AXIOMS. 1. If equal numbers are added to equal numbers, the sums will be equal. 2. If equal numbers are subtracted from equal numbers, the remainders will be equal. 3. If equals be multiplied by equals, the products will be equal. 4. If equals be divided by equals, the quotients will be equal. 5. If two numbers are each equal to the same number, they are equal to each other. 6. If the same number be added to and subtracted from another number, the latter number will not be changed. 7. If a number be both multiplied and divided by the same number, the former number will not be changed. 8. If two numbers be equally increased or diminished, the difference of the resulting numbers will be the same as the difference of the originals. 9. If two numbers are like parts of equal numbers, they are equal to each other. 10. The whole is greater than any of its parts. 11. The whole is equal to the sum of all its parts. SIGNS. 1. The sign +, called plus, is the sign of addition, and indicates that the number on the right hand is to be added to the one on the left. 2. The sign, called minus, is the sign of subtraction, and indicates that the number on the right is to be subtracted from that on the left. 3. The sign x, called into, is the sign of multiplication, and indicates that the numbers between which it is placed are factors of the same product. 4. The sign, divided by, the left-hand number to be divided by the right hand. 5. The sign =, equal to, indicates that the numbers between which it is placed are equal. 6. 52, 53, the 2 and 3 placed to the right, a little above a number, indicates the power to which it is to be raised. 7. √ √, indicate the extraction of the square and the cube root. 9 NOTATION AND NUMERATION. 1st. A figure standing alone, as 1, 2, 3, holds the units place, or is of the 1st order, and is read, one, two, three. 2d. A number having two figures, as 14, 26, the righthand figure holds the units place, and the left-hand figure that of tens, and they are read, fourteen, twenty-six. COR.-The right-hand figure of a number is called units, or the 1st order; the next figure to the left is called tens, or the 2d order; the third figure, hundreds, or the 3d order; the fourth figure, thousands, or the 4th order; and if a number be expressed with the nine figures in order, making 1 the right-hand figure, the figures will express their respective orders; thus, millions, thousands, units. hundreds of ∞ tens of units of 987 hundreds of or tens of + units of 6 5 4 321 If pointed in periods of three figures each, they may be read as follows: Nine hundred and eighty-seven millions six hundred and fifty-four thousand three hundred and twenty-one. REM. The figures designate the orders. hundreds of tens of units of 9 1 21 321 4,321 54,321 654,321 7,654,321 87,654,321 { eighty-seven millions 987,654,321 { nine hundred and eighty-seven millions twenty-one. three hundred and twenty-one. four thousand three hundred and twenty-one. fifty-four thousand three hundred and twenty-one. six hundred and fifty-four thousand three hundred and twenty-one. seven millions six hundred and fifty-four thousand three hundred and twenty-one. six hundred and fifty-four thousand three hundred and twenty-one. six hundred and fifty-four thousand three hundred and twenty-one. one. REM. The column of 1's is of the 1st order, the column of 2's is of the 2d order, the 3's the 3d order, the 4's the 4th order, etc. = 1 ten. = 1 hundred. = 1 thousand. COR.-The relation of any two consecutive orders is the same, for when in addition the sum of any column reaches 10, the lefthand figure belongs to the next column or order; hence, a table may be formed, thus, 10 units 10 tens 10 hundred 10 thousand 10 ten-thousand 10 hundred-thousand = 1 million. etc. 1 ten-thousand. = 1 hundred-thousand. etc. This method of numeration may be extended; thus, Etc. of Billions. Etc. Billions. of Millions. Hundreds of thousands 123456,789 Read the following notations: 1. 123. 2. 1,234. 3. 12,345. 4. 123,456. 5. 1,234,567. 6. 12,345,678. 7. 123,456,789. to spǝtpunн 123,456,789,123,456,789,123,456,789 REM. This is called the French Method of Numeration, and is generally followed; the English Method has six figures in each period, as follows: of Trillions. ∞ Tens Millions. Units Hundreds of Thousands. of Units. Hundreds of thousands 245,678,954. 365,421,783. 204,603,207. 100,200,300. 20,030,040. 3,004,005. Hundreds of ∞ Tens of Units of 123,456789,12 3 4 5 6 8. 9. 10. 11. 12. 13. 14. 12,302,105,401. Units. or Tens Units |