However, as was the tradition at the time, Bh–
askara did
not explain how he derived his results. It is conjectured
that Indian astronomers and mathematicians felt it nec-
essary to conceal their methods regarding proofs and
derivations as “trade secrets” of the art.
Bh–
askara’s accomplishments were revered for
many centuries. In 1817, H. J. Colebrook provided
English translations of both Lilavati and Bijaganita in
his text Algebra with Arithmetic and Mensuration.
bias A systematic error in a statistical study is called a
bias. If the sample in the study is large, errors produced
by chance tend to cancel each other out, but those from a
bias do not. For example, a survey on the shopping
habits of the general population conducted at a shopping
mall is likely to be biased toward people who shop pri-
marily at malls, omitting results from people who shop
from home through catalogs and on-line services. This is
similar to a loaded die, which is biased to produce a par-
ticular outcome with greater than one-sixth probability.
Surprisingly, American pennies are biased. If you
delicately balance 30 pennies on edge and bump the sur-
face on which they stand, most will fall over heads up.
If, on the other hand, you spin 30 pennies and let them
all naturally come to rest, then most will land tails.
See also
POPULATION AND SAMPLE
.
biconditional In
FORMAL LOGIC
, a statement of the
form “pif, and only if, q” is called a biconditional
statement. For example, “A triangle is equilateral if,
and only if, it is equiangular” is a biconditional state-
ment. A biconditional statement is often abbreviated as
piff qand is written in symbols as p↔q. It is equiva-
lent to the compound statement “pimplies q, and q
implies p” composed of two
CONDITIONAL
statements.
The truth-values of pand qmust match for the bicon-
ditional statement as a whole to be true. It therefore
has the following
TRUTH TABLE
:
The two statements pand qare said to be logically
equivalent if the biconditional statement p↔qis true.
See also
ARGUMENT
.
bijection See
FUNCTION
.
bimodal See
STATISTICS
:
DESCRIPTIVE
.
binary numbers (base-2 numbers) Any whole num-
ber can be written as a sum of distinct numbers from
the list of powers of 2: 1, 2, 4, 8, 16, 32, 64, … (Simply
subtract the largest power of 2 less from the given
number and repeat the process for the remainder
obtained.) For instance, we have:
89 = 64 + 25 = 64 + 16 + 9 = 64 + 16 + 8 + 1
No power of 2 will appear twice, as two copies of the
same power of 2 sum to the next power in the list.
Moreover, the sum of powers of 2 produced for a given
number is unique. Using the symbol 1 to denote that a
particular power of 2 is used and 0 to denote that it is
not, one can then encode any given number as a
sequence of 0s and 1s according to the powers of 2 that
appear in its presentation. For instance, for the number
89, the number 64 is used, but 32 is not. The number
16 appears, as does 8, but not 4 or 2. Finally, the num-
ber 1 is also used. We write:
89 = 10110012
(It is customary to work with the large power of 2 to the
left.) As other examples, we see that the code 100010112
corresponds to the number 128 + 64 + 32 + 16 + 8 +
–
4 +
2 + 1 = 139, and the code 101112to the number 16 + 8
+ 4 + 2 + 1 = 23. Numbers represented according to this
method are called binary numbers. These representa-
tions correspond precisely to the representations made
by choosing 2 as the
BASE OF A NUMBER SYSTEM
.
If one introduces a decimal point into the system
and interprets positions to the right of the point as neg-
ative powers of 2, then fractional quantities can also be
represented in binary notation. For instance, 0.1012
represents the quantity 2–1 + 2–3 = + = , and
0.010101…2the quantity + + + …, which,
1
64
1
16
1
4
5
8
1
8
1
2
pqp
↔
q
TTT
TFF
FTF
FFT
42 bias