2 × (3 + (3
2 + 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, multiplication, division, addition, subtraction.
See also EXPANDING BRACKETS.
order properties The REAL NUMBERS satisfy a number of basic properties with respect to the order relation <, meaning strictly less than. We have:
1. Trichotomy Law: For any two real numbers a and
b, exactly one of the following holds: a < b, b < a,
or a = b
2. Transitive Law: If a < b and b < c, then a < c.
3. Addition Law: If a < b, then a + c < b + c for real
number c.
4. Multiplication Law: If a < b and 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 precise, 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 DEDEKIND CUT, German mathematician JULIUS WILHELM RICHARD
DEDEKIND (1831–1916) was able to prove that the
real-number system does indeed satisfy this fifth property. (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 rational number x that satisfies x
2
≤ 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 –a and then –b to both sides.) More
generally, we have that if a < b and c < 0, then ac > bc.
From this and property 4, it follows that the square of
any nonzero number is positive: a 2 > 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 i
2 > 0, yielding the absurd statement –1 > 0.
It is often convenient to write a ≤ b to 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 GEORG CANTOR (1845–1918) noted
that the natural numbers 0, 1, 2,… satisfy the following 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 missing 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, ω is 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
2 + 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, multiplication, division, addition, subtraction.
See also EXPANDING BRACKETS.
order properties The REAL NUMBERS satisfy a number of basic properties with respect to the order relation <, meaning strictly less than. We have:
1. Trichotomy Law: For any two real numbers a and
b, exactly one of the following holds: a < b, b < a,
or a = b
2. Transitive Law: If a < b and b < c, then a < c.
3. Addition Law: If a < b, then a + c < b + c for real
number c.
4. Multiplication Law: If a < b and 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 precise, 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 DEDEKIND CUT, German mathematician JULIUS WILHELM RICHARD
DEDEKIND (1831–1916) was able to prove that the
real-number system does indeed satisfy this fifth property. (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 rational number x that satisfies x
2
≤ 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 –a and then –b to both sides.) More
generally, we have that if a < b and c < 0, then ac > bc.
From this and property 4, it follows that the square of
any nonzero number is positive: a 2 > 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 i
2 > 0, yielding the absurd statement –1 > 0.
It is often convenient to write a ≤ b to 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 GEORG CANTOR (1845–1918) noted
that the natural numbers 0, 1, 2,… satisfy the following 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 missing 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, ω is 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