Comparison of the two tables side by side show that

the two processes are identical.

The process of drawing a table for evaluating a

polynomial p(x) at a value x= h, or, equivalently, for

dividing p(x) by x– h, is called synthetic division. It is a

very simple and efficient process. As an example, the fol-

lowing table shows that the polynomial p(x) = 2x4+ 3x3

– 4x2+ 5x+ 4 has value 6 when evaluated at x= ,

and quotient 2x3+ 4x2– 2x+ 4 when divided by x– .

Returning to the theme of triangular numbers: Any

nested product of the form:

(9(…(9(9(9 + 1)+1)+1)…+1)+1)

is triangular. For instance, 9 + 1 = 10 = T4, 9(9 + 1) + 1

= 91 = T13, and 9(9(9 + 1) +1) + 1 = 820 = T40. This

follows from the relation 9Tn+ 1 = T3n+1 beginning

with T1= 1. This relation also shows that if the num-

ber of nines present in the nested product is n, then the

-th triangular number is produced.

net In geometry, a net is a diagram drawn on a page

which, when cut out and folded along the lines indi-

cated, can be used to construct a

POLYHEDRON

. For

example, the following diagram shows nets of a cube

and a

TETRAHEDRON

.

Not every arrangement of six squares on a page

produces a net for a cube. (For example, a row of six

squares does not fold to form a cube.) One can show

that there are 11 essentially different nets for a cube,

each representing a fundamentally different way of

unfolding the figure. There are just two different ways

to unfold a tetrahedron. Mathematicians have shown

that there are 261 different ways to unfold a four-

dimensional

HYPERCUBE

into a three-dimensional net of

eight connected cubes.

In terms of accounting, the word net refers to the

profit calculated after deducting all operating expenses.

For example, if a small private school receives

$100,000 in tuition payments per year as income but

incurs an annual operating expense of $97,000 (for

salaries, insurance fees, school supplies, and the like),

then the school is operating with an annual net profit

of $3,000.

In commerce, the term net denotes the weight of

goods excluding the weight of any wrapper or con-

tainer holding the goods. For example, many grocery

stores subtract the weight of the plastic containers used

to hold fresh food items from the total weight of the

item being purchased. Customers are charged only for

the net weight of the item.

Neumann, John von (1903–1957) Hungarian-American

Game theory, Logic, Analysis, Abstract algebra Born

on December 28, 1903, in Budapest, Hungary, pure

and applied mathematician John von Neumann is

remembered for his important contributions to a tre-

mendously wide range of topics. In pure mathematics,

he worked on problems in

SET THEORY

and devised a

set of axioms for the subject alternative to those pro-

posed by E

RNST

F

RIEDRICH

F

ERDINAND

Z

ERMELO

(1871–1953). He also made advances in functional

analysis,

OPTIMIZATION

theory, and

GROUP THEORY

,

and succeeded in providing an axiomatic system for the

theory of quantum mechanics. In applied mathematics,

von Neumann is best remembered for his 1944 text

3n– 1

––

2

1

–

2

1

–

2

350 net

hab c d

0ah (ah + b)h((ah + b)h+ c)h

aah+ b(ah + b)h+ c((ah + b)h+ c)h+ d= p(h)

23–454

01 2–12

24–246

1

–

2

Nets for a cube and a tetrahedron