b, then there is at least one point in this interval at
which the tangent to the curve is horizontal to the x-
axis. In more stringent mathematical language, his the-
orem reads:
For any function f(x) continuous in a closed
interval [a,b],differentiable in the open inter-
val (a,b),and satisfying f(a) = f(b) = 0, there
exists at least one value cbetween aand bsuch
that f′(c) = 0. That is, the zeros of a differen-
tiable function are always separated by zeros
of the derivative.
This result follows readily from the
EXTREME
-
VALUE
THEOREM
, which asserts that any such function attains
a maximum value at some location cin the interval (a,
b).The tangent to the curve must be horizontal at any
such apex. (A careful study of
MAXIMUM
/
MINIMUM
val-
ues establishes this.)
Although Rolle’s theorem is a special case of the
MEAN
-
VALUE THEOREM
, it is usual to prove Rolle’s the-
orem independently and use it as a first step toward
proving the more general result.
Roman numerals Based on a simple tally system sim-
ilar to the one used by the ancient Egyptians, merchants
of the Roman empire of about 500
B
.
C
.
E
. used letter
symbols for powers of 10 and for the intermediate val-
ues of 5, and simply grouped symbols together to repre-
sent all other quantities. The symbols used were:
The expression CLXXIII, for instance, represented the
number 100 + 50 + 10 + 10 + 1 + 1 + 1 = 173.
Although the order of the symbols was not important,
it became the convention to list symbols from largest to
smallest, left to right.
Initially the symbols D and M were not part of the
Roman system. The number 1,000 was written ( I ), and
further applications of round brackets allowed for the
expression of even greater quantities. For instance,
(( I )) represented 10,000, and ((( I ))) represented
100,000. Stonemasons introduced the symbols Dand
Mto simplify their work.
The Romans also introduced other ornamentations
to increase the value of a numerical symbol. For
instance, vertical bars were used to represent a 100-
fold increase, and a bar placed above the symbol repre-
sented a 1,000-fold increase. For instance,
|X| = 100 ×10 = 1,000
–
X = 1,000 ×10 = 10,000
|
–
X| = 100 ×1,000 ×10 = 1,000,000
There was no symbol for
ZERO
in the Roman system.
To avoid the four-fold repetition of symbols (as in
the expression CCCCXXXXIIIII for 444), a subtractive
principle was introduced in the 13th century:
The placement of a small value immediately
to the left of a higher value indicates that that
small value is to be subtracted from the
higher value.
Thus 4 could be written as IV, 90 as XC, and 444 as
CDXLIV. The subtractive principle was subject to
two rules:
1. The symbols V, L, and D cannot be used as the
numbers to be subtracted.
2. Only one symbol I, X, or C can be placed before a
higher number symbol.
Thus, for example, it was not permissible to write IIX
for eight. Although not a proper
PLACE
-
VALUE SYSTEM
,
with the subtractive principle in use, the order of the
symbols used was now important.
Performing operations of basic arithmetic with
Roman numerals is very awkward. For example, it is
not immediate what the solution to the following addi-
tion problem should be:
XLIV
+ XVII
+ XXIX
That European merchants were comfortable working
with the Roman numeral system for well over a millen-
nium suggests that scholars did not use the numeral
system to perform calculations, only to record the
I= 1
V= 5
X=10
L=50
C = 100
D = 500
M = 1000
450 Roman numerals