number is in theory found by successive divisions. Thus, to find the logarithm of a number to base 2, the number being greater than 1, we first divide repeatedly by 2 until we get a number between 1 and 2; then divide repeatedly by 10√2 until we get a number between 1 and 10√2; then divide repeatedly by 100√2; and so on. If, for instance, we find that the number is approximately equal to 23 × (10√2)5 × (100√2)7 × (1000√2)4, it may be written 23.574, and its logarithm to base 2 is 3.574.
For a further explanation of logarithms, and for an explanation of the treatment of cases in which an antilogarithm is less than 1, see Logarithm.
For practical purposes logarithms are usually calculated to base 10, so that log10 10 = 1, log10 100 = 2, &c.
IX. Units
89. Change of Denomination of a numerical quantity is usually called reduction, so that this term covers, e.g., the expression of £153, 7s. 4d. as shillings and pence and also the expression of 3067s. 4d. as £, s. and d.
The usual statement is that to express £153, 7s. as shillings we multiply 153 by 20 and add 7. This, as already explained (§ 37), is incorrect. £153 denotes 153 units, each of which is £1 or 20s.; and therefore we must multiply 20s. by 153 and add 7s., i.e. multiply 20 by 153 (the unit being now 1s.) and add 7. This is the expression of the process on the grouping method. On the counting method we have a scale with every 20th shilling marked as a £; there are 153 of these 20’s, and 7 over.
The simplest case, in which the quantity can be expressed as an integral number of the largest units involved, has already been considered (§§ 37, 42). The same method can be applied in other cases by regarding a quantity expressed in several denominations as a fractional number of units of the largest denomination mentioned; thus 7s. 4d. is to be taken as meaning 7412s., but £0, 7s. 4d. as £0741220 (§17). The reduction of £153, 7s. 4d. to pence, and of 36808d. to £, s. d., on this principle, is shown in diagrams A and B above.
For reduction of pounds to shillings, or shillings to pounds, we must consider that we have a multiple-table (§ 36) in which the multiples of £1 and of 20s. are arranged in parallel columns; and similarly for shillings and pence.
90. Change of Unit.—The statement “£153 = 3060s.” is not a statement of equality of the same kind as the statement “153 × 20 = 3060,” but only a statement of equivalence for certain purposes; in other words, it does not convey an absolute truth. It is therefore of interest to see whether we cannot replace it by an absolute truth.
To do this, consider what the ordinary processes of multiplication and division mean in reference to concrete objects. If we want to give, to 5 boys, 4 apples each, we are said to multiply 4 apples by 5. We cannot multiply 4 apples by 5 boys, for then we should get 20 “boy-apples,” an expression which has no meaning. Or, again, to distribute 20 apples amongst 5 boys, we are not regarded as dividing 20 apples by 5 boys, but as dividing 20 apples by the number 5. The multiplication or division here involves the omission of the unit “boy,” and the operation is incomplete. The complete operation, in each case, is as follows.
(i) In the case of multiplication we commence with the conception of the number “5” and the unit “boy”; and we then convert this unit into 4 apples, and thus obtain the result, 20 apples. The conversion of the unit may be represented as multiplication by a factor 4 apples1 boy, so that the operation is 4 apples1 boy × 5 boys = 5 × 4 apples1 boy × 1 boy = 5 × 4 apples = 20 apples. Similarly, to convert £153 into shillings we must multiply it by a factor 20£1, so that we get
20£1 × £153＝153 × 20£1 × £1＝153 × 20s.＝3060s.
Hence we can only regard £153 as being equal to 3060s. if we regard this converting factor as unity.
(ii) In the case of partition we can express the complete operation if we extend the meaning of division so as to enable us to divide 20 apples by 5 boys. We thus get 20 apples5 boys = 4 apples1 boy, which means that the distribution can be effected by distributing at the rate of 4 apples per boy. The converting factor mentioned under (i) therefore represents a rate; and partition, applied to concrete cases, leads to a rate.
In reference to the use of the sign × with the converting factor, it should be observed that “7 ℔4 ℔ ×” symbolizes the replacing of so many times 4 ℔ by the same number of times 7 ℔, while “74 ×” symbolizes the replacing of 4 times something by 7 times that something.
X. Arithmetical Reasoning
91. Correspondence of Series of Numbers.—In §§ 33-42 we have dealt with the parallelism of the original number-series with a series consisting of the corresponding multiples of some unit, whether a number or a numerical quantity; and the relations arising out of multiplication, division, &c., have been exhibited by diagrams comprising pairs of corresponding terms of the two series. This, however, is only a particular case of the correspondence of two series. In considering addition, for instance, we have introduced two parallel series, each being the original number-series, but the two being placed in different positions. If we add 1, 2, 3, . . . to 6, we obtain a series 7, 8, 9, . . . , the terms of which correspond with those of the original series 1, 2, 3, . . .
Again, in §§ 61-75 and 84-88 we have considered various kinds of numbers other than those in the original number-series. In general, these have involved two of the original numbers, e.g. 53 involves 5 and 3, and log2 8 involves 2 and 8. In some cases, however, e.g. in the case of negative numbers and reciprocals, only one is involved; and there might be three or more, as in the case of a number expressed by (a + b)n. If all but one of these constituent elements are settled beforehand, e.g. if we take the numbers 5, 52, 53, . . ., or the numbers 3√1, 3√2, 3√3, . . . or log10 1.001, log10 1.002, log10 1.003 . . . we obtain a series in which each term corresponds with a term of the original number-series.
This correspondence is usually shown by tabulation, i.e. by the formation of a table in which the original series is shown in one column, and each term of the second series is placed in a second column opposite the corresponding term of the first series, each column being headed by a description of its contents. It is sometimes convenient to begin the first series with 0, and even to give the series of negative numbers; in most cases, however, these latter are regarded as belonging to a different series, and they need not be considered here. The diagrams, A, B, C are simple forms of tables; A giving a sum-series, B a multiple-series, and C a series of square roots, calculated approximately.
92. Correspondence of Numerical Quantities.—Again, in § 89, we have considered cases of multiple-tables of numerical quantities, where each quantity in one series is equivalent to the corresponding quantity in the other series. We might extend this principle to cases in which the terms of two series, whether of numbers or
