is a possible error of as much as half a degree. When a
measurement is written in decimal notation, it is gener-
ally understood that the absolute error does not exceed
a half unit in the last digit. Thus, for example, a
recording of 2.3 indicates that the error does not
exceed ±0.05, whereas recording 2.30 indicates that the
error does not exceed ±0.005. In this context, the final
digit recorded is usually understood to be reliable.
If a number representing a measurement does not
have a decimal part, then a dot is sometimes used to
indicate up to which point the digits are reliable. For
example, in recording a measurement as 23
.00, we are
being told that the 3 is the final reliable digit and that
the error in this measurement could be as much as ±50
(half the 100s place). A recorded measurement of 230
.0,
on the other hand, indicates that the first zero is reli-
able and that the error in this measurement is at most
±5 (half the 10s place).
The digits up to, and including, the reliable digit
are called the significant figures of the measurement.
Thus the measurement 23
.00 has two significant figures,
for example, whereas the recorded measurement 230
.0
has three significant figures. If a result is expressed in
SCIENTIFIC NOTATION
, p×10n, it is generally under-
stood that all the digits of pare significant. For exam-
ple, in writing a value 0.0170 as 1.70 ×10–2, we are
indicating that the final 0 is the result of a measure-
ment, and so this digit is reliable. The quantity 0.0170
thus has three significant digits (and the error of this
measurement is at most ±0.00005). Similarly a
recorded measurement of 0.00030300, for example,
has five significant figures. (The initial three zeros of
the decimal expansion serve only to place the decimal
point correctly. The remaining five digits represent the
result of recording a measurement.)
When calculating with approximate values, it is
important to make sure that the result does not imply
an unrealistic level of precision. For example, if the
dimensions of the room are measured as 14.3 ft by
10.5 ft, multiplying length by width gives the area of
the room as 150.15 ft2. The answer presented this way
suggests a level of accuracy up to the nearest 1/100,
which is unreasonable given that the initial measure-
ments are made to the nearest 1/10. Generally, the
result of a calculation should be presented as no more
accurate than the least accurate initial measurement.
For example, in adding measurements 230
.0 and 1068,
the result should be recorded as 337
.0 (the number
3368 is rounded to the nearest 10). In multiplying 14.3
and 10.5, each with three significant figures, the result
should be written 15.
0 ft2(again three significant figures).
See also
PERCENTAGE ERROR
;
PRECISION
;
RELATIVE
ERROR
;
ROUND
-
OFF ERROR
.
Euclid (ca.300–260
B
.
C
.
E
.) Greek Geometry The geo-
meter Euclid is remembered as author of the most
famous text in the whole of mathematics, T
HE
E
LE
-
MENTS
. In 13 books, the work covers all that was known
in mathematics at his time, from elementary geometry
and number theory, to sophisticated theories of propor-
tion, irrationals, and solid geometry. But Euclid is revered
today primarily for his unique approach in organizing
the material he presented. Starting with a small set of def-
initions, “common notions,” and
AXIOM
s (basic state-
ments whose truth seems to be self-evident), Euclid
derived by pure logical reasoning some 465 propositions
(
THEOREM
s) in mathematics. This established standards
of rigor and powers of deduction that became the model
of all further work in mathematics for the two millennia
that followed. It can be said that The Elements defines
what
PURE MATHEMATICS
is about.
Close to nothing is known of Euclid’s life. It is
believed that he lived and taught in Alexandria, a Greek
city near the mouth of the Nile in what is now Egypt,
and may have been chief librarian of the great library at
the Alexandria Academy. Many ancient historical texts
describing the work of Euclid confuse the mathemati-
cian Euclid of Alexandria with philosopher Euclid of
Megara, who lived about 100 years earlier. Moreover,
Euclid was a very common name at the time, and there
were many prominent scholars from a variety fields
throughout this period. Because of the subsequent con-
fusion and the lack of specific information about the
mathematician Euclid, some historians have put for-
ward the theory that Euclid was not, in fact, a historical
figure, but the name adopted by a team of mathemati-
cians at the library of Alexandria who published a com-
plete work under the single name Euclid. (Compare this
with the fictitious N
ICOLAS
B
OURBAKI
of the 20th cen-
tury). This is not the popular view, however.
The Elements was deemed a standard text of study
for Greek and Roman scholars for 1,000 years. It was
translated into Arabic around 800
C
.
E
. and studied
extensively by Arab scholars. With the revival of scien-
tific interest during the Renaissance, Euclid’s work
168 Euclid