epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

14 The Nuts and Bolts of Proof, Third Edition
and B{x2,y2^ are any two points on a line (including its x- and y-intercepts),
the slope of the Hne is the number:
yi-yx
m = •
X2 -Xi
if Xi 7^ X2, and it is undefined if Xi = X2.
Proof: By hypothesis, if A is the x-intercept of the Hne, then A{p/q, 0),
where p7^0 (as .4 is not the point (0,0)), q^O (because division by 0 is not
defined), and p and q are integers. By hypothesis, if B is the y-intercept of
the line, then 5(0, r/s), where r 7^ 0 (as 5 is not the point (0,0)), 5 7^ 0 (because
division by 0 is not defined), and r and s are integers. Therefore, the slope of
the fine is the number:
--0 s _[^ ra
m =
0-l~ sp
where sp 7^ 0, r^ 7^ 0, and sp and rq are both integers. Thus, m, the slope
the line, is a rational number. •
EXAMPLE 3. The sum of the first n counting numbers is equal to
Wn+l)]/2.
Discussion: We can rewrite this statement in the more explicit (and less
elegant) form: "If n is an arbitrary counting number and one considers the
sum of the first n counting numbers {i.e., all the numbers from 1 to n,
including 1 and n), then their sum can be calculated using the formula
Wn+l)]/2.
Let us start by separating the hypothesis and the conclusion:
A. Consider the sum of the first n counting numbers {i.e., 1 + 2 + 3 +
\-n). {Implicit hypothesis: We are famihar with the properties and
operations of counting numbers.)
B. The sum above can be calculated using the formula [n(n + l)]/2; that is:
. ^ . n{n-\-l)
l+2 + 3 + ---+n = -^-y-^.
Before working on a proof, we might want to check that this equahty works
for some values of n, but we do need to keep in mind that these will be
examples and will not provide a proof.
When n = 5, we add the first 5 counting numbers so we have:
1 + 2 + 3 + 4 + 5-15.
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