”

“ For the unit is the originator of every number, since all multitude is made up by the composition and addition of units; nor is it possible without the unit to conceive the existence of number at all. But the unit itself is independent of multitude, apart from and superior to all number; forming, indeed, and making all, but receiving no increase from any.
13. Kindred to this is the triad; equally indivisible and perfect, the first of those sums which are formed of even and uneven numbers. ”

“ If we are interested, not so much in performing the operations, as in inquiring into what really takes place in a mind when several units are grasped together and made into a new unit,—for example, when twelve units are thought as one dozen,—the mathematician has a right to say: I leave all that to the psychologist or to the metaphysician; every one knows in a general way what is meant by a unit, and knows that units can be added and subtracted, grouped and separated ”

“ While measure is thus conventional there is still another conventional usage in the science of mathematics. In natural units bodies are the higher units of particles, the particle and the body are units of different orders, and the different orders of units in nature are thus coextensive with all the bodies of the universe. Thus there is an infinite system of orders of numbers; but man devises a numerical system where a definite plurality is considered as a higher unity, and such a system serves him a valuable purpose as a labor-saving device for the mental faculties. ”

“ DERIVED UNITS AND THE MEASURE OF A QUANTITY OF ENERGY
It must be understood that it is only by arbitrary convention that a dependency is established between a derived unit and the fundamental units. The laws of numbers in physics are often only laws of proportion. ”

“ Within them is embraced the whole of computation; but because every number, every aggregation of units, is also a new unit capable of being added or subtracted, there are also the operations of multiplication and division, which consists in one case of the addition of several equal numbers together, and in the other, of the subtraction of several equal numbers from a greater until that is exhausted. ”

“ If you set a unit before 20,000,000, it presently makes the number swell up to 120,000,000; and so powerful is a unit, by adding it to numbers, that it will infinitely enlarge them to such a vastness, that shall transcend the capacity of the best arithmetician to count them. ”

“ When we speak of final utility, we think of a series of "increments" supplied one after another, and in this case the successive increments become less and less important, since, after some have been supplied, the want of the kind of good that they represent is less keenly felt. The conception of the series of units is merely a means of isolating one unit from a total number and obtaining a mental measurement of its importance which corresponds with the effective importance of any unit in the entire quantity. ”

“ The double hand of ten fingers and the solar cycle of nearly twelve lunar periods first suggested to man the profound conception of an unit composed of equal units, and thereby originated the idea of a system of numbers, the first step towards mathematical thought. ”

“ It is such a Number, as divided by the rest of the Divisors, or their Product, the Remainder is 0; but divided by its own Divisor, the Remainder is an Unit. We require the Divisors proposed to be Primitive each to other, i.e. that no two or more of them can be reduced to lesser terms by any common Divisor: For if so, the Question may be possible in it self, but not revolvable by help of such Multipliers, such being impossible to be found. The reason is, because the Product of an Odd and Even Number is always Even, and that divided by an Even Number, leaves either Nothing, or an Even Number. ”

“ The result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured, and a certain other magnitude, which should, when possible, be a fixed unit or standard magnitude, or at least an intermediate unit of which the value can be ascertained in terms of the ultimate standard. But though a ratio is the required result, an equation is the mode in which the ratio is determined and expressed. ”

“ If 1⁄4π be commensurable with the unit, let it be (m/n) , m and n being integers: we know that 1⁄4π < 1. We have then
Now it is clear that m2/3n, m2/5n, m2/7n, etc. must at last become and continue severally less than unity. The continued fraction is therefore incommensurable, and cannot be unity. ”

“ As soon as distinct notions have been acquired of the manner in which a collection of ten units becomes a new unit of a higher order, our pupil may be led to observe the utility of this invention by various examples, before he applies it to the rules of arithmetic. ”

“ The power which is given by such an analysis, of expressing with more ease the mathematical laws of phenomena, depends in general on this, that since the calculus applies, not to the increments themselves of the proposed quantities, but to the limits of the ratios of those increments, we can always substitute for each increment any other magnitude more easy to consider, provided that their final ratio is the ratio of equality, or, in other words, that the limit of their ratio is unity. ”

“ A unit is an abstract quantity, represented by a certain standard, and more or less perfectly by copies of the standard.
A unit of mass is the matter of a standard of mass, or a multiple of that. For instance, in Britain the unit of the pound is defined by a piece of platinum preserved in the Standards Office. ”

“ That these may be capable of enumeration it is needful that they be more or less alike; and before any absolutely true numerical results can be reached, it is requisite that the units be absolutely equal. The only way in which we can establish a numerical relationship between things that do not yield us like impressions, is to divide them into parts that do yield us like impressions. Two unlike magnitudes of extension, force, time, weight, or what not, can have their relative amounts estimated only by means of some small unit that is contained many times in both ”

“ Hence, as there is but one kind of quantity in geometry, lines, surfaces, and solids must have identically the same unit of comparison, and must be always perfect measures of each other. The unit may be infinitely varied in size—it being the name or representative of any assumed magnitude to which it is applied—but it always represents a magnitude of a definite form, and hence a magnitude which has an extension in every direction from its centre, and consequently represents not only one in length, but also one in breadth, and one in thickness. ”

“ The upper number is called the numerator, the lower number is called the denominator, and both of these are called terms of the fraction. As long as the numerator is less than the denominator, the fraction is less than a unit: thus, 6/17 is less than a unit; for 6 divided into 6 parts gives 1 for each part, and must give less when divided into 17 parts. Similarly, the fraction is equal to a unit when the numerator and denominator are equal, and greater than a unit when the numerator is greater than the denominator. ”

“ As there is 2⁄5 in the denominator, we must take the part of which 2⁄5 will be a unit. That part is larger than a unit; it is 21⁄2 units; 21⁄2 is that of which 2⁄5 is 1. The above fraction then directs us to repeat 21⁄2 units 31⁄4 times. By extending our word ‘multiplication’ to the taking of a part of a time, all multiplications are also divisions, and all divisions multiplications, and all the terms connected with either are subject to be applied to the results of the other.
If 21⁄3 yards cost 31⁄2 shillings, how much does one yard cost? ”

“ Take, for example, the series of real numbers. Each is what it is, quite definitely and uncompromisingly; it does not pass over by imperceptible degrees into another; it is a hard, separate unit, and its distance from every other unit is finite, though it can be made less than any given finite amount assigned in advance. The question of the relation between the kind of continuity existing among the real numbers and the kind exhibited, e.g. by what we see at a given time, is a difficult and intricate one. ”

“ In all scientific calculations there is assumed an unit to start from, though in each branch of science it is a different unit; e.g. in barology, the pound-weight; in harmonics, the quarter-tone; in other branches of science, other units.68 Analytical research teaches us that the corresponding unit in Syllogism is the affirmative or negative proposition which is primary, immediate, indivisible. In Demonstration and Science it is the Noûs or Intellect.69 ”

“ The notion of order is one which has enormous importance in mathematics. Not only the integers, but also rational fractions and all real numbers have an order of magnitude, and this is essential to most of their mathematical properties. The order of points on a line is essential to geometry; so is the slightly more complicated order of lines through a point in a plane, or of planes through a line. Dimensions, in geometry, are a development of order. The conception of a limit, which underlies all higher mathematics, is a serial conception. ”

“ The statement that mathematics is the science of number would be untrue in two different ways. On the one hand, there are recognised branches of mathematics which have nothing to do with number—all geometry that does not use co-ordinates or measurement, for example: projective and descriptive geometry, down to the point at which co-ordinates are introduced, does not have to do with number, or even with quantity in the sense of greater and less. ”

“ The mathematical questions which have given birth to the Calculus of Variations consist generally in the investigation of the maxima and minima of certain indeterminate integral formulas, which express the analytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. ”

“ In this point of view the numerical solution of questions which we cannot resolve algebraically, and even the calculation of "Definite Integrals," whose general integrals we do not know, really make a part, in spite of all appearances, of the domain of arithmetic, in which we must necessarily comprise all that which has for its object the determination of the values of functions. ”

“ The notion of continuity depends upon that of order, since continuity is merely a particular type of order. Mathematics has, in modern times, brought order into greater and greater prominence. In former days, it was supposed (and philosophers are still apt to suppose) that quantity was the fundamental notion of mathematics. But nowadays, quantity is banished altogether, except from one little corner of Geometry, while order more and more reigns supreme. ”

“ And, indeed, it is the distinctive peculiarity of the calculus in general, that it proceeds on the hypothesis of that absolute equality of its abstract units, which no real units possess; and that the exactness of its results holds only in virtue of this hypothesis. The first ideas of number must necessarily then have been derived from like or equal magnitudes as seen chiefly in organic objects; and as the like magnitudes most frequently observed were magnitudes of extension, it follows that geometry and arithmetic had a simultaneous origin. ”

“ In the most simple questions, and when we have really to resolve only a single equation with a single unknown quantity, this numerical resolution is none the less a very imperfect method, even when it is strictly sufficient. ”

“ When I considered what people generally want in calculating, I found that it always is a number.
I also observed that every number is composed of units, and that any number may be divided into units. ”