Page 15 - Calculus Demystified

CHAPTER 1
Basics
2
• The integers are the positive and negative whole numbers and zero:
..., −3, −2, −1, 0, 1, 2, 3, ... . We denote the set of all integers by Z.
• The rational numbers are quotients of integers.Any number of the form p/q,
with p, q ∈ Z and q = 0, is a rational number. We say that p/q and r/s
represent the same rational number precisely when ps = qr. Of course you
know that in displayed mathematics we write fractions in this way:
1 2 7
+ = .
2 3 6
• The real numbers are the set of all decimals, both terminating and non-
terminating. This set is rather sophisticated, and bears a little discussion. A
decimal number of the form
x = 3.16792
is actually a rational number, for it represents

316792
x = 3.16792 = .
100000
A decimal number of the form

m = 4.27519191919 ...,
withagroupofdigitsthatrepeatsitselfinterminably,isalsoarationalnumber.
To see this, notice that
100 · m = 427.519191919 ...

and therefore we may subtract:
100m = 427.519191919 ...
m = 4.275191919 ...
Subtracting, we see that

99m = 423.244
or
423244
m = .
99000
So, as we asserted, m is a rational number or quotient of integers.
The third kind of decimal number is one which has a non-terminating dec-
imal expansion that does not keep repeating. An example is 3.14159265 ... .
This is the decimal expansion for the number that we ordinarily call π. Such
a number is irrational, that is, it cannot be expressed as the quotient of two
integers.