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upper one. The following example will assist the learner in
Proof by Subtraction, 109720158 Less. Here, we say: 8 from 16, eight; 6 from 8, two; 1 from 10, nine; 1 from 1, nought ; 2 from 7, five; 7 from 15, eight; 10 from 10, nought ; 1 from 2, one; 1 from 3, two.
Observe, that it is of little consequence whether the less number is placed under the greater, or the greater under the less. For, having proved the above example by addition, in order to prove it by subtraction, we perform the operation downwards: that is, we subtract the remainder from the number below it, saying, 8 from 16, eight; 3 from 8, five; 9 from 10, one; 1 from 1, nought; 5 from 7, two; 8 from 15, seven; 1 from 10, nine; 2 from 2, nought; 2 from 3, one.
Let all the subsequent examples in subtraction be proved both by addition and subtraction. Also, let the numbers be written in the form of those in the foregoing example.
3. From four hundred thousand and sixty-two, take ninety thousand eight hundred and seven.
4. From eight millions, three hundred and seventy thousand and sixty, take five millions, nine hundred and forty-three thousand and seven.
5. From nine millions, five hundred and ten thousand five hundred and eighty-seven, take six millions, eight hundred thousand seven hundred and sixty-nine.
6. From one hundred millions, one hundred and eight thousand and fifty-eight, take ninety-nine millions, seventeen thousand three hundred and forty-nine.
7. From eighty-three millions, four hundred and one thousand two hundred and sixty-three, take four millions, three hundred and one thousand eight hundred and twenty-four.
8. From sixteen millions, one hundred and thirty-nine thousand seven hundred and twenty-four, take seven millions, two hundred and nine thousand eight hundred and fifteen.
9. From forty-three millions, one hundred and ten thou
sand eight hundred and thirty-two, take eight millions, four hundred thousand eight hundred and fifty-three.
10. From ten millions, take ninety thousand and seven.
11. From three hundred and two millions, one hundred and eighty-seven thousand two hundred and ninety-one, take nine millions, ninety-six thousand three hundred and ninetytwo.
12. From one billion and forty-nine, take nine millions, ninety thousand and ninety-nine.
13. 9990003814 - 9890004905 =
20. 1000001799112 999999998107 = 87. A unit followed by ciphers we shall call an alt unit, (altus, Lat., high.) Thus, 10 is the first alt, (1 alt;) 100, (2 alt;) 1000, (3 alt,) &c.' The number which precedes thé word alt shows the number of ciphers on the right of the unit.
The first, second, third, fourth, &c. alt, we shall call the alt of any number consisting of 1, 2, 3, 4, &c. figures respectively. But (37) these alts are expressed by the binomials 9 +1; 99 + 1; 999 + 1; 9999 + 1, &c.; that is, the number of nines in the bin. value of an alt, corresponds with the number of figures in the number of which it is the alt.
88. The difference between a number and its alt is called its arithmetical complement, (ar. comp.) Therefore, when the sum of two numbers is an alt unit; each number is the ar. comp. of the other.
89. To find the ar. comp. of a number, (87,) we begin at the left, and subtract each of its figures, successively, from 9; taking care to subtract the right-hand figure from 10, which, being 9 + 1, completes the bin. value of the alt.
Thus : to find the ar. comp. of 23564, beginning with 2, we say: 7, 6, 4, 3, 6: consequently, the complement is 76436.
The numbers 23564 and 76436, which make the 5 alt, are the ar. complements of each other.
This is so easy, that the scholar will, by dividing any ordinary number into periods, see its ar. comp., as it were, intuitively, and read it with nearly the same facility as the number
itself. Thus, if we have the number 26543257, dividing it, by the eye merely, we read at once its ar. comp., which is, seventy-three millions; four hundred and fifty-six thousand, seven hundred and forty-three.
90. If to a number A, we add the ar. comp. of a less number B,-as this ar. comp. (88) is the alt of B-B, we evidently increase A by this alt and diminish it by B. If, then, from the result, we subtract the alt, we have the difference between A and B. Thus, to find A – B, Let A -92245812 and B 73456743 Then, A
92245812 ar. comp. B : 26543257 100 millions B or 8 alt - B.
A 18789069 Sum or A + 8 alt — B. Here, suppressing the + 8 alt, (100 millions,) by which the result is too great, we have A
Wherefore, to find the difference between two unequal numbers, we have the following rule: To the greater add the ar. comp. of the less, and, from the result, subtract a unit of the order of the alt of the less.
91. In finding by the ar. comp. the diff. between A and B, or (A — B,) when the alt of B is within the compass of A, that is, when A has more figures than B, -we may, in writing A, diminish it by a unit of the order of this alt; which is the same as to diminish the result, and more convenient. Should this order, or two, three, &c. consecutive orders, as we proceed to the left, be wanting in A, write 9 in each vacant order, and diminish the next digit towards the left by a unit. Thus:
Let A 56000902346 and B 9903077
55990999269 diff. or (A — B). Here, as the alt of B is the 7 alt, or 10 millions, and, as this order and the next on the left are both wanting in A, we write 9 in each of these two places, and diminish the next
figure 6 on the left by a unit; by which means (48) we have 599 units of this order instead of 600.
Proof. A 56000902346
55990999269 (A – B) as before. Let the following examples be performed and proved in like
1. 997325 987406 3. 1192076 - 999249 2. 520463 229074 4. 90000724 -999637 =
5. 10000085060 9988204 = 92. When B is a series of nines, -as the ar. comp. is, in this case always 1,~we have only to diminish A by the alt and add 1 to its unit figure.
1. 4250484 999999 3250485 Here, as the alt is 6 alt, we merely subtract 1 from the 4 on the left and add 1 to that on the right.
2. 80524631 - 9999999
4. 7000092406 - 9999999 93. A series of nines is the alt minus 1. Therefore, to add a series of nines to a number, is to increase it by the alt and diminish it by a single unit.
5. 9997854 + 99999 = 10097853 94. The signs + and ~, symbols of the opposite attributes of quantity, are further distinguished by the following appellatives : The sign +, which shows the increase of quantity, —the operation by which it is numerically formed, and, hence, may be said to affirm its positive or real existence, -is called the affirmative or positive sign: and the sign, the symbol of the operation by which it is diminished and ultimately destroyed, and which may, therefore, be said to deny it existence, is called the negative sign.
95. A number, having + on the left of it, is called a positive number, (real number.) A number, having – on the
left, is called a negative number, imaginary number ;) because, of itself, it expresses a want of the number of units, to render it equal to 0, which as a real number it would express. Thus, - 5+5= 0. Hence, the positive and negative value of the same quantity destroy each other. Numbers without sign are, of course, positive. Note. The expression on the left of the sign
= is called the first member, and that on the right, the second member of the equation.
96. Let a be the positive, and - a the negative value of any quantity or number: also, let d the difference between the two values. Then, if to each value we add a, this (84) will not affect d. Wherefore d is the difference between at a and a – a: but a- = 0, (95;) therefore, d is the diff. between a ta or 2 a and 0; consequently, d=2a; that is, the difference between the positive and negative value of a quantity is twice its positive value, or the double of that quantity.
Hence, to subtract the negative value of a quantity is the same as to add its positive value.
97. This has sometimes been familiarly illustrated thus : There are two persons, A and B, one of whom, A, has five dollars, (+5;) the other, B, has no money, and is in debt five dollars, ( 5.) Now, to discharge his debt-that is, to render him even with the world, or having nothing, (0) B would require five dollars, and, to be on a par with A, five more: therefore, A is 10 dollars better off than B: that is, the diff. between + 5 and -5 is 10. Also, as the cancelling or taking away of the debt of B is the same as to give B five dollars, we say, that to subtract 5 is the same as to add .+5. Therefore, when several negative quantities are to be subtracted, they must be added as positive quantities--that is, all their signs must be changed from minus to plus.
98. The scholar must not confound the idea of the subtraction of negative quantities with that of their addition. Thus,
- a is the addition of the negative value of a to its positive value; or, which is the same thing, the subtraction from a of its positive value. The expression of the subtraction of its negative value, without changing its sign, would stand thus :
(-a,) which looks awkward, and is much better written a + a.
Wherefore, (as negatives, or quantity destroyers,) the writing of negative quantities, in any expression, with their