I. Ten signs are used, one to stand for nothing, the rest for the first nine numbers. They are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first of these is called a cipher.
II. Higher numbers have not signs for themselves, but are signified by placing the signs already mentioned by the side of each other, and agreeing that the first figure on the right hand shall keep the value
which it has when it stands alone; that the second on the right hand shall mean ten times as many as it does when it stands alone; that the third figure shall mean one hundred times as many as it does when it stands alone; the fourth, one thousand times as many; and so on.
III. The right hand figure is said to be in the units’ place, the next to that in the tens’ place, the third in the hundreds’ place, and so on.
IV. When a number is itself an exact number of tens, hundreds, or thousands, &c., as many ciphers must be placed on the right of it as will bring the number into the place which is intended for it. The following are examples:
Fifty, or five tens, 50: seven hundred, 700.
Five hundred and twenty-eight thousand, 528000.
If it were not for the ciphers, these numbers would be mistaken for 5, 7, and 528.
V. A cipher in the middle of a number becomes necessary when any one of the denominations, units, tens, &c. is wanting. Thus, twenty thousand and six is 20006, two hundred and six is 206. Ciphers might be placed at the beginning of a number, but they would have no meaning. Thus 026 is the same as 26, since the cipher merely shews that there are no hundreds, which is evident from the number itself.
- If we take out of a number, as 16785, any of those figures which come together, as 67, and ask, what does this sixty-seven mean? of what is it sixty-seven? the answer is, sixty-seven of the same collections as the 7, when it was in the number; that is, 67 hundreds. For the 6 is 6 thousands, or 6 ten hundreds, or sixty hundreds; which, with the 7, or 7 hundreds, is 67 hundreds: similarly, the 678 is 678 tens. This number may then be expressed either as
- 1 ten thousand 6 thousands 7 hundreds 8 tens and 5;
- or 16 thousands 78 tens and 5; or 1 ten thousand 678 tens and 5;
- or 167 hundreds 8 tens and 5; or 1678 tens and 5, and so on.
- EXERCISES.
I. Write down the signs for—four hundred and seventy-six; two thousand and ninety-seven; sixty-four thousand three hundred and fifty; two millions seven hundred and four; five hundred and seventy-eight millions of millions.
II. Write at full length 53, 1805, 1830, 66707, 180917324, 66713721, 90976390, 25000000.
III. What alteration takes place in a number made up entirely of nines, such as 99999, by adding one to it?