zero by zero could, allegedly, have any desired value.
For example, one could argue that 0/0 = 17 by noting
that 0 ×17 = 0 is a true statement. There is no single
appropriate value for this quantity.)
A “unit” fraction, also known as an E
GYPTIAN
FRACTION
, is a proper fraction with numerator equal to
one. A “complex” fraction is one in which the numera-
tor or denominator, or both, is a fraction. For example,
is a complex fraction. The division-of-fractions
rule provides the means to simplify complex fractions.
See also
CANCELLATION
;
CONTINUED FRACTION
;
DECIMAL REPRESENTATION
;
PARTIAL FRACTIONS
;
PER
-
CENTAGE
;
RATIO
;
RATIONALIZING THE DENOMINATOR
;
REDUCED FORM
.
fractional part function See
FLOOR
/
CEILING
/
FRACTIONAL PART FUNCTIONS
.
frequency In
STATISTICS
, the absolute frequency of an
observed value is the number of times that value
appears in a data set. For example, in the sample 4, 6,
3, 4, 4, 1, 7, 3, 7, 9, 8, 4, 4, 5, 2, the absolute fre-
quency of the observation 4 is five, and that of 8 is one.
The relative frequency of an observed value is the pro-
portion of times it appears. This is computed by divid-
ing the absolute frequency of the observation by the
total number of entries in the data set. For example,
the relative frequency of 4 in the above data set is
5/15 = 1/3, and that of 8 is 1/15.
One can compute the relative frequency of entries
in an infinite data set by making use of a
LIMIT
. For
example, mathematicians have proved that 5.8% of
the powers of two 1, 2, 4, 8, 16, 32, 64, 128,… begin
with a 7. (The first power to do so is 246.) By this they
mean that if one were to examine the first Npowers of
two, approximately 5.8 percent of them begin with a
7. This approximation is made more exact by taking
larger and larger values of N.
In physics the term frequency is defined as the
number of cycles that occur per unit time in a system
that oscillates (such as a pendulum, a wave, a vibrating
string, or an alternating current). The symbol used for
frequency is usually f, although the Greek letter νis
often employed for the frequency of light or other elec-
tromagnetic radiation. A unit of frequency is called a
hertz (Hz).
frequency distribution See
STATISTICS
:
DESCRIPTIVE
.
frequency polygon See
STATISTICS
:
DESCRIPTIVE
.
frieze pattern (band ornament) A design on an infi-
nite strip that consists of repeated copies of a single motif
is called a frieze pattern. (In classical architecture, a frieze
is a horizontal structure, usually imprinted with decora-
tion, resting along the top of some columns. The modern
equivalent is a horizontal strip of wallpaper used to deco-
rate the top portion of a wall just below the ceiling.)
Mathematicians are interested in the
SYMMETRY
properties of frieze patterns. Each frieze pattern, by def-
inition, is symmetrical under a
TRANSLATION
Tin the
direction of the strip. A frieze pattern might also be
symmetrical about a horizontal
REFLECTION
(H),a ver-
tical reflection (V),a
ROTATION
of 180°about a point in
the design (R),a
GLIDE REFLECTION
(G),or some collec-
tion of these five basic transformations. The first frieze
pattern shown below possesses all five symmetries.
Not all combinations of T, H, V, R, and G, represent
the symmetries of a frieze pattern. For instance, a frieze
pattern cannot possess the symmetries Tand Halone,
for the composition of these two symmetries produces a
glide reflection, and so the pattern must possess symme-
try Gas well. Similarly, any frieze pattern that possesses
symmetries Hand Vmust also possess symmetry R,
since the combined effect of a vertical and a horizontal
reflection is a rotation. Reasoning this way, one can
prove that there are only seven combinations of symme-
try types a frieze pattern could possess. These are:
Talone T, H, and G
Tand V T, V, R, and G
Tand R T, H, V, R, and G
Tand G
3/5
–
4/7
206 fractional part function
A frieze pattern