ARITHMETIC NOTATION AND NUMERATION. T1. A single thing, as a dollar, a horse, a man, &c., is called a unit, or one. One and one more are called two, two and one more are called three, and so on. Words expressing how many (as one, two, three, &c.) are called numbers. This way of expressing numbers by words would be very slow and tedious in doing business. Hence two shorter methods have been devised. Of these, one is called the Roman* method, by letters; thus, I represents one; V, five; X, ten, &c., as shown in the note at the bottom of the page. The other is called the Arabic method, by certain characters, called figures. This is that in general use. * In the Roman method, by letters, I represents one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; and M, one thousand. As often as any letter is repeated, so many times its value is repeated, unless it be a letter representing a less number placed before one representing a greater; then the less number is taken from the greater; thus, IV represents four; IX, nine, &c., as will be seen in the following 10 is used instead of D to represent five hundred, and for every additional Ɔ an nexed at the right hand, the number is increased ten times. † CIO is used to represent one thousand, and for every C and Ɔ put at each end, the number is increased ten times. A line over any number increases its value o ze thousand times, In the Arabic method the first nine numbers have each a separate character to represent it; thus. Nine is the largest numoer wnica can be expressed by a single figure. There is another character, 0; it is called a cipher, naught, or nothing, because it denotes the absence of a thing. Still it is of frequent use in expressing numbers. By these ten characters, variously combined, any number may be expressed. The unit 1 is but a single one, and in this sense it is called a unit of the first order. All numbers expressed by one figure are units of the first order. T3. Ten has no appropriate character to represent it, but it is considered as forming a unit of a second or higher order, consisting of tens. It is represented by the same unit figure 1 as is a single thing, but it is written at the left hand of a cipher; thus, 10, ten. The 0 fills the first place, at the right hand, which is the place of units, and the 1 the second place from the right hand, which is the place of tens. Being put in a new place, it has a new value, which is ten times its simple value, and this is what is called a local value. Questions. ¶ 1. What is a single thing called? What is a number? Give some examples. How many ways of expressing numders shorter than writing them out in words? What are they called? Which is the method in general use? In the Arabic method, how many numbers have each a separate character? 2. How is one represented? Make the characters to nine. What are these nine characters called? Why? What is the simple value of igures? What is the largest number which can be represented by a single figure? What other character is frequently used? Why is it called naught? How many are the Arabic characters? What are umbers expressing single things called? There may be one, two, or more tens, just as there are one, two, or more units, or single things; it takes ten cents to make one tencent piece; just so it takes ten single things to make one ten. All figures in the second place express units of the 2d order, that is, units of tens. One ten and one unit, 11, are called eleven; one ten and two units, 12, twelve, &c. In this way the units of the 1st order are united with the tens, that is, with Note. Twenty, thirty, &c., are contractions for two tens, three tens, &c. the units of the 2d order, to form the numbers from 10 to 20, from 20 to 30, to 40, and so on to 99, which is the largest number that can be represented by two figures. The weeks in a year are 5 tens and 2 units, (5 of the second order and 2 of the first order now described,) and are expressed thus, 52, (fifty-two.) In the same manner express on your slate, or on the blackboard, the two orders united, so as to form all the numbers from 10 to 99. T4. Ten tens are called one hundred, which forms a unit of a still higher, or 3d order, and is expressed by writing two ciphers at the right hand of the unit 1, thus, Note. When there are no units or tens, we write ciphers in their places, which denote the absence of a thing, (¶2.) 100 one hundred. 200 two hundred. 300 three hundred, &c. Questions.¶3. How is ten represented? What is it considered as forming? Consisting of what? What place does the cipher fill? The one? Where is unit's place, and where ten's place, counting from the right? How much is the value of a figure increased by being removed from a lower to a higher place? In which place does it retain its simple value? In ten's place, what is its value called? What is 1 ten and 1 unit called? 1 ten and 2 units? How are the numbers from 10 to 99 expressed? Of what is the number forty made up? Ans. 4 tens and ao units. Sixty? What do you unite, to form the number twenty, hree? thirty-seven? seventy-five? &c. Of what are twenty, thirty, c., contractions? What is the largest, and what the least, number you can express by one figure? by two figures? |