number, the sum of these partial differences will be the difference of the two given numbers. But in cases where a figure of any order in the less number is greater than that of the same order in the larger number, the subtraction is not possible. If this figure be increased by 10 units of the same order, the subtraction becomes possible; and if the figure of the next superior order in the less number be increased by 1, the larger and smaller numbers having been equally increased, the difference of the two numbers will be unaltered. Let it be required to find the difference of 456 and 273. Here 456 consists of 4 hundreds, 5 tens, and 6 units, and 273 consists of 2 hundreds, 7 tens, and 3 units. Now 3 units taken from 6 units leaves 3 units, the first partial remainder. Next, 7 tens taken from 5 tens is impossible, but 10 tens added to 5 tens make 15 tens, and the subtraction becomes possible. Then 7 tens taken from 15 tens leave 8 tens, the second partial remainder. But as 10 tens are equal to 1 hundred, 1 hundred added to 2 hundreds make 3 hundreds, and 3 hundreds taken from 4 hundreds leave 1 hundred, the third partial remainder. Hence the sum of these partial remainders will be the difference of the two numbers 456 and 273. And the difference is 1 hundred, 8 tens, and 3 units, or 183. The preceding process may be briefly exhibited, and the names of the orders of the figures may be omitted in performing the opera tion. 456 183 difference. The number 3 taken from 6 leaves 3. 7 cannot be taken from 5, add 10 to 5, which make 15; then 7 taken from 15 leaves 8. Next add 1 to 2, which makes 3, and 3 taken from 4 leaves 1; and the difference of the two numbers is 183. The correctness of the process of subtraction may be readily verified; since of two numbers, the sum of their difference and the less number is equal to the greater. 6. The operation of the subtraction of one number from another is denoted by the sign —, called minus. Thus 8-5 means that 5 is to be subtracted from 8, and is read 8 minus 5; also 8 - 5 3 is read 8 minus 5 is equal to 3, or that the difference of 8 and 5 is 3. When the sum or difference of two or more numbers is required to be considered as one number, the two numbers connected by the symbol of addition or subtraction are included in a parenthesis or a brace, thus (5 + 2) and (5 — 2), or {5 +2} and {5 −2}, and sometimes by a line placed over them in this manner, 5 + 2 and 5 2. One number may be considered the complement of another when the sum of the two numbers make up any given number. 7. DEF.—The arithmetic complement of a number is defined to be the difference between any given number and the unit of the next superior order; as 6 is the arithmetic complement of 4, 47 of 53, 845 of 155, and so on, being the differences respectively of 4, 53, 155, and 10, 100, 1,000, the next superior units to these numbers. Conversely, also, 4, 53, 155 are the arithmetic complements of 6, 47, 845 respectively.' The arithmetic complement of a number may always be found by subtracting the figure in the unit's place from 10 and the rest of the figures of the number from 9. == Since from the definition, the arithmetic complement of 155 is 845 = 1000–155, whence it follows that 155+845 = 1000, or that the sum of any number and its arithmetic complement will always be equal to the unit of the next superior order. Again, since 1551000-845; if 155 or its equivalent in terms of the arithmetic complement be subtractive, it may be written 1845, by placing the subtractive unit before the left digit of the arithmetic complement, as is done in the characteristics of logarithms, when they are subtractive. 8. PROB.-If the arithmetic complement be added to any other number of the same number of figures, the sum will exceed the difference of the two numbers by an unit of the next superior order. If 155 be subtracted from 768, the remainder is 613, the difference between them. But if the arithmetic complement 845 of 155, the less number, be added to 768, the greater, the sum will be 1613, one unit (1000 in this case) of the next superior order greater than the difference of the two numbers. By removing this unit, the number left will be equal to 1 The arithmetic complement can be employed to find the difference of two numbers, as also the aggregate of several numbers when some of them are additive and some subtractive. It is employed with the greatest advantage in the arithmetic of logarithms, in cases where some logarithms are to be added and some to be subtracted in the same computation. Instead of finding the two sums and subtracting one from the other, the sum of the logarithms to be added and the arithmetic complements of the logarithms to be subtracted will give the correct difference of the sums of the additive and subtractive logarithms. The mode of writing the arithmetic complement is so simple that the arithmetic complement of a logarithm can be as readily written as the logarithm itself can be copied from the tables. the difference of 768 and 155; so that the difference of two numbers can be found by addition. The formal removal may be avoided by writing the arithmetic complement thus, 1845, with the subtractive unit on the left, which when added to 768, the sum will be 613, the additive and subtractive units being equal to zero.1 9. As numbers may be equal or unequal to one another, one number, or an aggregate of numbers, is said to be equal to, or equivalent to another number, when both are composed of the same number of units. And one number is said to be greater than another when there are more units in the former than in the latter; and the latter is said to be less than the former. The following axioms will be found to be useful in arithmetical reasonings. 1. Numbers are equal to one another which consist of the same number of units. 2. A whole number or integer is equal to the sum of all the parts of it. 3. Any whole number is greater than any part of it; and any part of a number is less than the number itself. 4. Numbers or aggregates of numbers which are equal to the same number are equal to one another. 5. If equal numbers be added to equal numbers, the sums are equal. 6. If equal numbers be added to unequal numbers or if unequal numbers be added to equal numbers, the sums are unequal. 7. If equal numbers be taken from, or subtracted from equal numbers, the remainders are equal. 8. If equal numbers be subtracted from unequal numbers, or if unequal numbers be subtracted from equal numbers, the remainders are unequal. 9. If the same number be added to, and subtracted from another, the value of the latter is unaltered.2 1 Ex. 1. Find the difference between 328011 and 23412 by means of the arithmetic complement. 328011 A. C. of 23412 is 176588 By addition 304599 is the difference of the two numbers. Ex. 2. Find the difference between the sum of 7970597, 1038037, and 3406424, and the sum of 3279120 and 8441042 by the arithmetic complement. EXERCISES. I. Find the sums of the following sets of numbers, and express the results in words: 1. 5, 12, 20, 15, 35, and 50. 2. 37, 8, 99, 105, 250, 700, and 2. 3. 5000, 500, 20, 90, 9000, and 1. 4. 3584, 2796, 6416, 7204, 1897, and 8103. 5. 10000, 50505, 171717, 1008, and 201. 6. 235762, 105501, 287788, and 799999. 7. 1234567, 8765433, 6894703, 3105297, 5712843, and 4187157. 8. 57, 1876423, 1587, 999, and 3334447. 9. 100, 314500, 1237, 94, and 3000000. 10. 5876432100, 1284, and 387142659. II. 1. Explain why in the addition of numbers the operation is begun at the unit's place. Is this necessary ? 2. Shew that 5 and 7 make 12; and state explicitly every definition and axiom in the proof. 3. Add 3041 to 7198; and explain the operation. 4. Arrange the nine digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, in three lines with three digits in each line, so that the sums of three digits taken in every possible direction may be always equal to 15. 5. Find the sum of all the numbers which can be formed with the figures 1, 2, 3, 4; all the four figures being taken to form each number. III. Find the differences of the following numbers, and verify the accuracy of the results: 1. 35 and 53. 2. 20000 and 999. 3. 478329 and 923874. 4. 1008425 and 100842. 5. 2784397 and 1234567. 6. 200000000 and 123456789. 7. 732584963 and 478342987. 8. 105723678 and 989456. 9. 10000001000 and 7077070077. IV. 1. Subtract thirty millions twenty-six thousand and three, from forty-five millions seven thousand and twenty-one; and find what number must be added to the difference to make one hundred millions. 2. Add together four million twenty thousand and seventy-nine, twelve million two thousand and seven, and one million and five thousand, and subtract 16538107 from the sum. 3. What is the difference between the aggregate of 1050, 325, 1769, 150801, and a million? Shew that the same difference is obtained by taking one of the numbers from a million, another from the remainder, and so on for the rest of the numbers. 4. Subtract four billions thirty millions seventy-five thousand and eleven from 8876521856201, and write the difference in words. 5. Find the remainder after subtracting the numbers 44444, 9999, 666, 77, 1, in succession from 1000000. 6. Find the difference between the sum of 31845, 814, 10345, and the sum of 10014, 569, 1845. 7. Subtract the sum of 48002, 6100, and 5018162, from the difference of 1000000000 and 1234567890. 8. Subtract the difference between sixty billions and sixty thousands from the sum of fifty trillions and fifty millions. V. 1. How may the accuracy of the process of subtraction be verified? Give an example. 2. Subtract 819 from 918, explaining the process. 3. Shew that the difference of 254 and 125 is the same as the difference when these numbers are each increased by 123. 4. What number subtracted from 670194 will leave 3825? 5. Half the sum of any two numbers is equal to the greater, and half the difference is equal to the less number. Apply this to find the two numbers whose sum is 521 and difference 175. 6. By how much does the sum of the numbers 27182818284 and 31415926535 exceed their difference? VI. 1. Find the excess of 8765427 above 7634289 by means of the arithmetic complement; and explain the principle. 2. Find the result by means of the arithmetic complement of the numbers 578421, 854325, 104231, which are to be added, and of 325410, 684531, 213418, which are to be subtracted. 3. Find the difference of the sums of 256, 1875, 34210, and of 4008, 214, 56, by means of the arithmetic complement. 4. Find the difference of the aggregate of the numbers 1234567, 8543021, 5432147, and of 3245785, 5180072, by means of the arithmetic complement. 5. Write five numbers of seven places of figures each under each other, and beneath these write the arithmetic complements of four of them. Shew that the sum of the nine lines of figures will consist of the fifth line with 4 in the eighth place reckoned from the right-hand figure. |