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Read Ebook: Encyclopaedia Britannica 11th Edition Arculf to Armour Philip Volume 2 Slice 5 by Various
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Ebook has 271 lines and 86413 words, and 6 pages
FIG. 2. FIG. 3.
FIG. 4
Thus, while arithmetical numbering refers to units, geometrical numbering does not refer to units but to the intervals between units.
Numerical quantities, to be added or subtracted, must be in the same denomination; we cannot, for instance, add 55 shillings and 100 pence, any more than we can add 3 yards and 2 metres.
One number is less or greater than another, according as the symbol of the former comes earlier or later than that of the latter in the number-series. Thus 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 series are shown below, the numbers being placed horizontally for convenience of printing, instead of vertically :--
This exhibits 9 as the sum of 4 and 5; it being understood that the sum of 4 and 5 means that we add 5 to 4. That this gives the same result as adding 4 to 5 may be seen by reckoning the series backwards.
It is convenient to introduce the zero; thus
indicates that after getting to 4 we make a fresh start from 4 as our zero.
To subtract, we may proceed in either of two ways. The subtraction of 4 from 9 may mean either "What has to be added to 4 in order to make up a total of 9," or "To what has 4 to be added in order to make up a total of 9." For the former meaning we count forwards, till we get to 4, and then make a new count, parallel with the continuation of the old series, and see at what number we arrive when we get to 9. This corresponds to the concrete method, in which we have 9 objects, take away 4 of them, and recount the remainder. The alternative method is to retrace the steps of addition, i.e. to count backwards, treating 9 of one series as corresponding with 4 of the other, and finding which number of the former corresponds with 0 of the latter. This is a more advanced method, which leads easily to the idea of negative quantities, if the subtraction is such that we have to go behind the 0 of the standard series.
yds. 0 1 2 3 ft. 0 1 2 0 1 2 0 1 2 0 1 2
The practical difficulty, of course, is that the addition of two numbers produces different results according to the scale in which we are for the moment proceeding; thus the sum of 9 and 8 is 17, 15, 13 or 11 according as we are dealing with shillings, pence, pounds or ounces. The difficulty may be minimized by using the notation explained in ? 17.
Although multiplication may arise in either of these two ways, the actual process in each case is performed by commencing with the unit and taking it the necessary number of times. In the above case of subdivision, for instance, each of the 5 shillings is separately converted into pence, so that we do in fact find in succession once 12d., twice 12d., ...; i.e. we find the multiples of 12d. up to 5 times.
If 1 boy receives 7 apples, then 3 boys receive 21 apples;
If 1s. is equivalent to 12d., then 5s. is equivalent to 60d.
The essential portions of these statements, from the arithmetical point of view, may be exhibited in the form of the diagrams A and B:--
or more briefly, as in C or C' and D or D':--
the general arrangement of the diagram being as shown in E or E':--
Multiplication is therefore equivalent to completion of the diagram by entry of the product.
It is to be considered that each column may extend downwards indefinitely.
Methods of division are considered later .
Properties not depending on the Scale of Notation.
Involution is a direct process, consisting of successive multiplications; the other two are inverse processes. 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.
If we divide n^p by n^p, the quotient is of course 1. This should be written n^0. Thus we may make the power-series commence with 1, if we make the index-series commence with 0. The added terms are shown above the line in the diagram in ? 43.
The following are the most important properties of numbers in reference to factors:--
If a number is a factor of another number, it is a factor of any multiple of that number.
If a number is a factor of two numbers, it is a factor of their sum or of their difference.
The number 1 is usually included amongst the primes; but, if this is done, the last paragraph requires modification, since 144 could be expressed as 1 ? 2^4 ? 3?, or as 1? ? 2^4 ? 3?, or as 1^p ? 2^4 ? 3?, where p might be anything.
The G.C.D. of three or more numbers is found in the same way.
Properties depending on the Scale of Notation.
by 10 if it ends in 0;
by 5 if it ends in 0 or 5;
by 2 if the last digit is even;
by 4 if the number made up of the last two digits is divisible by 4;
by 8 if the number made up of the last three digits is divisible by 8;
by 9 if the sum of the digits is divisible by 9;
by 3 if the sum of the digits is divisible by 3;
by 11 if the difference between the sum of the 1st, 3rd, 5th, ... digits and the sum of the 2nd, 4th, 6th, ... is zero or divisible by 11.
It must be noted that this is a definition of "n/a of," not a definition of "n/a," and that it is not necessary that n should be less than a.
Hence the former is greater than the latter; their sum is 41/28 of A; and their difference is 1/28 of A.
This is the most general expression of the relative magnitude of two quantities; i.e. the relation expressed by proportion includes the relations expressed by multiple, submultiple, fraction and ratio.
In the case of , and , the letters a, b, c, ... may denote either numbers or numerical quantities, while m and n denote numbers; in the case of and the letters denote numbers only.
If we denote the unit 1/n of A by X, then A is n times X, and p/n of n times X is p times X; i.e. p/n of n times is p times.
Ones. Sixths. 0 0 1 2 3 4 5 1 0 1 2 3 4 5 2 0 1 : :
A B Ones. Halves. Sixths. Ones. Thirds. Sixths. 0 0 0 0 0 0 1 1 2 1 0 1 0 1 1 2 0 2 1 1 0 0 1 0 0 : : : :
If we divide 1 by 5/7 we obtain, by this rule, 7/5. Thus the reciprocal of a number may be defined as the number obtained by dividing 1 by it. This definition applies whether the original number is integral or fractional.
In applications to money "per cent." sometimes means "per ?100." Thus "?3, 17s. 6d. per cent." is really the complex fraction
In actual practice, however, decimals only represent approximations, and the process has to be modified .
The Egyptians as a rule used only unit-fractions, other fractions being expressed as the sum of unit-fractions. The only known exception was the use of 2/3 as a single fraction. Except in the case of 2/3 and 1/2 , the fraction was expressed by the denominator, with a special symbol above it.
The Babylonians expressed numbers less than 1 by the numerator of a fraction with denominator 60; the numerator only being written. The choice of 60 appears to have been connected with the reckoning of the year as 360 days; it is perpetuated in the present subdivision of angles.
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