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When a magnitude decreases and becomes zero, its reciprocal obviously increases and becomes infinite. Since zero is not a magnitude, neither is infinity as here defined. In Art. 6 it was shown that b = / a implies also a = / 5; hence, from the definition / O = co follows
The construction for a product (Art. 5) shows that when one of its factors becomes o or co, the other remaining finite, the product itself is also o or co, so that for all finite values of a a / O = a X co = o,
From the definition of addition (Art. 3) it is also obvious that
11. Indeterminate Algebraic Forms. When a sum or product assumes one of the forms-Hoo — co, o Xoo, o/o, oo / co, it is said to be indeterminate, by which is meant: the form by itself gives no information concerning its own value.
(i). The form -- o — co. On a straight line ABP take at random two points A, B, so that AB is any real finite magnitude whatever. Take P, Q, R, on the same line,
and let Q pass into coincidence with A. P then passes out of finite range, R passes into coincidence with B and the difference A B =/a — / 6, whatever its original value, assumes the form / O – / O = co – o .
Hence, taken by itself, oo — co gives no information concerning its own value and is indeterminate.
(ii). The forms o X co, o/o, oo / co. In the figure of Art. 5, let M and / remain fixed, while MA and /B, being always parallel to one another, turn about M and / until MA coincides with, and /B becomes parallel to O.M. At this instant a becomes zero, ö infinite, and a X 6 assumes the form o X o ; and because the original value of a X & is anything we choose to make it, the expression O X co gives no information concerning its own value and is therefore an indeterminate form.
An expression, such as / a - / 5, that gives rise to an indeterminate form, may nevertheless approach a determinate value as it nears its critical stage. To find this value is described as evaluating the indeterminate form. (See Arts. 43 and 52.)
III. LAW OF SIGNS FOR REAL QUANTITIES.
12. In Addition and Subtraction. The sign +, by definition, indicates that the magnitude following it is to be added algebraically to what precedes, without having its character as a magnitude in any way changed; and the sign — indicates that the magnitude immediately following it is to be reversed in sense (taken in the opposite direction) and then added algebraically to what precedes. Any symbolic representative of quantity, a letter for example, unattended by either of the signs -- or —, but still thought of as part of an algebraic sum, is supposed to have the same relation and effect in such a sum as if it had before it the sign -i-. This usage necessitates the following law of signs in addition:
13. In Multiplication and Division. In Art. 8 it was agreed that a /b shall stand for the product a X / 5. This convention requires that the combination X / shall produce /. Looked at from another point of view, the symbol X indicates that the letter following it is to be used as a factor with its character as a magnitude unchanged, while / gives notice that the reciprocal of the magnitude immediately following it is to be used as a factor.
Any symbolic representative of quantity, a letter for example, unattended by either of the signs X or /, but still thought of as part of (factor in) an algebraic product, is supposed to have the same relation and effect in such product as if it had before it the sign X. The usage here described necessitates the following law of signs for X and / :
In practice the sign X is usually omitted, or replaced by a dot; thus: a X & = a 6 = a 6.
14. In Combination with each other: -i-, - with X, /. In the construction of a product any factor affected with the negative sign — must be laid off in the sense opposite to the one it takes when affected with the positive sign +, and the constructions involving negative factors lead to the following rule:
An odd number of negative factors produces a negative product;
An even number of negative factors produces a £ositive product.
For, suppose one factor, as a, to be affected with the negative sign. The construction of (— a) X & is then as follows: Lay off & in the positive sense, say to the right, and OJ, the real unit, in the positive sense, say upwards: then — a must extend downwards along /O produced. Join /B and draw AM parallel to B/ to intersect BO, produced backwards, in M. The product (– a) × b is thus the negative magni
Fig. 13. tude — m. .
When the product is in the form a X (– 5), — b must
be laid off in the negative sense towards M. – b = OB, , a in the positive sense towards A, a = OA, and it is easy to prove by proportion in the similar triangles thus formed, that the line through A, parallel to /B, intersects OB, in M. It is also obvious that -|- a X & = + m. Hence (— a) × b = a X (– 6) = — a X 6.
If both factors are affected with the negative sign, the construction is as follows: Draw O/3 = — 5 in the negative, J OA = — a in the negative,
O/= j in the positive sense, M AM parallel to B/, intersecting BO produced in M. Then OM = m is positive and by writing the proportions for the A similar triangles OB/and OMA
Pig. 14. it is easy to show that
Thus the product of one negative factor and any number of positive factors is negative, while every pair of negative factors yields only positive products. Hence the proposition, Q. E. D.
IV. ASSOCIATIVE LAW FOR REAL QUANTITIES.
15. In Addition and Subtraction. The sequence of the terms of a sum remaining unchanged, the terms may be added separately, or in groups of two or more, indiscriminately, without disturbing the value of the sum ; that is,
(a + 5) + c = a + (5 + c) = a + b + c,
where a, b, c may represent positive or negative magnitudes indiscriminately. This is made evident at once by laying off and comparing with one another the lines a + 6, c, and a, b + c, and a, b, c, taken in the proper sense and in the order indicated in the three groupings.