cube roots, for summing ARITHMETIC SERIES, and finding SUMS OF POWERS.
As an astronomical treatise, A
– ryabhatiya is written
as a series of 118 verses summarizing all Hindu mathematics and astronomical practices known at that time.
A number of sections are purely mathematical in context and cover the topics of ARITHMETIC, TRIGONOMETRY, and SPHERICAL GEOMETRY, as well as touch on the
theories of CONTINUED FRACTIONs, QUADRATIC equations, and SUMMATION. A
– ryabhata also described methods for finding integer solutions to linear equations of
the form by = ax + c using an algorithm essentially
equivalent to the EUCLIDEAN ALGORITHM.
Historians do not know how A
– ryabhata obtained
his highly accurate estimate for π. They do know, however, that A
– ryabhata was aware that it is an IRRATIONAL NUMBER, a fact that mathematicians were not
able to prove until 1775, over two millennia later. In
practical applications, however, A
– ryabhata preferred
to use √
—
10 ≈ 3.1622 as an approximation for π.
Scholars at the time did not think of sine as a ratio
of side-lengths of a triangle, but rather the physical
length of a half-chord of a circle. Of course, circles of
different radii give different lengths for corresponding
half-chords, but one can adjust figures with the use of
proportionality. Working with a circle of radius 3,438,
A
– ryabhata constructed a table of sines for each angle
from 1° to 90°. (He chose the number 3,438 so that the
circumference of the circle would be close to 21,600 =
360 × 60, making one unit of length of the circumference matching one minute of an angle.) Thus, in his
table, sine of 90° is recorded as 3,438, and the sine of
30°, for example, as 1,719.
With regard to astronomy, A
– ryabhatiya presents a
systematic treatment of the position and motions of the
planets. A
– ryabhata calculated the circumference of the
Earth as 24,835 miles (which is surprisingly accurate)
and described the orbits of the planets as ELLIPSEs.
European scholars did not arrive at the same conclusion until the Renaissance.
associative A BINARY OPERATION is said to be associative if it is independent of the grouping of the terms
to which it is applied. More precisely, an operation * is
associative if:
a * (b * c) = (a * b) * c
for all values of a, b, and c. For example, in ordinary
arithmetic, the operations of addition and multiplication are associative, but subtraction and division are
not. For instance, 6 + (3 + 2) and (6 + 3) + 2 are equal
in value, but 6 – (3 – 2) and (6 – 3) – 2 are not. (The
first equals 6 – 1 = 5, and the second is 3 – 2 = 1.) In
VECTOR analysis, the addition of vectors is associative,
but the operation of taking CROSS PRODUCT is not.
From the basic relation a * (b * c) = (a * b) * c, it
follows that all possible groupings of a finite number
of fixed terms by parentheses are equivalent. (Use an
INDUCTION argument on the number of elements present.) For example, that (a * b) * (c * d) equals
(a * (b * c)) * d can be established with two applications of the fundamental relation as follows: (a * b)
* (c * d) = ((a * b) * c) * d = (a * (b * c)) * d. As a
consequence, if the associative property holds for a
given set, parentheses may be omitted when writing
products: one can simply write a * b * c * d, for
instance, without concern for confusion.
These considerations break down, however, if the
expression under consideration contains an infinite
number of terms. For instance, we have:
0 = 0 + 0 + 0 + …
= (1 – 1) + (1 – 1) + (1 – 1) + …
If it is permissible to regroup terms, then we could write:
0 = 1 + (–1 + 1) + (–1 + 1) + (–1 + 1) + …
= 1 + 0 + 0 + 0 + …
= 1
This absurdity shows that extreme care must be taken
when applying the associative law to infinite sums.
See also COMMUTATIVE PROPERTY; DISTRIBUTIVE
PROPERTY; RING.
asymptote A straight line toward which the graph
of a function approaches, but never reaches, is called
an asymptote for the graph. The name comes from the
Greek word asymptotos for “not falling together” (a:
“not;” sym: “together;” ptotos: “falling”). For example, the function y = 1/x has the lines x = 0 and y = 0
as asymptotes: y becomes infinitely small, but never
reaches zero, as x becomes large, and vice versa. The
function y = (x + 2)/(x – 3) has the vertical line x = 3
asymptote 29
As an astronomical treatise, A
– ryabhatiya is written
as a series of 118 verses summarizing all Hindu mathematics and astronomical practices known at that time.
A number of sections are purely mathematical in context and cover the topics of ARITHMETIC, TRIGONOMETRY, and SPHERICAL GEOMETRY, as well as touch on the
theories of CONTINUED FRACTIONs, QUADRATIC equations, and SUMMATION. A
– ryabhata also described methods for finding integer solutions to linear equations of
the form by = ax + c using an algorithm essentially
equivalent to the EUCLIDEAN ALGORITHM.
Historians do not know how A
– ryabhata obtained
his highly accurate estimate for π. They do know, however, that A
– ryabhata was aware that it is an IRRATIONAL NUMBER, a fact that mathematicians were not
able to prove until 1775, over two millennia later. In
practical applications, however, A
– ryabhata preferred
to use √
—
10 ≈ 3.1622 as an approximation for π.
Scholars at the time did not think of sine as a ratio
of side-lengths of a triangle, but rather the physical
length of a half-chord of a circle. Of course, circles of
different radii give different lengths for corresponding
half-chords, but one can adjust figures with the use of
proportionality. Working with a circle of radius 3,438,
A
– ryabhata constructed a table of sines for each angle
from 1° to 90°. (He chose the number 3,438 so that the
circumference of the circle would be close to 21,600 =
360 × 60, making one unit of length of the circumference matching one minute of an angle.) Thus, in his
table, sine of 90° is recorded as 3,438, and the sine of
30°, for example, as 1,719.
With regard to astronomy, A
– ryabhatiya presents a
systematic treatment of the position and motions of the
planets. A
– ryabhata calculated the circumference of the
Earth as 24,835 miles (which is surprisingly accurate)
and described the orbits of the planets as ELLIPSEs.
European scholars did not arrive at the same conclusion until the Renaissance.
associative A BINARY OPERATION is said to be associative if it is independent of the grouping of the terms
to which it is applied. More precisely, an operation * is
associative if:
a * (b * c) = (a * b) * c
for all values of a, b, and c. For example, in ordinary
arithmetic, the operations of addition and multiplication are associative, but subtraction and division are
not. For instance, 6 + (3 + 2) and (6 + 3) + 2 are equal
in value, but 6 – (3 – 2) and (6 – 3) – 2 are not. (The
first equals 6 – 1 = 5, and the second is 3 – 2 = 1.) In
VECTOR analysis, the addition of vectors is associative,
but the operation of taking CROSS PRODUCT is not.
From the basic relation a * (b * c) = (a * b) * c, it
follows that all possible groupings of a finite number
of fixed terms by parentheses are equivalent. (Use an
INDUCTION argument on the number of elements present.) For example, that (a * b) * (c * d) equals
(a * (b * c)) * d can be established with two applications of the fundamental relation as follows: (a * b)
* (c * d) = ((a * b) * c) * d = (a * (b * c)) * d. As a
consequence, if the associative property holds for a
given set, parentheses may be omitted when writing
products: one can simply write a * b * c * d, for
instance, without concern for confusion.
These considerations break down, however, if the
expression under consideration contains an infinite
number of terms. For instance, we have:
0 = 0 + 0 + 0 + …
= (1 – 1) + (1 – 1) + (1 – 1) + …
If it is permissible to regroup terms, then we could write:
0 = 1 + (–1 + 1) + (–1 + 1) + (–1 + 1) + …
= 1 + 0 + 0 + 0 + …
= 1
This absurdity shows that extreme care must be taken
when applying the associative law to infinite sums.
See also COMMUTATIVE PROPERTY; DISTRIBUTIVE
PROPERTY; RING.
asymptote A straight line toward which the graph
of a function approaches, but never reaches, is called
an asymptote for the graph. The name comes from the
Greek word asymptotos for “not falling together” (a:
“not;” sym: “together;” ptotos: “falling”). For example, the function y = 1/x has the lines x = 0 and y = 0
as asymptotes: y becomes infinitely small, but never
reaches zero, as x becomes large, and vice versa. The
function y = (x + 2)/(x – 3) has the vertical line x = 3
asymptote 29