Imagine an equilateral TRIANGLE drawn inside
a CIRCLE. Find the PROBABILITY that a CHORD
chosen at random is longer than the side-length
of the triangle.
There are two possible answers:
1. Once a chord is drawn we can always rotate the picture of the circle so that one end of the selected
chord is placed at the top of the circle. It is clear
then that the length of the chord will be greater than
the side-length of the triangle if the other end-point
lies in the middle third of the perimeter of the circle.
The chances of this happening are 1/3, providing the
answer to the problem.
2. Rotating the picture of the circle and the selected
chord, we can also assume that the chord chosen is
horizontal. If the chord crosses the solid line
shown, then it will be longer than the side-length
of the triangle. One observes that this solid line is
half the length of the diameter. Thus the chances of
a chord being longer than the side-length of the triangle are 1/2.
Surprisingly, both lines of reasoning are mathematically
correct. Therein lies a PARADOX: the answer cannot
simultaneously be 1/3 and 1/2.
The problem here lies in defining what we mean
by “select a chord at random.” There are many different ways to do this: one could spin a bottle in the
center of the circle to select points on the perimeter
to connect with a chord, or one could roll a broom
across a circle drawn on the floor, or perhaps even
drop a wire from a height above the circle and see
where it lands. Each approach to “randomness”
could (and in fact does) lead to its own separate
answer. This paradox shows that extreme care must
be taken to pose meaningful problems in probability
theory. It is very difficult to give a precise definition
to “randomness.”
Bh – askara II (Bhaskaracharya) (1114–1185) Indian
Algebra, Arithmetic Born in Vijayapura, India, Bh – askara (often referred to as Bh – askara II to distinguish him
from the seventh-century mathematician of the same
name) is considered India’s most eminent mathematician of the 12th century. He revised and continued the
studies of the great BRAHMAGUPTA, making corrections
and filling in gaps in his work, and reached a level of
mastery of ARITHMETIC and ALGEBRA that was not
matched by a European scholar for several centuries to
come. Bh – askara wrote two influential mathematical
treatises: Lilavati (The beautiful), on the topic of arithmetic, and the Bijaganita (Seed arithmetic) on algebra.
Bh – askara was head of the astronomical observatory in Ujjain, the nation’s most prominent mathematical research center of the time. Although much of
Indian mathematics was motivated by problems and
challenges in astronomy, Bh – askara’s writings show a
keen interest in developing mathematics for its own
sake. For example, the text Lilavati, consisting of 13
chapters, begins with careful discussions on arithmetic
and geometry before moving on to the topics of
SEQUENCEs and SERIES, fractions, INTEREST, plane and
solid geometry, sundials, PERMUTATIONs and COMBINATIONs, and DIOPHANTINE EQUATIONs (as they are called
today). For instance, Bh – askara shows that the equation
195x = 221y + 65 (which he expressed solely in words)
has infinitely many positive integer solutions, beginning
with x = 6, y = 5, and x = 23, y = 20, and then x = 40,
and y = 35. (The x-values increase in steps of 17, and
the y-values in steps of 15.)
In his piece Bijaganita, Bh – askara develops the
arithmetic of NEGATIVE NUMBERS, solves quadratic
equations of one, or possibly more, unknowns, and
develops methods of extracting SQUARE and CUBE
ROOTs of quantities. He continues the discussions of
Brahmagupta on the nature and properties of the number ZERO and the use of negative numbers in arithmetic. (He denoted the negative of a number by placing
a dot above the numeral.) Bh – askara correctly points
out that a quantity divided by zero does not produce
zero (as Brahmagupta claimed) and suggested instead
that a/0 should be deemed infinite in value. Bh – askara
solves complicated equations with several unknowns
and develops formulae that led him to the brink of discovering the famous QUADRATIC formula.
Bh – askara also wrote a number of important texts
in mathematical astronomy and made significant
strides in the development of TRIGONOMETRY, taking
the subject beyond the level of just a tool of calculation
for astronomers. Bh – askara discovered, for example, the
famous addition formulae for sine:
sin(A + B) = sin A cos B + cos A sin B
sin(A – B) = sin A cos B – cos A sin B
Bh – askara II 41
a CIRCLE. Find the PROBABILITY that a CHORD
chosen at random is longer than the side-length
of the triangle.
There are two possible answers:
1. Once a chord is drawn we can always rotate the picture of the circle so that one end of the selected
chord is placed at the top of the circle. It is clear
then that the length of the chord will be greater than
the side-length of the triangle if the other end-point
lies in the middle third of the perimeter of the circle.
The chances of this happening are 1/3, providing the
answer to the problem.
2. Rotating the picture of the circle and the selected
chord, we can also assume that the chord chosen is
horizontal. If the chord crosses the solid line
shown, then it will be longer than the side-length
of the triangle. One observes that this solid line is
half the length of the diameter. Thus the chances of
a chord being longer than the side-length of the triangle are 1/2.
Surprisingly, both lines of reasoning are mathematically
correct. Therein lies a PARADOX: the answer cannot
simultaneously be 1/3 and 1/2.
The problem here lies in defining what we mean
by “select a chord at random.” There are many different ways to do this: one could spin a bottle in the
center of the circle to select points on the perimeter
to connect with a chord, or one could roll a broom
across a circle drawn on the floor, or perhaps even
drop a wire from a height above the circle and see
where it lands. Each approach to “randomness”
could (and in fact does) lead to its own separate
answer. This paradox shows that extreme care must
be taken to pose meaningful problems in probability
theory. It is very difficult to give a precise definition
to “randomness.”
Bh – askara II (Bhaskaracharya) (1114–1185) Indian
Algebra, Arithmetic Born in Vijayapura, India, Bh – askara (often referred to as Bh – askara II to distinguish him
from the seventh-century mathematician of the same
name) is considered India’s most eminent mathematician of the 12th century. He revised and continued the
studies of the great BRAHMAGUPTA, making corrections
and filling in gaps in his work, and reached a level of
mastery of ARITHMETIC and ALGEBRA that was not
matched by a European scholar for several centuries to
come. Bh – askara wrote two influential mathematical
treatises: Lilavati (The beautiful), on the topic of arithmetic, and the Bijaganita (Seed arithmetic) on algebra.
Bh – askara was head of the astronomical observatory in Ujjain, the nation’s most prominent mathematical research center of the time. Although much of
Indian mathematics was motivated by problems and
challenges in astronomy, Bh – askara’s writings show a
keen interest in developing mathematics for its own
sake. For example, the text Lilavati, consisting of 13
chapters, begins with careful discussions on arithmetic
and geometry before moving on to the topics of
SEQUENCEs and SERIES, fractions, INTEREST, plane and
solid geometry, sundials, PERMUTATIONs and COMBINATIONs, and DIOPHANTINE EQUATIONs (as they are called
today). For instance, Bh – askara shows that the equation
195x = 221y + 65 (which he expressed solely in words)
has infinitely many positive integer solutions, beginning
with x = 6, y = 5, and x = 23, y = 20, and then x = 40,
and y = 35. (The x-values increase in steps of 17, and
the y-values in steps of 15.)
In his piece Bijaganita, Bh – askara develops the
arithmetic of NEGATIVE NUMBERS, solves quadratic
equations of one, or possibly more, unknowns, and
develops methods of extracting SQUARE and CUBE
ROOTs of quantities. He continues the discussions of
Brahmagupta on the nature and properties of the number ZERO and the use of negative numbers in arithmetic. (He denoted the negative of a number by placing
a dot above the numeral.) Bh – askara correctly points
out that a quantity divided by zero does not produce
zero (as Brahmagupta claimed) and suggested instead
that a/0 should be deemed infinite in value. Bh – askara
solves complicated equations with several unknowns
and develops formulae that led him to the brink of discovering the famous QUADRATIC formula.
Bh – askara also wrote a number of important texts
in mathematical astronomy and made significant
strides in the development of TRIGONOMETRY, taking
the subject beyond the level of just a tool of calculation
for astronomers. Bh – askara discovered, for example, the
famous addition formulae for sine:
sin(A + B) = sin A cos B + cos A sin B
sin(A – B) = sin A cos B – cos A sin B
Bh – askara II 41