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XXXVIII. SIMPLE PROPERTIES OF NUMBERS.
Many persons are interested in questions of numbers, who are not algebraists, and have nothing to employ their talent upon. They may, by common rules, amuse themselves with multiplying numbers together, or finding how much 61 cwt. 23 lb. cost, at 5|rf. a pound, if they please. But there is nothing interesting in the result of all this; and accordingly, none but those to whom such proceedings are necessary have recourse to them.
Between the arithmetician, and a great many interesting properties of numbers, stand nothing but a few simple terms, which being once ex plained, a large field of amusement is open in verifying, upon simple numbers, the results which have been obtained in algebra. The following article contains some of these, in which nothing will be required but common multiplication and addition. In each succeeding volume, we shall add one or two more to the list. The reader may try high or low numbers, according to his facility in performing the operations.
I. The question often arises how to make up numbers out of other numbers; for instance, how to weigh any certain number of pounds with one set of weights. Suppose, for example, we wish to know what is the smallest set of weights which will do for any number of pounds under one hundred. The common system of arithmetic shows that nine ten-pound weights and nine pound weights will be sufficient: but here are eighteen weights in all. We shall now show that seven weights only are sufficient, and that it may even be done with five: that is to say, that every number of pounds under 128 lbs. may be weighed with seven weights, or, if necessary, every number of pounds under 122 lbs. may be weighed with five weights; and that a 6imple table might be constructed, which would make the process as easy as the common system, or easier.
Firstly. Suppose it is required that no weights shall be put in the scale with the goods weighed. Let the scale with the weights be called W; that with the goods, G. Then it will be sufficient to have weights of the following number of pounds,—
1 2 4 8 16 32 64
with these any number of pounds may be weighed under 128. For instance,—
To weigh Put into the scale W
71 pounds 64, 4, 2, and 1,
72 do. 64, and 8.
73 do. 64, 8, and 1,
74 do. 64, 8, and 2,
75 do. 64, 8, 2, and 1,
76 do. 64,-8, and 4,
77 do. 64, 8, 4, and J;
and so on. The reader may satisfy himself upon all the other cases with ease.
The law might be easily so framed that this method of weighing should enable a customer to secure himself against fraud. The necessary enactments would be as follows:—
1. Every tradesman must have his weights, beginning from one pound, each double of the preceding, and the same for ounces or grains.
2. Every one must have his weights marked A, B, C, D, &c. A being one pound, B two pounds (and so on). The same for ounces and grains.
3. Every one must keep a table (to be published by authority), and must show it to his customer, if required. The following is a specimen, and a card would contain all that is necessary for common purposes :—
72 G D
73 G D A
4. Stamped 1 lb. 1 oz. 1 gr. weights must be sold to all who will buy them: the stamp to be evidence of authenticity, as usual.
5. The tradesman must be bound to go through the process hereafter described, if required : of which it must be observed, that the trouble is great only when the quantity of goods in question is great. The purchaser takes his test-weight (say, one pound) to the shop or warehouse with him. With this one pound he can test every one of the shopkeeper's weights, as follows. Suppose he wants to buy 71 pounds of goods. On his own card, or on that of the seller, he sees G, C, B, A, which are the weights he has a right to. He produces his own test-weight (which call T),and requires the seller to show him that
T balances A
T and A . . . balance B
T, A, and B C
T, A,B, andC . . . . D T, A, B, C, and D ... E T, A,B, C,D,and E . . F T, A, B, C, D, E, and F . G If this be found true, and if only one of the preceding be tried twice, changing the scales the second time, then neither scales nor weights can be wrong.
It must be observed, that by this method the proper officers might make the usual examination of weights without carrying more weights about with them than one pound, one ounce, and one grain.
Secondly. Suppose it is allowed to place weights in both scales. Then, it is sufficient if each weight (the first being one pound) be three times as great as the other. That, is, any number of pounds under 122 may be weighed with weights of
1 8 9 27 81
A table might be formed as before. It is impossible to carry this principle further without introducing two weights of a sort.
The two mathematical theorems on which the preceding rests are as follows:—
1. Any number whatever may be made by adding together single terms out of the series
I 2 4 8 16 32 64 128 &c.
2. Any number may be made by adding and subtracting single terms out of the scries
1 3 9 27 81 243 729 &a.
The first of the preceding theorems explains a common puzzle, in which a number of cards is handed to the person to be mystified, and he is desired to think of a number, and tell on which cards it is to be found. The person in the secret then tells the number. By putting down the cards for all numbers up to 31 inclusive, as follows, the reader will easily find out the trick.
II. There is a very large class of amusing properties of numbers, which the greater part of readers never know, not because there is anything difficult in announcing them, but simply because a few mathematical terms stand in the way. To prove these theorems is a work of study; but to understand them is not. We shall in this article explain some terms, and then exhibit the use which can be made of them in abbreviating propositions which in common language would lose all their interest, on account of the cloud of words in which they would be wrapped. The reader will see whether he understands each term by looking at the questions which follow it.
1. Multiple. 30 is called a multiple of 10, because it contains an exact number of tens, namely 3 tens. It is also a multiple of 6, 5, 15, 2, 3, and 1 It is not, properly speaking, a multiple of itself; but it is usual to say that 30 is the first multiple of 30, as it contains itself once exactly. Thus the multiples of 3, are
3 6 9 12 15 18 21 &c. Question. Why is every multiple of 10 also a mulple of 5?
2. Prime. A prime number is one which is a multiple of no number but 1 (which all numbers are) and itself. Thus 13 is an exact number of ones, and of thirteens, but of nothing else: it is then what is called a prime number. The series of prime numbers, up to 41, is as follows:
I 2 3 5 7 11 13 17 19 23 29 31 37 41.
Question. Why is it that a prime number (excepting 2 and 5) cannot end with 0, 2, 4, 5, 6, or 8?
3. Square. A square number, or a second power, is a name given to a number which is made by multiplying a number by itself. Thus 9 is a square number (3 times 3); 10 is not. The series of square numbers is
1 4 9 16 25 36 49 64 81 100 &c. Questions. Why is it that a square number cannot end with 2, 3, or 7? Why is it that the square of an even number must be even, and of an odd number odd?
4. Cube. A cube number, or a thirdpower, is a name