2 ×(3 + (32+ 6) ×2) + 1 = 2 ×(3 + (9 + 6) ×2) + 1
= 2 ×(3 + 15 ×2) + 1
= 2 ×(3 + 30) + 1
= 2 ×33 + 1
= 66 + 1
= 67
Often schoolchildren are taught a mnemonic device
such as “Please Excuse My Dear Aunt Sally” to recall
the order of operations: parentheses, exponents, multi-
plication, division, addition, subtraction.
See also
EXPANDING BRACKETS
.
order properties The
REAL NUMBERS
satisfy a num-
ber of basic properties with respect to the order rela-
tion <, meaning strictly less than. We have:
1. Trichotomy Law: For any two real numbers aand
b, exactly one of the following holds: a< b, b < a,
or a= b
2. Transitive Law: If a< band b< c, then a< c.
3. Addition Law: If a< b, then a+ c< b+ cfor real
number c.
4. Multiplication Law: If a< band c> 0, then ac < bc.
5. Completeness Law: Any set of real numbers that is
bounded above has a least upper
BOUND
.
The first two properties are standard features of an
ORDERED SET
, and the next two properties explain the
extent to which addition and multiplication respect the
order relation. The final statement is a key property of
the real number system. In their attempts to make the
theory of
CALCULUS
mathematically rigorous and pre-
cise, mathematicians came to realize that one had to be
sure that no numbers are missing from the real number
line. (The
INTERMEDIATE
-
VALUE THEOREM
, for instance,
relies on this.) The fifth law above is designed to ensure
this. With his construction of a D
EDEKIND CUT
, Ger-
man mathematician J
ULIUS
W
ILHELM
R
ICHARD
D
EDEKIND
(1831–1916) was able to prove that the
real-number system does indeed satisfy this fifth prop-
erty. (The set of rational numbers, on the other hand,
satisfies the first four properties but not the fifth, and
so is not complete. Although, for example, every ratio-
nal number xthat satisfies x2≤2 is smaller than 3,
there is no smallest rational number that provides a
bound for all rationals of this type.)
From the five basic properties listed above, other
familiar properties of the real numbers follow. For
instance, one can show that if a< b, then –a> –b. (Use
property 3 adding –aand then –bto both sides.) More
generally, we have that if a< band c< 0, then ac > bc.
From this and property 4, it follows that the square of
any nonzero number is positive: a2> 0 for all a. This
final observation also shows that there is no analogous
order relation for the
COMPLEX NUMBERS
:
If the complex numbers were ordered, then we
would have i2> 0, yielding the absurd state-
ment –1 > 0.
It is often convenient to write a≤bto mean “less
than or possibly equal to.” For instance, 3 ≤5 and
12 ≤12. Often the order properties of the real numbers
are phrased in terms of this version of the relation.
See also
AXIOM OF CHOICE
;
WELL
-
ORDERED SET
.
ordinal numbers In common usage, the adjectives
that denote the position of individual objects in a
sequence, such as a “first,” “second,” or “107th,” are
called ordinal numbers. In the late 1800s German
mathematician G
EORG
C
ANTOR
(1845–1918) noted
that the natural numbers 0, 1, 2,… satisfy the follow-
ing simple order property:
If one writes down any finite number of natural
numbers, then there is always a unique earliest
natural number that was omitted from the list.
For example, in the list {0,1,4,7} the number 2 is the
first natural number missing from this list. Let us use
the notation a,b,c,…| for the first natural number miss-
ing from the collection a,b,c, … For example, we have:
0,1,4,7 | = 2
0,1,2,3,4 | = 5
0 | = 1
Cantor considered the possibility of allowing the
lists of natural numbers used to be infinite. He defined
a quantity ω(omega) as:
ω= 0,1,2,3,…|
that is, ωis the earliest number greater than any finite
natural number. It follows then that 0,1,2,3,…,ω| =
ω+ 1, the first number after ω, and 0,1,2,3,…,ω,
ω+1| = ω+ 2, and so forth. One can continue and
establish that
ordinal numbers 367