until it is known what numbers a and b stand for, no further step can be taken, and the process is considered to be algebraically complete. 23. When b is greater than a, the arithmetical operation denoted by a-b is impossible. For example, if a=3 and b=5, a-b will be 3-5, and we cannot take 5 from 3. But to subtract 5 is the same as to subtract 3 and 2 in succession, so that 3-5 3-3-2-0-2=-2. = We then consider that -2 is 2 which is to be subtracted from some other algebraical expression, or that - 2 is two units of the kind opposite to that represented by 2; and if-2 is a final result, the latter is the only view that can be taken. In some particular cases the quantities under consideration may be such that a negative result is without meaning; for instance, if we have to find the population of a town from certain given conditions; in this case the occurrence of a negative result would shew that the given conditions could not be satisfied, and so also in this case would the occurrence of a fractional result. SUBTRACTION. 24. Since subtraction is the inverse operation to that of addition, to subtract a positive quantity produces a decrease, and to subtract a negative quantity produces an increase. Hence to subtract a positive quantity we must subtract its absolute value, and to subtract a negative quantity we must add its absolute value. Thus, to subtract +4 from +10, we must decrease the amount by 4; we then get +10 - 4. Also to subtract 4 from +6, we must increase the amount by 4; we then get + 6 + 4. and Hence +10 (+4)+10-4+ 6, = = We therefore have the following rule for the subtraction of any term:-to subtract any term affix it to the expression from which it is to be subtracted but with its sign changed. 25. We have hitherto supposed that the letters used to represent quantities were restricted to positive values; it would however be very inconvenient to retain this restriction. In what follows therefore it must always be understood, unless the contrary is expressly stated, that each letter may have any positive or negative value. Since any letter may stand for either a positive or for a negative quantity, a term preceded by the sign + is not necessarily a positive quantity in reality; such terms are however still called positive terms, because they are so in appearance; and the terms preceded by the sign are similarly called negative terms. 26. On the supposition that b was a positive quantity, it was proved in Articles 22 and 24, that We have now to prove that the above laws being true for all positive values of b must be true also for negative values. Let b be negative and equal to c, where c is any positive quantity; then Hence, putting − c for + b, and + c for − b in (i), (ii), (iii), (iv), it follows that these relations are true for all negative values of b, provided are true for all positive values of c; and this we know to be the case. Hence the laws expressed in (A) are true for all values of b. 27. DEF. The difference between any two quantities a and b is the result obtained by subtracting the second from the first. The algebraical difference may therefore not be the same as the arithmetical difference, which is the result obtained by subtracting the less from the greater. The symbol a b is sometimes used to denote the arithmetical difference of a and b. DEF. One quantity a is said to be greater than another quantity b when the algebraical difference a-b is positive. From the definition it is easy to see that in the series 1, 2, 3, 4 &c., each number is greater than the one before it; and that, in the series 1, 2, 3, 4, &c., each number is less than the one before it. Thus 7, 5, 0, − 5, — 7 are in descending order of magnitude. EXAMPLES. Ex. 1. Find the sum of (i) 5 and -4, (ii) – 5 and 4, (iii) 5, -3 and Ans. 005 inches. Ex. 4. A thermometer which stood at 10 degrees centigrade, fell 20 degrees when it was put into a freezing mixture: what was the final reading? Ans. - 10. Ex. 8. Find the value of a + ( − b) − ( − c) when - 2, b = −3, c=-5. Ex. 9. Find the value of -(-a) + b − (−c) when Ans. -2, 4. Ans. - 4. Ans. 0. Ans. - 6. MULTIPLICATION. 28. In Arithmetic, multiplication is first defined to be the taking one number as many times as there are units in another. Thus, to multiply 5 by 4 is to take as many fives as there are units in four. As soon, however, as fractional numbers are considered, it is found necessary to modify somewhat the meaning of multiplication, for by the original definition we can only multiply by whole numbers. The following is therefore taken as the definition of multiplication: "To multiply one number by a second is to do to the first what is done to unity to obtain the second.” Thus 4 is 1+1+1+1; .. 5 × 4 is 5 + 5 + 5 + 5. Again, to multiply by, we must do to done to unity to obtain ; that is, we must divide four equal parts and take three of those parts. the parts into which is to be divided will be 5 what is into Each of and 7x4 So also, (-5) × 4 − (− 5) + (− 5) + ( − 5) + ( − 5) = With the above definition, multiplication by a negative quantity presents no difficulty. For example, to multiply 4 by - 5. Since to subtract 5 by one subtraction is the same as to subtract 5 units successively, -5-1-1-1-1-1; .. 4 × (-5)=-4-4-4-4-4 = - - 20. Again, to multiply - 5 by 4. Since -4-1-1-1-1; .. (− 5) × (− 4) = − (− 5) − (− 5) − (− 5) − (− 5) We can proceed in a similar manner for any other numbers, whether integral or fractional, positive or negative. Hence we have the following rule: To find the product of any two quantities, multiply their absolute values, and prefix the sign + if both factors be positive or both negative, and the sign - if one factor be positive and the other negative. The rule by which the sign of the product is determined is called the Law of Signs. This law is sometimes enunciated briefly as follows: Like signs give +, and unlike signs give -. |