458 semi-magic square

falsehood. Although the statement given above is

clearly true, the truth-value of “This sentence is false,”

for example, is unclear: it cannot be true, for then the

statement says it is false, nor can it be false, for then

the statement is true! (This example forms the basis for

the

LIAR

’

S PARADOX

.) The statement “This sentence is

true” is equally problematic: it can be both true and

false. Self-referential statements are forbidden in formal

systems of logic.

Austrian mathematician K

URT

G

ÖDEL

(1906–78)

made clever use of a self-referential statement to prove

his famous incompleteness theorems.

See also

FORMAL LOGIC

; G

ÖDEL

’

S INCOMPLETENESS

THEOREM

;

HALTING PROBLEM

; J

OURDAIN

’

S PARADOX

.

semi-magic square A square array of numbers in

which the sum of the numbers in any row or column is

the same is called a semi-magic square. (The array is

dubbed “fully magic” if, in addition, the numbers in

each of the two diagonals also add to this same sum.)

The common sum of the rows and columns of the

array is called the magic sum of the array. For exam-

ple, the array

is a semi-magic square with magic sum 10. (This array,

however, is not a

MAGIC SQUARE

).

Semi-magic squares have a number of remarkable

properties in the theory of

MATRIX

algebra. For

instance, suppose Aand Bare semi-magic squares of

the same size with magic sums aand b, respectively.

Regard each as a matrix. Then:

1. The matrix sum A+ Bis again a semi-magic square,

with magic sum a+ b.

2. The matrix product AB is again a semi-magic

square, with magic sum ab.

3. The inverse A–1, if it exists, is still semi-magic, with

magic sum 1/a.

These claims can be proved by setting Jto be the

square matrix with all entries equal to one and noting

that a square matrix Ais semi-magic with magic sum a

if, and only if, AJ = aJ = JA.

There is no such thing as a “semi-magic rectangle”

if one insists that all entries be positive numbers. Sup-

pose, for instance that an array has nrows and m

columns and the entries in each row and column sum

to a. Then, by adding together each of the nrows, the

sum of the entries in the entire array must be na. By the

same token, adding together each of the mcolumns

shows that the sum of the entries in the entire array

must also be ma. Consequently, we must have n= m. If

one permits zero or negative entries, then semi-magic

rectangles are possible, but the magic sum must neces-

sarily be zero.

sequence (progression) A set of numbers arranged in

a list, with each number in the list unambiguously spec-

ified, is called a sequence. For example, the sequence 3,

5, 7, 9, 13 lists five specific numbers, and the sequence

1, 4, 7, 10, 13, 16, … indicates an infinite list of num-

bers, with each number being 3 greater than its prede-

cessor. The numbers in a sequence are called its terms

or elements.

Sometimes the terms of a sequence are specified as a

formula. For example, the sequence 2, 5, 10, …, n2+ 1,

… indicates that the nth term of the sequence is given as

one more than the number squared. A sequence with

finitely many terms is called a finite sequence; one with

infinitely many terms is an infinite sequence. A sequence

with the nth term given by anis denoted {an}, or some

times (an). For example, { } represents the sequence

1, , , ,…; {{–1}n} the sequence –1,1,–1,1,…; and

{ } the sequence x,,,,….

A sequence might also be specified via a

RECUR

-

RENCE RELATION

. For example, the sequence of

FIBONACCI NUMBERS

{FN} is given by Fn+2 = Fn+1 + Fn

with F1= 1 and F2= 1.

A sequence a1, a2, a3, … is said to converge to a

number Lif the numbers anget closer and closer to Las

nbecomes large. Such a sequence is called a

CONVER

-

GENT SEQUENCE

, and the quantity L to which it con-

verges is called the

LIMIT

of the sequence. For instance,

the sequence ,,,,,… converges to the value 1.

Convergent sequences play an important role in

mathematics. For example, an

IRRATIONAL NUMBER

can be thought of as a limit of a sequence of rational

5

–

6

4

–

5

3

–

4

2

–

3

1

–

2

x2

–

4!

x2

–

3!

x2

–

2!

xn

–

n!

1

–

4

1

–

3

1

–

2

1

–

n