Here are some concepts you will need to be familiar with.
1. Ellipses -- Three dots (. . .) are used to indicate continuation of a pattern. For
example, the list of natural numbers (also called positiveintegers,
or counting numbers) from 1 to 20, inclusive, might be indicated
by 1, 2, 3, . . . , 20. Two or three numbers are written at the beginning
to indicate the pattern, and the last member of the list is written to
indicate the end. Note the punctuation pattern (the positions of the
commas). If no ending number is written, as for instance in
1, 2, 3, . . . , this indicates an unending list - in this example, the list
of all natural numbers. No last member is indicated since there is no
largest positive integer. (Proof - For any positive integer n, the number
n+1 is a larger integer.)
2. Sets -- A set is any specified collection of objects. (See pages 49 and 50 of the text
for a short review of sets, along with the standard notation.) The set of all
natural numbers is denoted by N = {1, 2, 3, ...}. The set of all integers, positive,
negative, or zero, is Z= {. . ., -3, -2, -1, 0, 1, 2, 3, . . .}. Note the use of two
sequences of three dot.
3. Rational Numbers -- A rational number is a ratio of integers a/b, where b ? 0. For
instance, 4/5, 7/3, -3/8, 2 (= 2/1), 0 (= 0/3), and 6.3 (= 63/10) are
rational numbers. As two of the examples suggest, all integers
are rational numbers. The set of rational numbers is denoted Q.
4. Real Numbers and Irrational Numbers -- A real number is any number
corresponding to a point on a ‘real line’ (e.g., on the x-axis of a coordinate
system). The set of all real numbers is denoted by R.
An irrational number is any real number that is not rational. (Note that this
is a technical definition, not a commentary on the mental state of such a
number.) For example, p and e are irrational.
5. Even and Odd -- Integers are classified as even or odd. Here are the definitions --
(a) An integer n is even if there is an integer k such that n = 2k. For
instance, -8, 0, and 24 all are even since -8 = 2(-4), 0 = 2(0), and
24 = 2(12), and -4, 0, and 12 are integers.
(b) An integer n is odd if there is an integer k such that n = 2k + 1 .
For instance, 1 = 2(0) + 1, 17 = 2(8) + 1, and -29 = 2(-15) + 1, so
1, 17, and -29 are odd.
6. Exercises --
(a) Prove that 6, -48, and 100 are even, while -71 and 47 are odd.
(b) The definitions of even and odd are stated only for integers. This is
because if we try to apply these definitions to other real numbers, we
find that all non-integers are neither even nor odd. For instance, prove
that this is the case for the real number 1.7.
7. The term Divides -- An integer n is said to divide an integer m if there is an
integer k such that m = k? n . (Note carefully that the definition is stated in
terms of multiplication, not division!) For example, 7 divides 21 because
21 = 3(7) , and 3 is an integer.
8. Exercises -- Prove that
(a) -4 divides 20;
(b) Every integer divides zero.
9. Prime Numbers -- A positive integer n is prime if n > 1 and the equation
n = k(m), where k and m are positive integers, implies either k = 1 and
m = n, or k = n and m = 1 . In short, a positive integer is prime if it is
larger than 1 and its only positive integer factors are 1 and itself. For
instance, 5 is prime, while 6 = 2(3) is not.
10. Absolute Value -- The absolute value of a real number x, denoted | x| , is defined
If x > 0, then | x| = x ;
if x < 0, then | x| = -x ;
if x = 0, we have | 0| = 0 .
For instance, since 2 > 0, | 2|
= 2 . And since -4 is negative, | -4|
= -(-4) = 4 .
11. Exercise -- Prove that for every real number x, |
x| >=0 .
12. Square Root -- The square root of a non-negative real number x, sqrt(x), is the
non-negative real number whose square is x . Thus, for example,
sqrt(4) = 2 .
(Note that sqrt(x) is not defined if x < 0. And sqrt(4)
equals 2, not ± 2 .)
13. Exercise -- Prove that for every real number x, sqrt(x2) = | x| . (Thus, for example,
sqrt(-3)2 = 3 .)