When a magnitude decreases and becomes zero, its reciprocal obviously increases and becomes infinite. Since zero

=

/a implies also a = | b ;

is not a magnitude, neither is infinity as here defined.
In Art. 6 it was shown that b
hence, from the definition / o = ∞ follows

The construction for a product (Art. 5) shows that when one of its factors becomes o or ∞, the other remaining finite, the product itself is also o or∞, so that for all finite values of a

From the definition of addition (Art. 3) it is also obvious that

II. Indeterminate Algebraic Forms. When a sum or product assumes one of the forms +∞ — ∞,0X∞,0/0, ∞∞, it is said to be indeterminate, by which is meant: the form by itself gives no information concerning its ownvalue.

(i). The form + ∞ 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,

AP=/ AQ=|a, BP=| BR=|b,

and let pass into coincidence with A.

P then passes out

of finite range, R passes into coincidence with В and the difference

In the figure

until

(ii). The forms oX ∞, ∞ / ∞, ∞ / ∞0. of Art. 5, let M and J remain fixed, while MA and JB, being always parallel to one another, turn about M and MA coincides with, and JB becomes parallel to OM. At this instant a becomes zero, b infinite, and a × b assumes the form o × ∞ ; and because the original value of a b is anything we choose to make it, the expression OX∞ gives no information concerning its own value and is therefore an indeterminate form.

and / ∞

=

Since / o ∞ o, we may replace o/o by OX∞ and ∞ / ∞ by ∞ X 0. The two forms o/o and ∞∞ are therefore also indeterminate.

An expression, such as ab, 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 +. This usage necessitates the following law of signs in addition:

− (− a) = +a, − (+ a) = a, and by a, unattended by any sign, is understood + a.

13. In Multiplication and Division. In Art. 8 it was agreed that a / b shall stand for the product a × /b. This convention requires that the combination X / shall produce /. Looked at from another point of view, the symbol 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 × b

ab: = a. b.

14. In Combination with each other: +, 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 positive product.

For, suppose one factor, as a, to be affected with the negative sign. The construction of (— a) Xb is then as follows: Lay off b 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 JO produced. Join JB and draw AM parallel to BJ to intersect BO, produced backwards, in M. The product (— a) × b is thus the negative magnitude

m..

When the product is in the form a X (-b), - b must be laid off in the negative sense towards M, - b= OB1, a in

OA,, and it is easy to

the positive sense towards A1, α = prove by proportion in the similar triangles thus formed, that the line through A, parallel to JB, intersects OB, in M. It is also obvious that + axb=+m. Hence

If both factors are affected with the negative sign, the construction is as follows: Draw OB= — b in the negative, a in the negative,

B -6

-α

A

Fig. 14.

m

(a)

OA =

OJ=j in the positive sense, м AM parallel to BJ, intersecting BO produced in M. Then OMm is positive and by writing the proportions for the similar triangles OBJ and OMA it is easy to show that (b) = + a × b.

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+b)+c=a+(b+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 + b, c, and a, b + c, and a, b, c, taken in the proper sense and in the order indicated in the three groupings.