ARITHMETIC. DEFINITIONS-NUMERATION. 1. QUANTITY is anything that will admit of increase or diminution. Time, space, weight, and motion, are quantities. MATHEMATICS is the science of quantity. Questions. What is quantity? Give some examples of quantity. What is mathematics? 2. An UNIT is a single quantity which is used to compare quantities of the same kind with each other. NUMBERS express how many of these units are considered. Thus if we say a box contains ten pounds of tea, one pound of tea is the unit, and ten the number. In six yards of cloth, one yard of cloth is the unit, and six the number; in twenty dollars, one dollar is the unit, and twenty the number. The amount of tea, cloth, and money, is in each case compared with its own unit, which is one pound of tea for the first, one yard of cloth for the second, and one dollar for the third. Q. What is a unit? What do numbers express? In ten pounds of tea, what is the unit? What is the number? In six yards of cloth? In twenty dollars? In fifty miles? In each case what is the amount of tea, cloth, &c., compared with? 3. ARITHMETIC, which is a branch of Mathematics, treats of quantity when it is expressed by numbers. Arithmetic then is the science of numbers. Numbers are expressed by certain characters which are called FIGURES. Only ten of these characters are used. They are, O which is called zero, a cipher, or Naught, 4 01239 6 7 8 The nine figures 1, 2, 3, 4, 5, 6, 7, 8, 9, are called digits or significant figures. Q. What is arithmetic? Of what does it treat? Of what science is it a branch? How are numbers expressed? How many figures are there? Write them upon your slate. Which of these figures are called digits? What else are they called? 4. Although every number might be represented by a distinct figure, we have no separate character to express a number greater than nine. We express numbers greater than nine, by combining the characters already known. Q. Have we a separate character for numbers greater than nine? Why not? How do we express such numbers? 5. To express ten, we use the two characters 1 and 0, placing the 0 on the right of the 1. Thus 10, which is ten. The position occupied by the figure on the right is called the units' place, that by the figure on the left the tens' place. The O is in the units' place, the 1 in the tens' place. The same figure is therefore ten times greater in the tens' place than in the units' place. In like manner, we can express two tens, or twenty, three tens, or thirty, &c., as far as nine tens, or ninety, by placing 0 on the right of 2, 3, 4, 5, &c. Q. How is ten expressed? Which figure occupies the units' place in 10? Which the tens' place? How much greater is the same figure in the tens' place than in the units' place? How is twenty expressed? Thirty? Forty? Fifty? &c. Which figure occupies the tens' place in 20? Which the units' place? In 30? In 60? In 90? Write these numbers upon your slate. 6. To express the intermediate numbers between 10 and 20, 20 and 30, &c., we place in the tens' and units' places, the number of tens and units of which the number is composed. Thus, Eighteen contains 1 ten and 8 units, and is written 18 25 37 44 99 Ninety-nine is the greatest number which is expressed by two figures. Q. How are the intermediate numbers between 10 and 20 expressed? How many tens and units in eighteen? How is it written? How many tens and units in forty-four? Write this number? How many tens and units in fifty-six? In seventy-two? In eighty-eight? In ninety-nine? What is the greatest number expressed by two figures? 7. To express one hundred, we place two ciphers on the right of 1. Thus 100, which is one hundred. The place occupied by the 1 in this case is called the hundreds' place. It is ten times greater in the hundreds' place than in the tens' place, and one hundred times greater than in the units' place. In like manner, 200, 300, 400, 900, represent two hundred, three hundred, four hundred, &c. We may express any intermediate number by writing down the number of hundreds, tens, and units, of which it is composed, placing the hundreds in the hundreds' place, the tens in the tens' place, and the units in the units' place. Thus, One hundred and fifty-seven, contains 1 hundred, 5 tens, and 7 units, and is written Three hundred and sixty-nine contains 3 hundreds, 6 tens, and 9 units hunds. Citens. units. 369 Nine hundred and ninety-nine contains 9 hundreds, 9 tens, and 9 units 999 Nine hundred and ninety-nine is the largest number which is expressed by three figures. Q. How is one hundred expressed? What place does 1 occupy in this number? How much greater is it in the hundreds' place than in the tens' place? Than in the units' place? How many hundreds, tens, and units in 157? In 369? In 846? In 999? What is the largest number which is expressed by three figures? 8. By methods precisely similar we may express one thousand, two thousand, &c., by placing three ciphers on the right of 1, 2, 3, &c. Thus, 1000, 2000, 3000, &c., and all intermediate numbers may be written by writing down in the thousands', hundreds', tens', and units' places, the number of thousands, hundreds, &c., of which they are composed. We conclude, that any number, whatever be its magnitude, may be expressed by means of the ten characters which have been used; and this is done by giving different values to these characters, according to the positions which they occupy. Thus, 5 in the units' place, expresses 5 units. 50 is 5 tens, since the 5 is in the tens' place, or 50 units. 500 is 5 hundreds, or 50 tens, or 500 units. 5000 is 5 thousands, or 50 hundreds, or 500 tens, or 5000 units. The addition of a cipher having the effect in each case to increase the number tenfold. We may thus form a table, showing the values of the same numbers according to their position, so as to embrace any numbers whatever. Such a table is called a Numeration Table. It teaches the manner of reading figures. Q. How may thousands be expressed by figures? Intermediate numbers? May any number be expressed by the ten known figures? How? How does the addition of a cipher to the right of a number affect it? What does 5 express? 50? 500? 5000? What is a numeration table? What does it teach? 130, 614, 276, 432. 7, 324, 853, 047, 895. 32, 011, 405, 983, 021. 520, 475, 320, 108, 502. 7, 919, 932, 753, 210, 875. 23, 791, 859, 921, 509, 321. 944, 608, 785, 264, 321, 840. To find the value of any number in this table, we have to see under what denomination the figures which compose it are placed. Thus, 5 is written under units, and is read 5 units. In the number 64, the 6 is under tens, and the 4 under units; the number is then 6 tens and 4 units, or sixty-four. The number 453,210 has the 4 under hundreds of thousands, which is 4 hundred thousand; the 5 under tens of thousands, which is 50 thousand; the 3 under thousands, which is 3 thousand; the 2 under hundreds, which is 2 hundred; the 1 under tens, which is 1 ten; the 0 under units, |