366 order of a matrix

and it is a greatest lower bound if lis larger than any

other lower bound for aand b. In our example, the

greatest lower bound of {A,B} and {B,C} is their inter-

section {B}. A partially ordered set is called a lattice if

every pair of elements has a least upper bound and a

greatest lower bound. The existence of unions and

intersections shows that the set of all subsets of any

given set is a lattice. The set of natural numbers under

the relation “is a factor of” is also a lattice: the least

upper bound of any two natural numbers is their low-

est

COMMON MULTIPLE

, and the greatest lower bound is

their

GREATEST COMMON FACTOR

. Many results from

NUMBER THEORY

can be interpreted as statements

about this order relation of the natural numbers.

Any totally ordered set is a lattice. For example,

the least upper bound of any two real numbers aand b

is simply the one that is the larger of the two, and their

greatest lower bound is the smaller number.

See also

BOUND

;

ORDER PROPERTIES

.

order of a matrix (dimension of a matrix) An m×n

MATRIX

, that is, a matrix with mrows and ncolumns, is

said to be of order m×n(read as “mby n”). An n×n

matrix is sometimes called a square matrix of order n.

In

GROUP THEORY

an element gof a

GROUP

is said

to be of order nif nis the first positive integer such that

gn= e, assuming there is such an integer. (Here eis the

IDENTITY ELEMENT

of the group.) If, in some mathemat-

ical work, one is thinking of matrices as elements of a

group, one usually reserves the word order for a group

theoretic meaning, and uses the word dimension to

describe the size of the matrix. For example, the matrix

is a square matrix of dimension 3 and order 2, since

A2= I.

See also

IDENTITY MATRIX

.

order of magnitude See

SCIENTIFIC NOTATION

.

order of operation (operational precedence) In eval-

uating arithmetic computations involving more than

one type of operation, mathematicians have assigned an

order of precedence as to which operations are exer-

cised first.

It is agreed that any computation involving

ADDI

-

TION

alone is computed in the order as read from left

to right. For instance, 8 + 5 + 2 is computed as 13 + 2,

which is 15. (Although the

ASSOCIATIVE

property of

addition shows that the order of computation in this

case does not matter.) As

SUBTRACTION

can be viewed

as the addition of negative quantities, any computation

involving both addition and subtraction is thus com-

puted in the same manner, as read from left to right.

For instance, 2 – 5 + 7 is computed as (–3) + 7, which

is 4. (Again, the associative property shows that fol-

lowing this convention is not vital.)

MULTIPLICATION

can be viewed as “repeated addi-

tion” and so, in some sense, is a more potent operation

than addition and subtraction. It is given precedence

over these operations.

DIVISION

, which can be viewed

as multiplication by

RECIPROCAL

quantities, is given the

same status. Thus given any computation involving all

four operations, one is expected to compute all multi-

plications and divisions that appear first (read in a left-

to-right manner) and all additions and subtractions

second. For instance, one computes:

2 + 4 ×3 ÷6 – 3 ×3 + 5

as 2 + 12 ÷6 – 9 + 5, which equals 2 + 2 – 9 + 5, which

is zero. (Reading strictly from left to right produces the

incorrect answer of 5.)

As

EXPONENT

s can be viewed as an act of perform-

ing repeated multiplications, all powers that appear in

a computation are given greater precedence over multi-

plications and divisions, and so must be computed

first. For instance, 2 + 62÷9 is computed as 2 + 36 ÷9

= 2 + 4 = 6.

Often parentheses or

BRACKETS

are introduced to

change the order of operations in a computation. Math-

ematicians follow the convention that if parentheses are

present, one must compute the quantities inside the

parentheses first (using the above rules). If multiple sets

of parentheses are present, this requires evaluating the

innermost parentheses first. For instance, we compute:

2 ×(3 + (32+ 6) ×2) + 1

as