William Fleetwood Sheppard

“ Continued Fractions.—The theory of continued fractions (q.v.) gives a method of expressing a number, in certain cases, as a continued product. A continued fraction, of the kind we are considering, is an expression of the form where b, c, d, . . . are integers, and a is an integer or zero. ”

“ Notation of Numerical Quantities.—Over a large part of the civilized world the introduction of the metric system (§ 118) has caused the notation of all numerical quantities to be in the denary scale. In Great Britain and her colonies, however, and in the United States, other systems of notation still survive, though there is none which is consistently in one scale, other than the denary. The method is to form quantities into groups, and these again into larger groups; but the number of groups making one of the next largest groups varies as we proceed along the scale. ”

“ On the whole, the grouping method refers mainly to concrete numbers and the counting method to abstract numbers. If we sort objects into groups of ten, and find that there are five groups of ten with three over, we regard the five and the three as names for the actual sets of groups or of individuals. The three, for instance, are regarded as a whole when we name them three. If, however, we count these three as one, two, three, then the number of times we count is an abstract number. Thus number in the abstract is the number of times that the act of counting is performed in any particular case. ”

“ All numbers, for instance, which represent physical measurements, are limited in their accuracy not only by our powers of measurement but also by the accuracy of the measure we use as our unit. Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations. ”

“ If we consider that we are dealing with numerical quantities, we must recognize the fact that, while addition and subtraction might in the first instance be limited to such quantities, multiplication and division necessarily introduce the idea of pure number. (ii) If on the other hand we regard ourselves as dealing with pure number throughout, then, as multiplication is continued addition, we ought to include in our classification involution as continued multiplication. ”

“ One result is the formation of a number-series as a mere succession of names without any corresponding ideas of number; the series not being necessarily correct.
When numbering begins, the names of the successive numbers are attached to the individual objects; thus the numbers are originally ordinal, not cardinal.
The conception of number as cardinal, i.e. as something belonging to a group of objects as a whole, is a comparatively late one, and does not arise until the idea of a whole consisting of its parts has been formed. This is the quantitative aspect of number. ”

“ Factors, Primes and Prime Factors.—If we take the successive multiples of 2, 3, . . . as in § 36, and place each multiple opposite the same number in the original series, we get an arrangement as in the adjoining diagram. If any number N occurs in the vertical series commencing with a number n (other than 1) then n is said to be a factor of N. Thus 2, 3 and 6 are factors of 6; and 2, 3, 4, 6 and 12 are factors of 12.
A number (other than 1) which has no factor except itself is called a prime number, or, more briefly, a prime. ”

“ An average is found by adding together several measurements of the same kind and dividing by the number of measurements. In calculating an average it should be observed that the addition of any numerical quantity (positive or negative) to each of the measurements produces the addition of the same quantity to the average, so that the calculation may often be simplified by taking some particular measurement as a new zero from which to measure. ”

“ The £1 is termed the unit. A numerical quantity, therefore, represents a certain unit, taken a certain number of times. If we take £3 twice, we get £6; and if we take 3s. twice, we get 6s., i.e. 6 times 1s. Thus arithmetical processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout. If we retain the unit, the arithmetic is concrete; if we ignore it, the arithmetic is abstract. But in the latter case it must always be understood that there is some unit concerned, and the results have no meaning until the unit is reintroduced. ”

“ Calculation of Square Root.—The calculation of the square root of a number depends on the formula (iii) of § 60. To find the square root of N, we first find some number a whose square is less than N, and subtract a2 from N. If the complete square root is a + b, the remainder after subtracting a2 is (2a + b) b. We therefore guess b by dividing the remainder by 2a, and form the product (2a + b) b. If this is equal to the remainder, we have found the square root. If it exceeds the square root, we must alter the value of b, so as to get a product which does not exceed the remainder. ”

“ Multiplication of a Mixed Number.—When a mixed quantity or a mixed number has to be multiplied by a large number, it is sometimes convenient to express the former in terms of one only of its denominations. Thus, to multiply £7, 13s. 6d. by 469, we may express the former in any of the ways £7·675, 307/40 of £1, 1531/2s., 153·5s., 307 sixpences, or 1842 pence. Expression in £ and decimals of £1 is usually recommended, but it depends on circumstances whether some other method may not be simpler.
A sum of money cannot be expressed exactly as a decimal of £1 unless it is a multiple of 3/4d. ”

“ The second power and third power are usually called the square and cube respectively. If we know p and N, n is called the pth root of N, so that n is the second (or square) root of n2, the third (or cube) root of n3, the fourth root of n4, . . . If we know n and N, then p is the logarithm of N to base n.
The calculation of powers (i.e. of N when n and p are given) is involution; the calculation of roots (i.e. of n when p and N are given) is evolution; the calculation of logarithms (i.e. of p when n and N are given) has no special name. ”

“ The actual limit of the number that can be “seen”—i.e. seen without counting or adding—depends for any individual on the shape and arrangement of the objects, but under similar conditions it is not the same for all individuals. It has been suggested that as many as six objects can be seen at once; but this is probably only the case with few people, and with them only when the objects have a certain geometrical arrangement. The limit for most adults, under favourable conditions, is about four. ”

“ Hindu treatises on arithmetic show the use of fractions, containing a power of 10 as denominator, as early as the beginning of the 6th century A.D. There was, however, no development in the direction of decimals in the modern sense, and the Arabs, by whom the Hindu notation of integers was brought to Europe, mainly used the sexagesimal division in the ′ ′′ ′′′ notation. Even where the decimal notation would seem to arise naturally, as in the case of approximate extraction of a square root, the portion which might have been expressed as a decimal was converted into sexagesimal fractions. ”

“ Arithmetic is supposed to deal with cardinal, not with ordinal numbers; but it will be found that actual numeration, beyond about three or four, is based on the ordinal aspect of number, and that a scientific treatment of the subject usually requires a return to this fundamental basis.
One difference between the treatment of ordinal and of cardinal numbers may be noted. Where a number is expressed in terms of various denominations, a cardinal number usually begins with the largest denomination, and an ordinal number with the smallest. ”

“ It must be remembered that the number expressing any quantity in terms of a unit is inversely proportional to the magnitude of the unit, i.e. the number of new units is to be found by multiplying the number of old units by the ratio of the old unit to the new unit. ”

“ Recurring Decimals are a particular kind of series, which arise from the expression of a fraction as a decimal. If the denominator of the fraction, when it is in its lowest terms, contains any other prime factors than 2 and 5, it cannot be expressed exactly as a decimal; but after a certain point a definite series of figures will constantly recur. ”

“ A quantity expressed in two or more denominations is usually called a compound number or compound quantity. The former term is obviously incorrect, since a quantity is not a number; and the latter is not very suggestive. For agreement with the terminology of fractional numbers (§ 62) we shall describe such a quantity as a mixed quantity. The letters or symbols descriptive of each denomination are visually placed after or (in actual calculations) above the figures denoting the numbers of the corresponding units; but in a few cases, e.g. in the case of £, the symbol is placed before the figures. ”

“ The modern method is to deal with fractions which have 100 as denominator; such fractions are called percentages. They only apply accurately to divisions by 2, 4, 5, 10, 20, 25 or 50; but they have the convenience of fitting in with the denary scale of notation, and they can be extended to other divisions by using a mixed number as numerator. ”

“ One number is less or greater than another, according as the symbol (or ordinal) of the former comes earlier or later than that of the latter in the number-series. Thus (writing ordinals in light type, and cardinals in heavy type) 9 comes after 4, and therefore 9 is greater than 4. To find how much greater, we compare two series, in one of which we go up to 9, while in the other we stop at 4 and then recommence our counting. ”

“ The attention may be directed in succession to the different objects, so that the perception is rhythmical; the distinctive rhythm thus aiding the perception of the particular number.
In consequence of this limitation of the power of perception of number, it is practically impossible to use a pure denary scale in elementary number-teaching. If a quinary-binary system (such as would naturally fit in with counting on the fingers) is not adopted, teachers unconsciously resort to a binary-quinary system. ”

“ Product of Decimals.—To multiply two decimals exactly, we multiply them as if the point were absent, and then insert it so that the number of figures after the point in the product shall be equal to the sum of the numbers of figures after the points in the original decimals. ”

“ On the other hand, it would seem that, for most educated people, sixteen and seventeen or twenty-six and twenty-seven, and even eighty-six and eighty-seven, are single numbers, just as six and seven are, and are not made up of groups of tens and ones. In other words, the denary scale, though adopted in notation and in numeration, does not arise in the corresponding mental concept until we get beyond 100. ”

“ If we take several different units, and write down their successive multiples in parallel columns, preceded by the number-series, we obtain a multiple-table such as the following:—
It is to be considered that each column may extend downwards indefinitely.
37. Successive Multiplication.—In multiplication by repetition the unit is itself usually a multiple of some other unit, i.e. it is a product which is taken as a new unit. When this new unit has been multiplied by a number, we can again take the product as a unit for the purpose of another multiplication ”

“ If we precede the series of convergents by 0/1 and , then the numerator (or denominator) of each term of the series 0/1, 1/0, a/1, ab+1/b . . ., after the first two, is found by multiplying the numerator (or denominator) of the last preceding term by the corresponding quotient and adding the numerator (or denominator) of the term before that. If a is zero, we may regard 1/b as the first convergent, and precede the series by 1/0 and 1/1.
(ii) Each convergent is a fraction in its lowest terms.
(iii) The convergents are alternately less and greater than the true value. ”

“ The calculation of a logarithm can be performed by successive divisions; evolution requires special methods.
The above definitions of logarithms, &c., relate to cases in which n and p are whole numbers, and are generalized later.
44. Law of Indices.—If we multiply np by nq, we multiply the product of p n’s by the product of q n’s, and the result is therefore np+q. Similarly, if we divide np by nq, where q is less than p, the result is np−q. Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa. ”

“ At each stage of the division the number of digits in the reduced dividend is decreased by one. The final dividend being 0000, we have x − 231 = 0, and therefore x = 231.
107. Methods of Division.—What are described as different methods of division (by a single divisor) are mainly different methods of writing the successive figures occurring in the process. In long division the divisor is put on the left of the dividend, and the quotient on the right; and each partial product, with the remainder after its subtraction, is shown in full. ”