SECTION 1

ARITHMETIC

PART 1

NOTATION AND ENUMERATION

DEFINITIONS

1. A unit is the standard by which anything is measured. For example if it were desired to measure the length of a room, it could be done with a tape line, a rule, a stick of some convenient length, or anything else that would form a basis of comparison. Suppose, for example, a broom

stick were used; the broomstick would be laid on the floor, with one end a against the wall at one Iside of the room and with the other end b extending towards the opposite side of the room, as shown in Fig. 1. The point b would be marked and the broomstick would be shifted so as to occupy the position bc, then to the position cd, and so on until the other wall was reached. Each of the lengths ab, bc, etc.

a

b

2

FIG. 1.

is equal in length to the broomstick, which is in this case the unit of length. The word that denotes how many times the room is longer than the broomstick is number. In other words number means an aggregation, or collection, of units; a number may also mean a single unit or a part of a unit-it refers in all cases to the complete measurement.

As another example, consider the compensation that a man receives for his work. For working a certain number of hours he receives a certain number of dollars; here two units are involved,

...we general classes, according to their ... and concrete numbers.

Addo ot applied to any object or qua ace dred seventeen, etc.

Adora amber is one relating to a particular kind of object or usare dollar eleven pounds, seven hours, etc. decunding to was numbers are also divided into two Kothapet

bike numbers are numbers having the same units: for exAnswoldowns and three hours are like numbers, since they all be the same wit one hour; fourteen, seventeen and we are add umbes the unit of each being one.

numbers since their units are unlike, being respectively one dollar, one hour, and one pound.

NOTATION

In order that numbers may be recorded, some method of writing and reading them must be devised.

10. Notation is the word used to express the act of and the result obtained by writing numbers.

11. Numeration is the act or process of reading numbers that have been written.

12. Notation is accomplished in three ways: (1) by words; (2) by letters; (3) by figures. The first method, by words, is never used in computation, only the second and third methods having ever been employed for this purpose. The second method, called the Roman notation, is very seldom used at the present time, its only use is for numbering, as the indexes, chapters, etc. of books, and in a few cases for dates. The third method, the Arabic notation, employs certain characters, called figures, and is in universal use today.

THE FIRST OR WORD METHOD

13. This is really only the names of the different numbers. The first twelve numbers have distinct names; beyond these all are formed according to a definite system that is readily learned. The names of the first twelve numbers are: one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. The next seven numbers all end in teen, which means and ten, the first part of the word being derived from the words three, four, etc. up to nine; they are thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, and nineteen. Thirteen means three and ten, fourteen means four and ten, etc. The name of the next number is twenty, and the names of the next nine numbers are all compound words, having for their first element the word twenty and for their second element the words one, two, etc. to nine. Thus, twenty-one, twenty-two, etc. to twenty-nine. The word twenty means two tens. The next number is thirty, which means three tens, and the next nine are thirty-one, thirty-two, etc. to thirty-nine. Then follow forty, fifty, sixty, seventy, eighty, and ninety, with their

ine. The number following e numbers are then repeated, added, one-hundred-one, onewenty-one, one-hundred-twenty...y-nine, the next number being scontinued in this manner to nineA number being one-thousand; then e-thousand-two, etc., one-thousandand-one-hundred-two, etc. to nine...ety-nine, the next number being tenca-thousand-one, etc. to nine-hundredety-nine, the next number being called on is then carried on one-million-one, etc. on, which in the United States, France, er countries is called a billion; a thousand In Great Britain and a few other coun..on million, a trillion a million-billion, etc.

THE ROMAN NOTATION

ection uses the letters I, V, X, L, C, D, and ae letters when standing alone are: I is one, &ity, Cis one-hundred, D is five-hundred, Normerly other letters were used, but these Numbers are expressed by letters wy od now. collowing principles:

poris its value.

x.wo, It is three, XX is two tens or twenty, any, CC is two hundred, MM is two

ve repeated; only I, X, C, and M are ever any number, except when they precede a

se of greater value, their difference is
greater value, their sum is denoted.
X, IX is nine, XII is twelve, XL is forty,
, etc. Only one letter of lower value is
higher value, and in general, the only
Further, I precedes only V and X, and
Occasionally, however, C is used in

this manner before M; as for example, in the date nineteen-hundred-four, when instead of writing MDCCCCIV the shorter form MCMIV is used. Pope Leo XIII wrote the date eighteenhundred ninety-five MDCCCVC instead of MDCCCXCV. On the Columbus memorial in front of the Union Station, Washington, D. C., the year of his birth is expressed by MCDXXXVI, which, of course, is read fourteen-hundred-thirty-six.

III. A bar placed over a letter multiples its value by one-thousand. Thus, X is ten-thousand, L is fifty-thousand, XCDXVII is ninety-thousand five-hundred seventeen. In the last expression, it might be thought that D was preceded by X and C; this is not the case, however, as XC is treated as a single letter having a value of ninety-thousand.

The following table illustrates the foregoing principles and applications of the Roman notation:

VIII is eight

XII is twelve
XVIII is eighteen

XXIX is twenty-nine
XXXV is thirty-five

XLIV is forty-four
LXXVI is seventy-six

XCII is ninety-two

CXLIV is one-hundred forty-four

CCCCL is four-hundred fifty

IXLX is nine-thousand sixty
C is one-hundred-thousand
D is five-hundred-thousand
M is one-million

THE ARABIC NOTATION

15. The Arabic notation employs ten characters, called figures, to represent numbers; these characters together with

The first figure (0) is called naught, cipher, or zero, and has no value. The other nine figures are called digits, and each has the value denoted by its name.