3xa
2 + a
3 are used in elementary ALGEBRA. These are
both special cases of the general binomial theorem that
asserts, for any positive integer n, we have:
Here each number
is a COMBINATORIAL
COEFFICIENT, also called a binomial coefficient.
The binomial theorem is proved by examining the
process of EXPANDING BRACKETS, thinking of the quantity (x + a)
n as a product of n factors: (x + a)(x + a)…
(x + a). To expand the brackets, one must select an entry
from each set of parentheses (“x” or “a”), multiply
together all the selected elements, and add together all
possible results. For example, there is one way to obtain
the term x
n : select x from every set of parentheses.
There are n ways to create a term of the form x
n–1 a:
select a from just one set of parentheses, and x from the
remaining sets. In general there are
ways to select k
as and n – k xs. Thus, in the expansion, there will be
terms of the form x
n–k a
k .
The combinatorial coefficients are the entries of
PASCAL’S TRIANGLE. The binomial theorem applied to
(1 + 1) n explains why the elements of each row of Pascal’s triangle sum to a power of two:
Applying the theorem to (1 – 1) n explains why the
alternating sum of the entries is zero:
Applying the theorem to (10 + 1)
n explains why the
first few rows of Pascal’s triangle resemble the powers
of 11:
11 2 = (10 + 1)
2 = 100 + 2 × 10 + 1
11
3 = (10 + 1)
3 = 1,000 + 3 × 100 + 3 × 10 + 1
11
4 = (10 + 1)
4 = 10,000 + 4 × 1,000 + 6 × 100 + 4
× 10 + 1
(The correspondence would remain valid if we did not
carry digits when computing higher powers of 11.)
The binomial theorem can be used to approximate
high powers of decimals. For example, to estimate
(2.01)
10 we observe:
2.01
10 = (2 + 0.01)
10
= 2
10 + 10 × 2
9
× 0.01 + 2
8 + 45 × 2
8
× 0.01
2 + …
≈ 1024 + 10 × 512 × 0.01 + 45 × 256 × 0.00001
= 1024 + 51.2 + 1.152
≈ 1076
In 1665 SIR ISAAC NEWTON, coinventor of CALCULUS, discovered that it is possible to expand quantities
of the form (x + a)
r where r is not equal to a whole
number. This leads to the generalized binomial theorem:
If r is an arbitrary real number, and |x| < |a|,
then:
The formula is established by computing the TAYLOR
SERIES of f(x) = (x + a)
r at x = 0. In 1826 Norwegian
mathematician NIELS ABEL proved that the series converges for the range indicated. Notice that if r is a positive integer, then the theorem reduces to the ordinary
binomial theorem. (In particular, from the n + 1’th
place onward, all terms in the infinite sum are zero.)
The combinatorial coefficients
arising in the
binomial theorem are sometimes called binomial coefficients. The generalized combinatorial coefficients appear
in expansions of quantities of the type (x + y + z)
n and
(x + y + z + w)
n , for example.
See also COMBINATION.
bisection method (dichotomous line search, binary line
search) Often one is required to find a solution to an
n
k
⎛
⎝
⎜
⎞
⎠
⎟
(
)(
)
!
r r
r
x a
r
+
−
−
+
−3 3
1
2
3
L
(
)
(
)
!
x a x rx a
r r
x a
r
r
r
r
+ = +
+
−
−
−
1
2 2
1
2
0 1 1
0
1
2
= −
=
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟ − ±
⎛
⎝
⎜
⎞
⎠
⎟
(
)
n
n
n
n
n
n
L
2
1 1
0
1
n
n
n
n
n
n
= +
=
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟ + +
⎛
⎝
⎜
⎞
⎠
⎟
(
)
L
n
k
⎛
⎝
⎜
⎞
⎠
⎟
n
k
⎛
⎝
⎜
⎞
⎠
⎟
n
k
n
k n k
⎛
⎝
⎜
⎞
⎠
⎟ =
−
!
!(
)!
n
n
xa
a
n
n
+ + −
⎛
⎝
⎜
⎞
⎠
⎟
+
−1
1
L
(
)
x a
n
k
x a
x
n x a
n x a
n
nkk
k
n
n
n
n
+
=
⎛
⎝
⎜
⎞
⎠
⎟
=
+
⎛
⎝
⎜
⎞
⎠
⎟
+
⎛
⎝
⎜
⎞
⎠
⎟
−
=
−
−
∑
0
1
22
1
2
bisection method 45
2 + a
3 are used in elementary ALGEBRA. These are
both special cases of the general binomial theorem that
asserts, for any positive integer n, we have:
Here each number
is a COMBINATORIAL
COEFFICIENT, also called a binomial coefficient.
The binomial theorem is proved by examining the
process of EXPANDING BRACKETS, thinking of the quantity (x + a)
n as a product of n factors: (x + a)(x + a)…
(x + a). To expand the brackets, one must select an entry
from each set of parentheses (“x” or “a”), multiply
together all the selected elements, and add together all
possible results. For example, there is one way to obtain
the term x
n : select x from every set of parentheses.
There are n ways to create a term of the form x
n–1 a:
select a from just one set of parentheses, and x from the
remaining sets. In general there are
ways to select k
as and n – k xs. Thus, in the expansion, there will be
terms of the form x
n–k a
k .
The combinatorial coefficients are the entries of
PASCAL’S TRIANGLE. The binomial theorem applied to
(1 + 1) n explains why the elements of each row of Pascal’s triangle sum to a power of two:
Applying the theorem to (1 – 1) n explains why the
alternating sum of the entries is zero:
Applying the theorem to (10 + 1)
n explains why the
first few rows of Pascal’s triangle resemble the powers
of 11:
11 2 = (10 + 1)
2 = 100 + 2 × 10 + 1
11
3 = (10 + 1)
3 = 1,000 + 3 × 100 + 3 × 10 + 1
11
4 = (10 + 1)
4 = 10,000 + 4 × 1,000 + 6 × 100 + 4
× 10 + 1
(The correspondence would remain valid if we did not
carry digits when computing higher powers of 11.)
The binomial theorem can be used to approximate
high powers of decimals. For example, to estimate
(2.01)
10 we observe:
2.01
10 = (2 + 0.01)
10
= 2
10 + 10 × 2
9
× 0.01 + 2
8 + 45 × 2
8
× 0.01
2 + …
≈ 1024 + 10 × 512 × 0.01 + 45 × 256 × 0.00001
= 1024 + 51.2 + 1.152
≈ 1076
In 1665 SIR ISAAC NEWTON, coinventor of CALCULUS, discovered that it is possible to expand quantities
of the form (x + a)
r where r is not equal to a whole
number. This leads to the generalized binomial theorem:
If r is an arbitrary real number, and |x| < |a|,
then:
The formula is established by computing the TAYLOR
SERIES of f(x) = (x + a)
r at x = 0. In 1826 Norwegian
mathematician NIELS ABEL proved that the series converges for the range indicated. Notice that if r is a positive integer, then the theorem reduces to the ordinary
binomial theorem. (In particular, from the n + 1’th
place onward, all terms in the infinite sum are zero.)
The combinatorial coefficients
arising in the
binomial theorem are sometimes called binomial coefficients. The generalized combinatorial coefficients appear
in expansions of quantities of the type (x + y + z)
n and
(x + y + z + w)
n , for example.
See also COMBINATION.
bisection method (dichotomous line search, binary line
search) Often one is required to find a solution to an
n
k
⎛
⎝
⎜
⎞
⎠
⎟
(
)(
)
!
r r
r
x a
r
+
−
−
+
−3 3
1
2
3
L
(
)
(
)
!
x a x rx a
r r
x a
r
r
r
r
+ = +
+
−
−
−
1
2 2
1
2
0 1 1
0
1
2
= −
=
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟ − ±
⎛
⎝
⎜
⎞
⎠
⎟
(
)
n
n
n
n
n
n
L
2
1 1
0
1
n
n
n
n
n
n
= +
=
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟ + +
⎛
⎝
⎜
⎞
⎠
⎟
(
)
L
n
k
⎛
⎝
⎜
⎞
⎠
⎟
n
k
⎛
⎝
⎜
⎞
⎠
⎟
n
k
n
k n k
⎛
⎝
⎜
⎞
⎠
⎟ =
−
!
!(
)!
n
n
xa
a
n
n
+ + −
⎛
⎝
⎜
⎞
⎠
⎟
+
−1
1
L
(
)
x a
n
k
x a
x
n x a
n x a
n
nkk
k
n
n
n
n
+
=
⎛
⎝
⎜
⎞
⎠
⎟
=
+
⎛
⎝
⎜
⎞
⎠
⎟
+
⎛
⎝
⎜
⎞
⎠
⎟
−
=
−
−
∑
0
1
22
1
2
bisection method 45