one hund. and twenty-three; eighty-two thous. and thirty-one; eighty-one thous. three hund. and two; eighty-two thous. one hund. and three. Observe that the foregoing twenty-four numbers are expressed by the same four digits and a cipher. In all these twenty-four numbers, the digit 8 occupies the fifth place of integers. If the 8 be moved from place to place, like the others, then the four digits and cipher will suffice to express ninety-six different numbers. And if a significant figure be employed instead of the cipher, then the five digits will be capable of expressing one hundred and twenty different numbers. See Principle I. page 3. For one hundred thousand, we write digit 1 in sixth place of integers; thus, 100000 EXERCISE VI. Write in figures the following numbers. Six hundred thousand; seven hund. thous.; eight hund. thous.; nine hund. thous.; two hund. thous.; three hund. thous.; four hund. thous.; five hund. thous. Eight hund. and ninety-five thous. four hund. and twelve ; three hund. and eleven thous. and thirteen; five hund. thous. and twenty-nine. one Nine hund. and forty-three thous. six hund.; six hund. and nine thous. and seventy; four hundred and eighty thous. one hund. and six; seventy thous. three hund. and sixty-one; six hund. and fifty-four thous. three hund. and twenty-one; hund. and sixty-four thous. three hund. and fifty-six; four hund. and twenty-one thous. five hund. and thirty-six; three hund. and fifty-four thous. six hund. and twenty-one; six hund. and fourteen thous. five hund. and thirty-two; six hund. and fifty-three thous. two hund. and forty-one. Any six significant figures will express seven hundred and twenty different numbers. Examples of this will be given under the head "Numeration." For one million, or one thousand thousand, we write digit 1 in seventh place of integers; thus, 1000000 EXERCISE VII. Write in figures the following numbers. One million one hund. thous. and sixty-four; two millions; three millions; four millions; five millions; six millions; seven millions; eight millions; nine millions; two millions four hund. and six thous. five hund. and three; seven millions and fifty-four hundreds; * *NOTE TO TEACHER. This, and subsequent similar examples, are purposely put in this form, that the pupil may remember that a thousand is ten hundreds; two thousand, twenty hundreds, and so on. As the places required for the notation of a number become so numerous eight millions ten hund. and twelve; twelve hund. and fifty thous. six hund. and seventy; five million one hund. and fortysix thous. three hund. and sixteen ; eighteen hund. thous. eighteen hund. and five; three million four hund. and fifteen eleven hund. and ten thous. thous. five hund. and fourteen; and eleven. For ten millions, or ten thousand thousand, we write digit 1 in eighth place of integers; thus, 10000000. EXERCISE VIII. Write in figures the following numbers. Eleven millions; twelve millions; thirteen millions; fourteen millions; fifteen millions; sixteen millions; seventeen millions; eighteen millions; nineteen millions; twenty millions; thirty millions; forty millions; fifty millions; sixty millions; seventy millions; eighty millions; ninety millions; eighty-seven million six hund. and fifty-four thous. three hund. and twenty-one; thirty million five hund. and eight thous. and forty; twelve million twelve thous. and twelve; forty million sixty thous. nine hund. and sixty-four; fourteen million one hund. thous. one hund. and forty-one; ninety million sixty-one hund. and four; thirty-six million one hund. and eighty thous. and twenty-nine. For one hundred millions, we write digit 1 in ninth place of integers; thus, 100000000. EXERCISE IX. Write in figures the following numbers. Two hund. millions; three hund. mill.; four hund. mill.; five hund. mill.; mill. ; six hund. seven hund. mill.; eight hund. mill.; nine hund. mill. Seven hund. and one mill. eight hund. and forty-three thous. two hund. and six; five hund. mill. four hund. and fifteen thous. and eight; three hund. mill. fifty thous. and seven; four hund. and ten mill. and sixteen; five hund. and forty mill. three hund. and ten; seven hund. and two mill. seven thous. forty tens; six hund. mill. three hund. and forty-two thous.; nine hund. and three mill. five hund. and sixty thous. nine hund. and fifty-three; eight hund. and ten mill. five hund. and one thous. and thirty-seven; one hund. and ten mill. one hund. and ten thous. one hund. and ten; three hund. and five mill. sixty-one thous. four hund. and eight. The explanations and examples which have been given shew that integers are represented by digits arranged according to the following table. as to embarrass the young pupil, he may be assisted either by permitting him to fill all the necessary places with ciphers at first, and then to substitute for them the significant figures, one by one, or by allowing him to rule vertical lines to separate spaces for the places of the digits. FIRST NOTATION TABLE: FOR INTEGERS ONLY. Hundreds of millions are expressed by digits in ninth place of integers. Millions digits in eighth place of integers. DECIMAL POINT. Places of integers are counted from decimal point towards the left. NUMERATION. process to notation. By reversing the above notation table expressing numbers by digits, numeration is the opposite expressed by digits. And, since notation is the art of Numeration is the art of reading in words numbers we obtain the FIRST NUMERATION TABLE : FOR INTEGERS ONLY. Digits in ninth place of integers represent Hundreds of millions. Digits in eighth place of integers Tens of millions. Millions. Hundreds of thousands. Tens of thousands. Thousands. Hundreds of units. Tens of units. Units, or single ones. DECIMAL POINT. Digits to the left of decimal point, represent the value for which it stands. First, notice the place of each digit, and thus ascertain In reading off a number, two steps are necessary. Second, read off those values in succession, beginning at the left of the line. In the following line of figures, 342080697. 3 in ninth place of integers stands for Three hund. mill. Eighty thous. Forty mill. integers. 6 in third place of integers stands for Six hund. 9 second Nine tens (ninety). Therefore the whole line of digits represents the number, three hund. and forty-two mill. eighty thous. six hund. and ninety-seven. EXERCISE X. 3216. Read, or write in words, the following numbers. 314. 406' 500⚫ 512. 9001. 8010 30106. 5400. 16016. 3126. 3162. 3261. 3612. 3621. 2316. 2163. 2631. 2613. 1263. 1236. 1362. 1623. 6321. 6312. 6213. 6231. 6132. 2361. 2136. 1326. 1632. 6123. Observe that the last twenty-four numbers are expressed by the same four digits differently placed. This shews the advantage of giving to each digit a local value, that is, a value depending on its place. See Principle I. p. 3. Arithmetical Symbols. To save time and space in arithmetic, many symbols are employed instead of words. For example, instead of writing the words "is equal to," we employ the sign of equality, which is formed by two horizontal and equal lines, one under the other, in this way =; so that "100. ten tens," means that "100 is equal to ten tens." Seeing, then, that 100 - ten tens; 200. thirty tens; we may read the number 316 in two ways; either as "three hundred and sixteen," or as "thirty-one tens and six." We may therefore read the number 3562 in four different ways.* ten hund. or one hundred tens;, twenty hund. or two hundred tens; three thous. five hund. and sixty-two; thirty-five hund. and sixty-two; three thous. fifty-six tens and two; three hund. and fifty-six tens and two. EXERCISE XI. Read, or write in words, the following numbers, in as many ways *NOTE TO TEACHER. The pupil should be well grounded in this method of reading the same number in a variety of denominations; for not only will the practice ensure an intelligent dexterity in notation and numeration, but it will also be of essential service hereafter in simplifying the processes of Long Division, one of the most perplexing operations to the learner. In every example of numeration, the pupil should be required to declare the separate local value of each individual digit. |