Calculus: Analytic Geometry and Calculus, with Vectors

4 Analytic geometry in two dimensions

pates a and b is Ib - al, that is, b - a whenb z a and a - b when b 5 a.

The fundamental fact that the distance from a to b is less than or equal to

the distance from a to 0 plus the distance from 0 to b is expressed by the

inequality

(1.14) la - bI5 Ial + IbI.

Replacing b by -b in this inequality gives the inequality

(1.15) j a + bI s at + Ibi.

Problem 41 at the end of this section shows how this can be proved.

We learn in the arithmetic and algebra of real numbers that x2 = 0

when x = 0 and that x2 > 0 when x 5;,-' 0. If N is a positive number,

then there,ore two values of x for which x2 = N; the positive one of these

numbers by 1/N, and the negative one is denoted by - 1/N.

Thus 42 = 16, (-4)2 = 16, 16 = 4, and - = -4. Since

(-4)2 = 16 and 16 = 4, we see that

(1.16) 1/(-4)2= =4=I-4I.

This is a special case of the formula = Ixl, which holds for each real

number x. In particular, - = 0.

There are times when special properties of the number zero must be

taken into account. The facts that 0 + a = a and 0 for each

number a seem to be thoroughly understood by all arithmeticians, but

the role of zero in division may require comment here. It is a funda-

mental fact that we write x = b/a to represent the number x that satisfies

the equation ax = b, provided there is one and only one number x that

satisfies the equation. If a 0 0 and b = 0, then 0 is the one and only

number x that satisfies the equation and therefore

0=

a

(a 54 0).

Thus 0/a = 0 provided a 0 0. If a = b = 0, then each number satisfies

the equation and therefore

0

6

is meaningless.

If a = 0 and b 0 0, then no number x satisfies the equation and therefore

b

6

is meaningless

when b 0 0. Thus we see that b/a is meaningless when a = 0, whether

b is 0 or not; division by zero is taboo. To look at the matter another

way, we observe that if a 0 0, then the equation ax = ay implies that