ON THE PROPERTIES OF NUMBERS.

PROPOSITION 1.

Digits in our system of notation increase in value from right to left in a tenfold ratio.

PROPOSITION 2.

In any series of digits expressing a number, the value of any digit is greater than the value of all the digits on its right.

This property results also from value according to place; and that the proposition is true is obvious, for if we take the smallest digit (1) and place it on the left of the largest (9) we form 19; the 1 expresses 10 units, while the 9 expresses but 9 units; and let us add. what numbers of nine we may, the unit will constantly retain its greater value: e. g. 19, 199, 1999, etc. Not only is the left hand digit higher in value than all upon its right, but the same remark applies to each digit, in reference to those on its right.

PROPOSITION 3.

If the sum of the digits in any number be a multiple of 9, the whole number is a multiple of 9.

This is one of several peculiar properties of the number 9, all arising from its being just one less than the radix of our system of notation, and

hence the highest number expressed by a single character; and these properties will belong to the highest number so expressed in any system. We might go a step further in reference to this property, and say that it belongs to any number that will divide the radix of the system, and leave one as a remainder.

If we carefully examine the genesis of numbers, we must see that so far as the number 9 is concerned, this is an accidental property, resulting from our scale of notation. We constantly express each successive number from unity to 9, inclusive, by a digit of greater value than any preceeding it; but when we pass 9 we express the next number, 10, by a unit and a cipher. The number is one greater than 9, and the sum of its digits is 1. Eleven is 2 greater, and the sum of its digits is 1+1=2. Thus we proceed, the sum of the digits constantly expressing the excess over 9, until we reach 18, or twice 9. Nineteen is 1 and 9, and it is one over twice 9. 20 is 2 over twice 9, and the sum of its digits is 2. The same course continued to millions, would but produce the same recurring result. Nine is 1 less than 10; twice 9 are 2 less than 20; 3×9 are 3 less than 30; and so on; and hence the 1 of 10, 2 of 20, 3 of 30, etc., come just in the proper place to keep up the excess above 9 and its multiples. If the multiples

1

of 9 did not constantly fall at each product, one further behind the corresponding multiples of 10, the two of 20, 3 of 30, etc., would not fall in the right place, to keep up the regular order of the series.

PROPOSITION 4.

If the sum of the digits in any number be a multiple of 3, the number is a multiple of 3.

The same reasoning applied to the number 9 to show the correctness of the preceding proposition, will show the correctness of this. Ten, the sum of whose digits is 1, is 1 over 3 times 3; 11, the sum of whose digit is 2, is 2 more than 3 times 3; 12, the sum of whose digit is 3, is a multiple, etc., etc.

PROPOSITION 5.

Dividing any number by 9 or 3, will leave the same remainder as dividing the sum of its digits by 9 or 3.

This proposition follows as a matter of course from the two next preceding it; and we shall adduce no other proof of its correctness. Like the former, it is an accidental property of the highest number expressed by a single digit in any systein, and of all its factors. If 9 were the basis of our system, these properties would belong to 8, 4, and 2; if 8, then 7 only, since 7 has no factors; and

if 7 were the basis, then 6, 3, and 2 would possess these properties; and if 12 were the basis, then 11 only would possess such properties; for it would in that case be expressed by a single digit, and would be the highest number so expressed. Twelve would be written with a unit and a cipher as 10 now is; and 11 being prime, it would be the only number that would divide the radix of the system and leave 1 as a remainder.

As early as 1657, Dr. Wallis, of England, applied this principle to prove the correctness of operations in the elementary rules of Arithmetic, and the practice has been continued to the present time. The operation is performed thus:

We cast the nines out of each number separately, and set the excess on the right. We then cast the nines out of the sum total 305160, and also out of the sum of the excesses 8+1+8+7, and they are equal: both being 6, and we hence infer that the work is

Add 79864-7

32075-8

83214-0

61840-1

48167-8

305160-6

right. To cast out the nines, the number may be divided by 9; but a better way is to add the digits together in each number, rejecting 9 whenever it occurs, and carrying forward only the excess. Thus 7 and 8 are 15; 9 being rejected, we carry 6 to 6 is 12; rejecting 9, we carry 3 to 4=7; the num

ber carried in each place is the excess over 9; and where 9 occurs it is passed over.

In Subtraction cast out the nines from the minu end and subtrahend, and also from the remainder If the excess in the remainder is equal to the difference of excesses in the minuend and subtrahend, the work is right.

Here, as we can not take 8 from 6, we take from 9 and add 6; the result, 7 agrees with the excess above 9 in the difference of the numbers.

From 6894321=6
Take 2960864-8

Leaves 3933457-7

In Multiplication, find the excess in the factors, and if the excess in the product of these two excesses equals the excess in the product of the factors the operation is correct.

This is often called proving by the cross; and instead of placing the excesses after marks of equality, they are placed in the angles of a cross as on the right hand of the above operation.