RULES FOR THE REAL NUMBERS
ONLY
(AND NOT NECESSARILY FOR OTHER
TYPES OF NUMBERS)
3. Rules of inequalities
(7) (a) Given two numbers a and b, exactly one
of the following holds:
(i) a < b
(ii) a = b
(iii) a > b
(b) a > b is the same as b < a.
(c) If a < b, then a + c < b + c
(d) If a < b, then a − c < b − c
(e) If a < b and c > 0, then ca < cb
Each symbol a, b, c represents any real number,
variable, or formula.
Please remember that integers and rationals are
particular kinds of real numbers, so all of the properties below also apply to them.
4. Useful formulas
(8) (a) Difference of squares:
1. Rules of arithmetic
a2 − b2 = (a + b)(a − b)
(1) Addition
(a) Associativity: (a + b) + c = a + (b + c)
(b) Commutativity: a + b = b + a
(c) Identity element: a + 0 = 0 + a = a
(d) Inverse: For each a, there is a corresponding number −a such that
(b) (a + b)2 = a2 + 2ab + b2
(c) (a − b)2 = a2 − 2ab + b2
(d) Geometric sum: If a 6= 1, then
1 − an
1 + a + a2 + · · · an−1 =
1−a
(e) Binomial sum: If n is a nonnegative integer, then
X
n k n−k
n
(a + b) =
k=n
a b
,
k
a + (−a) = (−a) + a = 0
(2) Subtraction
(a) Definition: a − b = a + (−b)
(3) Multiplication
(a) Associativity: (ab)c = a(bc)
(b) Commutativity: ab = ba
(c) Identity element: a · 1 = 1a = a
(d) Inverse:
For each a 6= 0, there is a
corresponding number a−1 = 1/a such
that
k=0
where
n
n!
=
k
(n − k)!k!
n(n − 1) · · · (n − k + 1)
=
.
k(k − 1) · · · 1
Remember that
aa−1 = a−1 a = 1
(e) a · 0 = 0a = 0
(4) Division
(a) Definition: a/b =
(5) Distributivity
(a) a(b + c) = ab + ac
0! = 1 and n! = n(n − 1)!
5. The distance function
a
b
= ab−1
(1) The distance between two real numbers a and
b is equal to
d(a, b) = |a − b| = |b − a|,
where
2. Rules of equations
(6) (a)
(b)
(c)
(d)
(e)
(
x
|x| =
−x
If a = b, then a + c = b + c
If a = b, then a − c = b − c
If a = b, then ca = cb
If a = b and c 6= 0, then a/c = b/c
In general, if a = b, then in any formula
that contains a, you can replace each a
by b.
if x ≥ 0
if x ≤ 0
(2) Triangle inequality:
|a + b| ≤ |a| + |b|
(3) Scaling:
|ab| = |a||b|
Warning: If ca = cb, it is not necessarily true that
a = b.
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