equation of the form f(x) = 0 even if there are no clear
algebraic means for doing so. (For instance, there are
no general techniques helpful for solving
+
+
– 5 = 0. The bisection method provides the
means to find, at least, approximate solutions to such
equations. The method is based on the fact that if two
function values f(a) and f(b) of a CONTINUOUS FUNCTION have opposite signs, then, according to the INTERMEDIATE VALUE THEOREM, a ROOT of the equation f(x) =
0 lies between a and b. The method proceeds as follows:
1. Find two values a and b (a < b) such that f(a) and
f(b) have opposite signs.
2. Set m = (a + b)/2, the midpoint of the interval, and
compute f(m).
3. If f(m) = 0, we have found a zero. Otherwise, if f(a)
and f(m) have opposite signs, then the zero of f lies
between a and m; repeat steps 1 and 2 using these
new values. If, on the other hand, f(m) and f(b) have
opposite signs, then the zero of f lies between m and
b; repeat steps 1 and 2 using these new values. In
either case, a new interval containing the zero has
been constructed that is half the length of the original interval.
4. Repeated application of this procedure homes in on
a zero for the function.
To solve the equation f(x) =
+
+
– 5
= 0, for example, notice that f(1) = –0.268 < 0 and
f(2) = 0.650 > 0. A zero for the function thus lies
between 1 and 2. Set m = 1.5. Since f(1.5) = 0.217 > 0
we deduce that, in fact, the zero lies between 1 and
1.5. Now set m = 1.25 to see that the zero lies between
1.25 and 1.5.
One can find the location of a zero to any desired
degree of accuracy using this method. For example,
repeating this procedure for the example above six
more times shows that the location of the zero lies in
the interval [1.269,1.273]. This shows that to three significant figures the value of the zero is 1.27.
The bisection method will fail to locate a root if the
graph of the function touches the x-axis at that location
without crossing it. Alternative methods, such as NEWTON’S METHOD, can be employed to locate such roots.
bisector Any line, plane, or curve that divides an
angle, a line segment, or a geometric object into two
equal parts is called a bisector. For example, the equator
is a curve that bisects the surface of the EARTH. A
straight line that divides an angle in half is called an
angle bisector, and any line through the MIDPOINT of a
line segment is a segment bisector. If a segment bisector
makes a right angle to the segment, then it is called a
perpendicular bisector.
Bolyai, János (1802–1860) Hungarian Geometry
Born on December 15, 1802, in Kolozsvár, Hungary,
now Cluj, Romania, János Bolyai is remembered for
his 1823 discovery of NON-EUCLIDEAN GEOMETRY, an
account of which he published in 1832. His work
was independent of the work of NIKOLAI IVANOVICH
LOBACHEVSKY (1792–1856), who published an account
of HYPERBOLIC GEOMETRY in 1829.
Bolyai was taught mathematics by his father
Farkas Bolyai, himself an accomplished mathematician,
and had mastered CALCULUS and mechanics by the time
he was 13. At age 16 he entered the Royal Engineering
College in Vienna and joined the army engineering
corps upon graduation four years later.
Like many a scholar throughout the centuries,
Farkas Bolyai had worked, unsuccessfully, on the challenge of establishing the PARALLEL POSTULATE as a logical consequence of the remaining four of EUCLID’S
POSTULATES. He advised his son to avoid working on
this problem. Fortunately, János Bolyai did not take
heed and took to serious work on the issue while serving as an army officer. During the years 1820 and
1823 Bolyai prepared a lengthy treatise outlining the
details of a new and consistent theory of geometry for
which the parallel postulate does not hold, thereby settling once and for all the problem that had troubled
scholars since the time of EUCLID: the parallel postulate
cannot be proved a consequence of the remaining postulates of Euclid.
In Bolyai’s system of hyperbolic geometry it is
always the case that, for any point P in the plane, there
are an infinite number of distinct lines through that
point all PARALLEL to any given line not through P. (In
ordinary geometry, where the parallel postulate holds,
there is only one, and only one, line through a given
point P parallel to a given direction. This is PLAYFAIR’S
AXIOM.) In his new geometry, angles in triangles sum to
less than 180°, and the ratio of the circumference of a
circle to its diameter is greater than π.
√x + 3
√x + 2
√x
√x + 3
√x + 2
√x
46 bisector
algebraic means for doing so. (For instance, there are
no general techniques helpful for solving
+
+
– 5 = 0. The bisection method provides the
means to find, at least, approximate solutions to such
equations. The method is based on the fact that if two
function values f(a) and f(b) of a CONTINUOUS FUNCTION have opposite signs, then, according to the INTERMEDIATE VALUE THEOREM, a ROOT of the equation f(x) =
0 lies between a and b. The method proceeds as follows:
1. Find two values a and b (a < b) such that f(a) and
f(b) have opposite signs.
2. Set m = (a + b)/2, the midpoint of the interval, and
compute f(m).
3. If f(m) = 0, we have found a zero. Otherwise, if f(a)
and f(m) have opposite signs, then the zero of f lies
between a and m; repeat steps 1 and 2 using these
new values. If, on the other hand, f(m) and f(b) have
opposite signs, then the zero of f lies between m and
b; repeat steps 1 and 2 using these new values. In
either case, a new interval containing the zero has
been constructed that is half the length of the original interval.
4. Repeated application of this procedure homes in on
a zero for the function.
To solve the equation f(x) =
+
+
– 5
= 0, for example, notice that f(1) = –0.268 < 0 and
f(2) = 0.650 > 0. A zero for the function thus lies
between 1 and 2. Set m = 1.5. Since f(1.5) = 0.217 > 0
we deduce that, in fact, the zero lies between 1 and
1.5. Now set m = 1.25 to see that the zero lies between
1.25 and 1.5.
One can find the location of a zero to any desired
degree of accuracy using this method. For example,
repeating this procedure for the example above six
more times shows that the location of the zero lies in
the interval [1.269,1.273]. This shows that to three significant figures the value of the zero is 1.27.
The bisection method will fail to locate a root if the
graph of the function touches the x-axis at that location
without crossing it. Alternative methods, such as NEWTON’S METHOD, can be employed to locate such roots.
bisector Any line, plane, or curve that divides an
angle, a line segment, or a geometric object into two
equal parts is called a bisector. For example, the equator
is a curve that bisects the surface of the EARTH. A
straight line that divides an angle in half is called an
angle bisector, and any line through the MIDPOINT of a
line segment is a segment bisector. If a segment bisector
makes a right angle to the segment, then it is called a
perpendicular bisector.
Bolyai, János (1802–1860) Hungarian Geometry
Born on December 15, 1802, in Kolozsvár, Hungary,
now Cluj, Romania, János Bolyai is remembered for
his 1823 discovery of NON-EUCLIDEAN GEOMETRY, an
account of which he published in 1832. His work
was independent of the work of NIKOLAI IVANOVICH
LOBACHEVSKY (1792–1856), who published an account
of HYPERBOLIC GEOMETRY in 1829.
Bolyai was taught mathematics by his father
Farkas Bolyai, himself an accomplished mathematician,
and had mastered CALCULUS and mechanics by the time
he was 13. At age 16 he entered the Royal Engineering
College in Vienna and joined the army engineering
corps upon graduation four years later.
Like many a scholar throughout the centuries,
Farkas Bolyai had worked, unsuccessfully, on the challenge of establishing the PARALLEL POSTULATE as a logical consequence of the remaining four of EUCLID’S
POSTULATES. He advised his son to avoid working on
this problem. Fortunately, János Bolyai did not take
heed and took to serious work on the issue while serving as an army officer. During the years 1820 and
1823 Bolyai prepared a lengthy treatise outlining the
details of a new and consistent theory of geometry for
which the parallel postulate does not hold, thereby settling once and for all the problem that had troubled
scholars since the time of EUCLID: the parallel postulate
cannot be proved a consequence of the remaining postulates of Euclid.
In Bolyai’s system of hyperbolic geometry it is
always the case that, for any point P in the plane, there
are an infinite number of distinct lines through that
point all PARALLEL to any given line not through P. (In
ordinary geometry, where the parallel postulate holds,
there is only one, and only one, line through a given
point P parallel to a given direction. This is PLAYFAIR’S
AXIOM.) In his new geometry, angles in triangles sum to
less than 180°, and the ratio of the circumference of a
circle to its diameter is greater than π.
√x + 3
√x + 2
√x
√x + 3
√x + 2
√x
46 bisector