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.

then

For, suppose one factor, as a, to be affected with the negative sign. The construction of (— a) × 6 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: 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

M

-a

A

Fig. 13.

B

m.

When the product is in the form a X (-b), - 6 must be laid off in the negative sense towards M, b = OB1, 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 JB, intersects OB, in M. It is also obvious that + axb=+m. Hence

(−a)Xb=ax(−b)=-a Xb.

If both factors are affected with the negative sign, the construction is as follows: Draw OB:

B -6

-a

A

Fig. 14.

m

(a)

OA =

- b in the negative,

a in the negative,

OJ j in the positive sense, M AM parallel to BJ, intersecting BO produced in M. Then OM m 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.

In introducing negative magnitudes the law of combinations of the signs + and −, as described in Art. 12, must be observed. The complete symbolical statement of the associative law for addition and subtraction is contained in the formula:

±(±a ±b) = ±(± a) ± (±b),

wherein the order of occurrence of the signs + and must be the same in the two members of the equation.

-

16. In Multiplication and Division. Construct upon OA and O M, Fig. 15, the products

'=

OL = axb, OM= axb,

OE=bXc,

ON=(ab) X c, ON,= ax (bXc).

In this construction the lines marked fi, fa, fa, through L, N and B, E and J, Care, by the rule for constructing products

(Art. 5), parallel to each other, as are also the lines h,, h through A, N, and J, E, and the lines k,, k, through A, M and J, B1. Hence by proportion, as determined by the parallels f2, fi

babbXc: (a X b) Xc, (Prop. 13.)

and as determined by the parallels k„, k1,

j:a::b: axb,

.. jabXc: (a X b) X c.

But the parallels h2, h, determine the proportion

j:a::bc:a × (b × c).

(Prop. 13.) (Prop. 1.)

Therefore N, N, are one and the same point (Prop. 6) and X(XaXb) Xc=XaX(Xb × c).

Thus the sign X is distributive over the successive factors of a product; that is, as here follows when a=j,

X (bXc)=(X b) X (X c).

The same is true of the sign /; for the product of any magnitude by its reciprocal is j (Art. 6), and two products that have their factors respectively equal are obviously themselves equal (Art. 5); that is,

and

[axa=j, bx / b=j,

(/a Xa) × (b×|b)=jXj=j.

(Art. 7.)

Whence, by the associative law in multiplication just proved, and because the means in a proportion between like magnitudes may be interchanged (Prop. 11),

Thus the associative law for the sign / is the same as that for the sign X, and its general statement is

× (×a× b) = × (× a) × (b),

wherein the order of occurrence of the signs X and in the two members must be the same, and the law of their combination (Art. 13) must be observed.

V.

COMMUTATIVE LAW FOR REAL QUANTITIES.

17. In Addition and Subtraction. Let a and b represent any two lengths taken in the same sense along a straight line such that a= OA and b= AB; then

Let P' be so taken that OB′=b; then B’B=a. Therefore

a+b=OB=b+ a..

If AB be negative and equal to -b, lay off a from O to the right, from the extremity of a to the left. B will fall to the right or left of O according as b is less than or

In the former OB will be positive, in the latter negative. If now B' be so taken that B'O=b, then B’B = a.

If a b c and d be any fourth magnitude, then

=

c±d=d+c=a+b±d=±d±b+a.

Hence

Obviously any algebraic sum may have its terms commuted in like manner. Stated symbolically, this law is

±a±b=±b±a,

wherein each letter carries with it like signs on both sides of the equation.