I.

EXERCISES IN NOTATION AND NUMERATION.

1.-NOTATION.

NOTATION is the art of expressing numbers by figures or symbols, appropriated for that purpose.

2. Unit, or unity, is the name given to that quantity which is to be reckoned as one, when other quantities of the same kind are to be measured.

3. Number is the relation of a quantity to its unit; the notion of number being suggested by successive repetitions of the individual unit, or number, is the name by which we signify how many objects or things are considered, whether one or more. When, for instance, we speak of one ship, two steamers, three masts, or four yards, the number of things referred to will be one, two, three, or four, according to the case; and so one, two, three, four, and the rest are called numbers.

In the common system of arithmetic all numbers, however large or small, can be expressed by the following symbols or characters, called figures, viz. :

The first nine of these are called significant figures or digits,* and sometimes represent units, sometimes tens, hundreds, or higher classes. When placed singly they denote the simple numbers subjoined to the characters; where several are placed together the first figure on the right is taken for its simple value, the next, or figure standing in the second place, expresses ten times its simple value, or signifies so many tens; thus 94 expresses ten times nine units, together with four units more; the third, or figure standing in the third place, expresses one hundred times its simple value, or signifies so many hundreds; thus 943 expresses one hundred times nine units, together with four times ten units, and also three units more, and so on by a ten-fold increase for each additional figure that follows it. The value which thus belongs to a figure in consequence of its position or place is called its local

Names frequently throw light on the origin of things. It is interesting to notice that the namo digits is plainly significant of the early rude method of counting on the fingers; and that the name calculation as plainly refers to the primitive practice of reckoning with pebbles (calculus, a pebble),

value. Therefore, all numbers have a simple or intrinsic value, and also a local value.

5. It appears, then, that in common arithmetic we proceed towards the left from units to tens of units, from tens of units to tens of tens of units, or hundreds of units, from hundreds of units to tens of hundreds of units or thousands of units; from thousands of units to tens of thousands of units; from tens of thousands of units to tens of tens of thousands of units, that is to hundreds of thousands of units, thence to tens of hundreds of thousands of units, or millions of units, thence to tens of millions of units, hundreds of millions of units, &c., till we come to millions of millions of units, which are called billions of units, and so on to trillions, quadrillions, &c.*

6. When any of the denominators, units, tens, hundreds, &c., is wanting, it becomes necessary to supply its place with the last sign or character, viz., o, which is termed cypher, or nothing-the word cypher in the Arabic signifying vacuity. This character, which indicates the absence of all number, is a most important one, inasmuch as its introduction serves to preserve the proper position of the significant figure, thus the number forty thousand three hundred and twenty would be expressed in figures by 40320, because the denominations units and thousands are wanting, and the absence of each is indicated by the cypher which occupies its place.

RULE I.

To write in figures a number expressed in words.-Write down a row of noughts, or cyphers, and, as if these blanks were numbers, mark off the periods by cutting off the last three, then the next three, then the next, and so on; then

* It is worth while to remark that as regards billions there is a difference between the French and English practice; in French, a billion (or milliard) is one thousand million, in English a billion is a million of millions, and accordingly the word is seldom used in our language, for such large numbers are rarely of any practical use. The old books use a scale of numbers of this kind

A million of millions is a billion,

A million of billions is a trillion,

and so forth; but these names are never used in practice, and can hardly be said to belong to the language of arithmetic or to English speech. It may be worth a passing notice, too, that no distinct ideas are conveyed by any of these terms; beyond a very moderate extent our notions of the value of numbers become confused. The number of ones in a million even, is hard to conceive; it is a thousand thousand, and would take more than twenty-three days to count through, kept at it for twelve hours a day, and counted one every second.

commencing at the first cypher on the left, put under each the proper figure in the number proposed, taking care that it be in its proper place; if any vacancies appear under the corresponding cyphers, fill them up with noughts.

Thus, let it be required to put into figures the number five hundred and four million, eighty-two thousand and thirty-five. We know the place of millions has six places to the right of it, we therefore put a 000,000,000 nought for the millions, and write six noughts after it, and, as 504,082,035 we see, from hundreds being the leading word in the written expression, that the first period will be a complete period, we prefix two noughts more. The requisite number of noughts, divided as proposed, is as in the margin, and under them we now have to write, in their proper places, the figures 5, 4, 8, 2, 3, 5, and then fill up the gaps with noughts; we thus find the number, when written, to be 504,082,035.

EXAMPLES FOR PRACTICE.

Express the following numbers in Figures:

1. Sixty-three; eighty-one; ninety-nine; forty; thirteen.

2. Two hundred; three hundred and three; five hundred and ninety-eight; eight hundred and eighty-eight.

3. Four thousand; one thousand, seven hundred and eighty-three; six thousand and eighty-three; seven thousand, nine hundred and thirty; nine thousand and nine. 4. Twenty-seven thousand, five hundred and four; eighty-nine thousand and sixty-four; thirty-three thousand.

5. One hundred thousand; six hundred and seventy-six thousand and fifty; six hundred and three thousand, two hundred and forty.

6. Twenty thousand, six hundred; ninety thousand and ninety-two; two hundred and four thousand, six hundred and forty-one; eight hundred thousand and eight hundred.

7. Three million, six thousand and four; five million, thirty thousand and forty; seven million, seven hundred thousand and six; ten million, ten thousand and ten.

8. Seven million, three thousand; eleven million, one hundred and eight thousand, one hundred and six; fifty-four million, fifty-four thousand and eighty eight; six hundred and thirteen million, twenty thousand, three hundred and three.

9. Seventy million, seven hundred and four thousand, and thirty-two; forty-five million, three hundred and eighty-seven thousand, and twenty-five; three hundred and forty-nine million, four thousand and sixty-five; one hundred million, ten thousand and one.

10. Eight hundred and forty-two million, two hundred and forty-eight thousand, four hundred and eighty-four; nine hundred and nine million, nine thousand and ninety-nine; two hundred and twenty-two million, and forty; three hundred and five million, forty thousand and eight.

11. Seven hundred million, seven hundred thousand and seven hundred; two hundred and two million, two hundred and two thousand, two hundred; nine hundred million, and nine hundred; one hundred million, ten thousand and one.

2.-NUMERATION.

NUMERATION is generally applied to the converse process of expressing

in words a number which is already expressed in symbols.

8. To express in words the numbers denoted by a line of figures.

RULE II.

1o. Divide them into periods of three figures each, beginning at the right hand. 2°. Then, commencing at the left hand, read the figures of each period in the same manner as those of the right hand period are read, and at the end of each period pronounce the name.

NOTE.-A glance at the numeration table shows that the leading figure of each set is hundreds of something; that of the first set, on the right, is hundreds of units, or simply hundreds; that of the next set, hundreds of thousands; that of the next set, hundreds of millions, and so on. And by thus finding out the local value of the leading figure in each period, the number may be read with ease. When any of the figures is o, a little extra care is, however, necessary.

Ex. 1. Express in words 68547329.

EXAMPLES.

millions. thousands. units.

The number 68547329, when divided into periods as proposed, is 68, 547, 329, pointing to the 3 you say hundreds, and passing to the 5 you say hundreds of thousands; the incomplete period 68, must, therefore, be 68 millions; and the entire number 68 millions, 547 thousand, 329, or expressing the whole in words it is, sixty-eight million, five hundred and forty-seven thousand, three hundred and twenty-nine.

Ex. 2. Express in words 460305007.

millions. thousands. units.

The number 460305007 being divided into periods is 460, 305, 007, and is read, four hundred and sixty millions, three hundred and five thousand and seven.

Ex. 3. Express in words 999999999.

millions. thousand. units.

Divided into periods this is 999, 999, 999, and is read, nine hundred and ninetynine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nine.

Ex. 4. Express in words 561234826479365.

billions. thousand. millions. thousand. units.
561,

234, 826, 479, 365

and is read, five hundred and sixty-one billions, two hundred and thirty-four thousand eight hundred and twenty-six million, four hundred and seventy-nine thousand three hundred and sixty-five.

9.

The process of finding a number which shall be equal to the sum of two or more numbers is called addition. The number found, or the answer, is called the sum, and numbers which are added are called addends,