to Beatrice, and tears the third piece into thirds
again. He hands one piece to Andrea, a second
to Beatrice, and tears the third remaining piece
into thirds again. He repeats this process indefinitely. Eventually, John will give all the paper
away—half to Andrea and half to Beatrice. But
the quantity of paper Andrea receives and the
quantity Beatrice receives can also be computed as a third, plus a third of a third, plus a
third of a third of a third, and so forth. Thus it
must be the case that
+
+
equals .
A repeating decimal can be thought of as a sum of a
geometric sequence. For example, the decimal 0.111…
equals the series
+
+
, which,
according to the formula above, is . (Consequently the
repeating decimal 0.999… equals 9 times this, 9 × ,
which is 1: 0.999… = 1.)
The sum S of just the first n terms of a geometric
series, a + ar + ar
2 +…+ ar
n–1 can be computed from the
formula S – rS = a – ar
n
. Provided that r ≠ 1, this gives:
(If –1 < r < 1, then r
n
→ 0 as n grows. This again shows
that a + ar + ar
2 + ar
3 + … = a ·
=
.) For
example, the sum 1+
+
+ … +
equals
. A similar calculation solves a
famous “chessboard puzzle”:
Legend has it that the game of chess was
invented for an Indian maharaja who became
so delighted with the game that he wanted to
reward the inventor with whatever he desired.
The inventor asked for nothing more than one
grain of rice on the first square of the chessboard, two grains on the second square, four
on the third, and so forth, each square containing double the number of grains than the previous square. Given that there are 64 squares on
a chessboard, how many grains of rice did the
inventor in fact request?
According to the formula, the inventor asked for 1 + 2
+ 4 + 8 +…+ 2
63 grains of rice. This equals
= 2
64 – 1 = 18,446,744,073,709,551,615 grains, which
is the equivalent of about 25 billion cubic miles of rice,
an inconceivable quantity. The inventor fooled the
maharaja into making a promise he could not possibly
honor.
See also ARITHMETIC SEQUENCE; CONVERGENT SEQUENCE; SERIES.
geometric series See GEOMETRIC SEQUENCE.
geometric transformation A specified procedure
that shifts points in the plane to different positions
(and thereby changing the location, and possibly the
shapes, of geometric figures) is called a geometric transformation. More precisely, a geometric transformation
is a FUNCTION that associates with each point of the
plane some other point in the plane. (One may require
the function to be one-to-one and onto.)
For example, the function that shifts each point of
the plane one unit to the right is a geometric transformation (called a translation). This transformation preserves the shapes of all geometric figures.
Any geometric transformation that preserves distances between points in the plane (and hence the
shape and size of geometric figures) is called an isometry or a rigid motion. One that multiplies all distances
between points by a constant factor (called the dilation
factor) is called a similitude, and a transformation that
takes straight lines to straight lines is called a LINEAR
TRANSFORMATION. All isometries are linear transformations, for example. These ideas also extend to transformations in three-dimensional space.
While he never made explicit use of the concept in
his writings, the idea of a geometric transformation
came from the work of the Greek geometer EUCLID (ca.
300–260 B.C.E.).
We list here some classical examples of geometric
transformations.
Reflection in a Line
Given a line l in the plane, a reflection in this line takes
a point P on one side of l to the corresponding point P′
1
1 2
1 2
64
⋅
−
−
1
1
1
2
1
1
2
1023
512
10
⋅
−
−
=
1
––
512
1
––
4
1
––
2
a
––
1 – r
1 – 0
––
1 – r
a ar ar
ar
a
r
r
n
n
+ +
+ +
=
−
−
−
2
1
1
1
L
1
––
9
1
––
9
+ ...
1
––
1000
1
––
100
1
––
10
1
––
2
+ ...
1
––
27
1
––
9
1
––
3
224 geometric series
again. He hands one piece to Andrea, a second
to Beatrice, and tears the third remaining piece
into thirds again. He repeats this process indefinitely. Eventually, John will give all the paper
away—half to Andrea and half to Beatrice. But
the quantity of paper Andrea receives and the
quantity Beatrice receives can also be computed as a third, plus a third of a third, plus a
third of a third of a third, and so forth. Thus it
must be the case that
+
+
equals .
A repeating decimal can be thought of as a sum of a
geometric sequence. For example, the decimal 0.111…
equals the series
+
+
, which,
according to the formula above, is . (Consequently the
repeating decimal 0.999… equals 9 times this, 9 × ,
which is 1: 0.999… = 1.)
The sum S of just the first n terms of a geometric
series, a + ar + ar
2 +…+ ar
n–1 can be computed from the
formula S – rS = a – ar
n
. Provided that r ≠ 1, this gives:
(If –1 < r < 1, then r
n
→ 0 as n grows. This again shows
that a + ar + ar
2 + ar
3 + … = a ·
=
.) For
example, the sum 1+
+
+ … +
equals
. A similar calculation solves a
famous “chessboard puzzle”:
Legend has it that the game of chess was
invented for an Indian maharaja who became
so delighted with the game that he wanted to
reward the inventor with whatever he desired.
The inventor asked for nothing more than one
grain of rice on the first square of the chessboard, two grains on the second square, four
on the third, and so forth, each square containing double the number of grains than the previous square. Given that there are 64 squares on
a chessboard, how many grains of rice did the
inventor in fact request?
According to the formula, the inventor asked for 1 + 2
+ 4 + 8 +…+ 2
63 grains of rice. This equals
= 2
64 – 1 = 18,446,744,073,709,551,615 grains, which
is the equivalent of about 25 billion cubic miles of rice,
an inconceivable quantity. The inventor fooled the
maharaja into making a promise he could not possibly
honor.
See also ARITHMETIC SEQUENCE; CONVERGENT SEQUENCE; SERIES.
geometric series See GEOMETRIC SEQUENCE.
geometric transformation A specified procedure
that shifts points in the plane to different positions
(and thereby changing the location, and possibly the
shapes, of geometric figures) is called a geometric transformation. More precisely, a geometric transformation
is a FUNCTION that associates with each point of the
plane some other point in the plane. (One may require
the function to be one-to-one and onto.)
For example, the function that shifts each point of
the plane one unit to the right is a geometric transformation (called a translation). This transformation preserves the shapes of all geometric figures.
Any geometric transformation that preserves distances between points in the plane (and hence the
shape and size of geometric figures) is called an isometry or a rigid motion. One that multiplies all distances
between points by a constant factor (called the dilation
factor) is called a similitude, and a transformation that
takes straight lines to straight lines is called a LINEAR
TRANSFORMATION. All isometries are linear transformations, for example. These ideas also extend to transformations in three-dimensional space.
While he never made explicit use of the concept in
his writings, the idea of a geometric transformation
came from the work of the Greek geometer EUCLID (ca.
300–260 B.C.E.).
We list here some classical examples of geometric
transformations.
Reflection in a Line
Given a line l in the plane, a reflection in this line takes
a point P on one side of l to the corresponding point P′
1
1 2
1 2
64
⋅
−
−
1
1
1
2
1
1
2
1023
512
10
⋅
−
−
=
1
––
512
1
––
4
1
––
2
a
––
1 – r
1 – 0
––
1 – r
a ar ar
ar
a
r
r
n
n
+ +
+ +
=
−
−
−
2
1
1
1
L
1
––
9
1
––
9
+ ...
1
––
1000
1
––
100
1
––
10
1
––
2
+ ...
1
––
27
1
––
9
1
––
3
224 geometric series