Calculus: Analytic Geometry and Calculus, with Vectors

8 Analytic geometry in two dimensions

44 Let h be positive and let X (lambda) be greater than 1. Observe that

Figure 1.191 shows the correct positions of six points having the coordinates

A 0 P1 B PO

-h 0 x-i h

+1h

Figure 1.191

a2+i

%2-1 h

P2

shown there when X = 2. Make a new figure which shows where Po, Pi, and P2

should be when X = 10. Make another new figure which shows where Po, P1,

and P2 should be when X = 191.

45 Referring to Figure 1.191 and supposing that h > 0 and X > 1 as before,

show that Po is the mid-point of the line segment with end points at P1 and P2.

46 When an appropriate time comes, we shall prove that there is a positive

number, denoted by the symbol -\/2, whose square is 2. Everyone should know

that is not rational, and students possessing requisite time and acumen

should become familiar with a proof. We prove the fact by obtaining a con-

tradiction of the assumption that is rational and hence that is repre-

sentable in the form = m/n, where m and n are positive integers. We use

the fact that 28 = 22.7 and the more general fact that each positive integer n

is representable in the form n = 2qs, where q is a nonnegative integer and s is

one of the odd integers 1, 3, 5, 7,. If we suppose that = m/n, then

(1) 2

(M)2 2pr12 22pr2

n \2gs/ 22gs21

where p and q are nonnegative integers and r and s are odd integers. In case

q > p, (1) gives

(2) 21+25-2p52 = r2)

and this is false because the left side is divisible by 2, while the right side, being

the square of an odd integer, is odd and is not divisible by 2. In case p > q,

(1) gives

(3) s2 = 22p-2q-'r2,

and this is false because the right side is divisible by 2 while the left side is not.

This proves that is not rational; the assumption that is rational leads

to false conclusions. Remark: It is possible to give different proofs of this result

and of the more general fact that if n is a positive integer which is notone of the

perfect squares 1, 4, 9, 16, 25, 36, 49, .. , then is irrational. The standard

proofs depend, in one way or another, upon fundamental facts about factoring

positive integers.

(^47) Persons who make desk calculators do their menial arithmetical chores
can get very good approximations to square roots by use of an excellent method
which involves some very interesting arithmetical ideas. When we want to
approximate the square root of a positive number I given in decimal form, we
put 6 in the form .4 = 102"a, where n is an integer and 1 a < 100, and use
the fact that = 10" V. To obtain good approximations to V_a, we start
with a given first approximationx, for which 1 xl 10 and calculate some