REVIEW I.

Numeration is a system of naming numbers.

The names of numbers from one to twelve are all different. The names of numbers between twelve and twenty are formed by adding ten to names of numbers less than ten; as, thirteen, fourteen, etc.

Tens are counted like simple units. Thus, two tens, three tens, called twenty, thirty, etc.

Ten tens are called a hundred, just as ten units are called a ten. That is, we regard a ten as a single group, and call it a unit of the second order; a hundred as a group of ten tens, and call it a unit of the third order.

The names of numbers between any two tens, as from twenty to thirty, are formed by adding to the name of the smaller number of tens the names of numbers less than ten ; as, twenty-one, twenty-two, etc.

Hundreds are counted like tens; as, one hundred, two hundred, etc., up to ten hundreds, which is called a thousand.

The names of numbers between any two hundreds are formed by adding to the name of the smaller number of hundreds the names of the numbers less than a hundred; as, one hundred one, one hundred two, etc.

A thousand is regarded as a single group of ten hundreds, and is called a unit of the fourth order.

Thousands are counted precisely like hundreds, tens, and simple units, up to ten thousands.

The names of numbers between any two thousands are formed by adding to the name of the smaller number of thousands the names of numbers less than a thousand; as, one thousand one, one thousand two, etc.

Ten thousand is regarded as a unit of the fifth order; a hundred thousand as a unit of the sixth order.

Ten hundred thousand is called a million, and is regarded as a unit of the seventh order; and so on, always forming from ten units of the same order one unit of the next higher order.

It is necessary to observe that we say a unit of thousands, a ten of thousands, a hundred of thousands, just as we say a simple unit, a ten of units, a hundred of units. That is, we regard the thousands as a second class of units composed of three orders, just as the first class of units is composed of three orders.

Likewise we regard the million as a third class of units, containing the units of millions, the tens of millions, the hundreds of millions.

The following table presents the names and the succession of the units of the different orders and different classes:

upon the following principle: Ten units of any order are equal in value to one unit of the next higher order; and one unit of any order is equal in value to ten units of the next lower order.

Decimal fractions are formed by dividing the unit into ten equal parts, called tenths; the tenth into ten equal

parts, called hundredths; the hundredth into ten equal parts, called thousandths; the thousandth into ten equal parts, called ten-thousandths, etc.

By comparing these different decimal fractions, we see that the formation of decimal fractions depends, like that of whole numbers, upon the following principle: Ten units of any order are equal in value to one unit of the next higher order; and one unit of any order is equal in value to ten units of the next lower order.

Notation is a system of writing numbers.

The common system of notation employs ten figures, or digits: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first nine figures represent the first nine numbers; the last, which is called zero, or naught, is used to denote the absence of units of the order in which it stands. These ten figures express all numbers by the artifice of making the value of each figure increase tenfold for every place that it is moved to the left.

To write a number in figures, we write successively the number of units of each order from left to right, beginning at the highest order, taking care, if the number contain a decimal fraction, to put a full point at the right of the units' figure, and to supply by zero the units of any order that may be lacking. If the number contain no whole number, we put a zero in the units' place, and the decimal point to the right of the zero.

To read a number written in figures, we divide the number into periods of three figures each, from right to left. This done, we begin to read at the last period on the left, and read as if the figures of that period stood alone, adding the name of the period. Then the next period to the right is read, with the name of that period, and so on.

If the number contain a decimal fraction, we first read the whole number; then the decimal as a whole number, taking care to add the fractional name of the lowest place.

CHAPTER III.

ADDITION.

36. ADDITION means putting together. The Saint George's cross + is read plus, and means that the numbers between which it stands are to be added together.

The result obtained by adding together two or more numbers is called their sum.

=

The sign of equality stands for the words "equals," or "equal."

Thus, 8+1=9 is read, eight plus one equals nine.

37. Add 2 to each number from 0 to 9 ; add 3 to each number from 3 to 9.

Add 4 to each number from 4 to 9; add 5 to each number from 5 to 9.

Add 6 to each number from 6 to 9; 7 to 7, to 8, and to 9; 8 to 8, and to 9; 9 to 9.

Repeat these additions until thoroughly familiar with them.

38. Name, as fast as you can talk, the even numbers, 2, 4, 6, etc., up to 102.

Name the odd numbers, 1, 3, 5, etc., up to 101.

39. Name every third number, 0, 3, 6, etc., up to 102. Name every third number, 1, 4, 7, etc., up to 103. Name every third number, 2, 5, 8, etc., up to 101.

40. Name every alternate even number, 0, 4, 8, 12, etc., up to 100.

Name every alternate even number, 2, 6, 10, etc., up to 102. Name every alternate odd number, 1, 5, 9, etc., up to 101. Do the same, beginning 3, 7, 11, etc., and go to 103.

41. Name every fifth number under 100, beginning 5, 10, 15, etc.; beginning 1, 6, 11; beginning 2, 7, 12; beginning 3, 8, 13; and beginning 4, 9, 14.

42. In like manner, add by sixes up to a number exceeding a hundred, beginning 0, 6, 12; beginning 1, 7, 13; beginning 2, 8, 14; and so on.

43. Add by sevens up to a number exceeding 100, beginning 0, 7, 14; 1, 8, 15; and so on; by eights, beginning 0, 8, 16; 1, 9, 17; and so on; by nines, beginning 0, 9, 18; 1, 10, 19; and so on. *

44. For the addition of numbers in general, the following mode has been found most convenient.

Write the numbers in columns, units under units, tens under tens, tenths under tenths, etc.

Add the digits in the right-hand place; set the units of the sum in that place, but carry the tens mentally to the next place to the left, to be added to the digits there, and so proceed.

The study of three or four examples will make the process understood.

*The teacher may take ten small cards. On each side of the first write 0; of the second, 1; etc. Shuffle, and dictate the numbers on all but one for the class to add. Subtract the reserved number from 45; the remainder is the sum. With two sets of cards at once, subtract the reserved number from 90. For advanced classes, use cards with larger numbers; and complemental cards, which may be obtained of the publishers, furnishing unlimited examples, with the answers to all.