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§1.2 Elementary Algebra

Contents

§1.2(i) Binomial Coefficients

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Notes:
See Chrystal (1959, pp. 62–70).
Keywords:
binomials
Referenced by:
§26.3(iv), Subsection 1.2(i)
Permalink:
http://dlmf.nist.gov/1.2.i
Errata (effective with 1.0.5):
The condition for (1.2.2), (1.2.4), and (1.2.5) was corrected. These equations are true only if n is a positive integer. Previously n was allowed to be zero.

Reported 2011-08-10 by Michael Somos

In (1.2.1) and (1.2.3) k and n are nonnegative integers and k\leq n. In (1.2.2), (1.2.4), and (1.2.5) n is a positive integer.

1.2.1\binom{n}{k}=\frac{n!}{(n-k)!k!}=\binom{n}{n-k}.
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Defines:
\binom{m}{n}: binomial coefficient
Symbols:
!: n!: factorial, k: integer, m: nonnegative integer and n: nonnegative integer
A&S Ref:
3.1.2
Referenced by:
§1.2(i)
Permalink:
http://dlmf.nist.gov/1.2.E1
Encodings:
TeX, pMML, png

Binomial Theorem

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Keywords:
binomial theorem
1.2.2(a+b)^{n}=a^{n}+\binom{n}{1}a^{{n-1}}b+\binom{n}{2}a^{{n-2}}b^{2}+\dots+\binom% {n}{n-1}ab^{{n-1}}+b^{n}.
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Symbols:
\binom{m}{n}: binomial coefficient and n: nonnegative integer
A&S Ref:
3.1.1
Referenced by:
§1.2(i), §1.2(i), Subsection 1.2(i)
Permalink:
http://dlmf.nist.gov/1.2.E2
Encodings:
TeX, pMML, png
1.2.3\binom{n}{0}+\binom{n}{1}+\dots+\binom{n}{n}=2^{n}.
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Symbols:
\binom{m}{n}: binomial coefficient and n: nonnegative integer
A&S Ref:
3.1.6
Referenced by:
§1.2(i)
Permalink:
http://dlmf.nist.gov/1.2.E3
Encodings:
TeX, pMML, png
1.2.4\binom{n}{0}-\binom{n}{1}+\dots+(-1)^{n}\binom{n}{n}=0.
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Symbols:
\binom{m}{n}: binomial coefficient and n: nonnegative integer
A&S Ref:
3.1.7
Referenced by:
§1.2(i), §1.2(i), Subsection 1.2(i)
Permalink:
http://dlmf.nist.gov/1.2.E4
Encodings:
TeX, pMML, png

where \ell is n or n-1 according as n is even or odd.

In (1.2.6)–(1.2.9) k and m are nonnegative integers and n is unrestricted.

1.2.7\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}.
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Symbols:
\binom{m}{n}: binomial coefficient, k: integer and n: nonnegative integer
A&S Ref:
3.1.4
Permalink:
http://dlmf.nist.gov/1.2.E7
Encodings:
TeX, pMML, png

§1.2(ii) Finite Series

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Notes:
See Chrystal (1959, pp. 482–483, 489).
Permalink:
http://dlmf.nist.gov/1.2.ii

Arithmetic Progression

1.2.10a+(a+d)+(a+2d)+\dots+(a+(n-1)d)=na+\tfrac{1}{2}n(n-1)d=\tfrac{1}{2}n(a+\ell),
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Symbols:
n: nonnegative integer
A&S Ref:
3.1.9
Permalink:
http://dlmf.nist.gov/1.2.E10
Encodings:
TeX, pMML, png

where \ell = last term of the series = a+(n-1)d.

Geometric Progression

1.2.11a+ax+ax^{2}+\dots+ax^{{n-1}}=\frac{a(1-x^{n})}{1-x},x\not=1.
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Symbols:
n: nonnegative integer
A&S Ref:
3.1.10
Permalink:
http://dlmf.nist.gov/1.2.E11
Encodings:
TeX, pMML, png

§1.2(iii) Partial Fractions

Let \alpha_{1},\alpha_{2},\dots,\alpha_{n} be distinct constants, and f(x) be a polynomial of degree less than n. Then

1.2.12\frac{f(x)}{(x-\alpha_{1})(x-\alpha_{2})\cdots(x-\alpha_{n})}=\frac{A_{1}}{x-% \alpha_{1}}+\frac{A_{2}}{x-\alpha_{2}}+\dots+\frac{A_{n}}{x-\alpha_{n}},

where

1.2.13A_{j}=\frac{f(\alpha_{j})}{\prod\limits_{{k\not=j}}(\alpha_{j}-\alpha_{k})}.
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Symbols:
j: integer, k: integer and A_{j}: coefficient
Permalink:
http://dlmf.nist.gov/1.2.E13
Encodings:
TeX, pMML, png

Also,

1.2.14\frac{f(x)}{(x-\alpha_{1})^{n}}=\frac{B_{1}}{x-\alpha_{1}}+\frac{B_{2}}{(x-% \alpha_{1})^{2}}+\dots+\frac{B_{n}}{(x-\alpha_{1})^{n}},

where

1.2.15B_{j}=\frac{f^{{(n-j)}}(\alpha_{1})}{(n-j)!},
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Symbols:
!: n!: factorial, j: integer, n: nonnegative integer and B_{j}: coefficient
Permalink:
http://dlmf.nist.gov/1.2.E15
Encodings:
TeX, pMML, png

and f^{{(k)}} is the k-th derivative of f (§1.4(iii)).

If m_{1},m_{2},\dots,m_{n} are positive integers and \deg f<\sum_{{j=1}}^{n}m_{j}, then there exist polynomials f_{j}(x), \deg f_{j}<m_{j}, such that

To find the polynomials f_{j}(x), j=1,2,\dots,n, multiply both sides by the denominator of the left-hand side and equate coefficients. See Chrystal (1959, pp. 151–159).

§1.2(iv) Means

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Notes:
See Hardy et al. (1967, pp. 12–15).
Keywords:
arithmetic mean, geometric mean, harmonic mean, means, weighted means
Referenced by:
§1.7(iii)
Permalink:
http://dlmf.nist.gov/1.2.iv

The arithmetic mean of n numbers a_{1},a_{2},\dots,a_{n} is

1.2.17A=\frac{a_{1}+a_{2}+\dots+a_{n}}{n}.
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Defines:
A: arithmetic mean
Symbols:
n: nonnegative integer
A&S Ref:
3.1.11
Permalink:
http://dlmf.nist.gov/1.2.E17
Encodings:
TeX, pMML, png

The geometric mean G and harmonic mean H of n positive numbers a_{1},a_{2},\dots,a_{n} are given by

1.2.18G=(a_{1}a_{2}\cdots a_{n})^{{1/n}},
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Defines:
G: geometric mean
Symbols:
n: nonnegative integer
A&S Ref:
3.1.12
Permalink:
http://dlmf.nist.gov/1.2.E18
Encodings:
TeX, pMML, png
1.2.19\frac{1}{H}=\frac{1}{n}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\dots+\frac{1}{a_% {n}}\right).
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Defines:
H: harmonic mean
Symbols:
n: nonnegative integer
A&S Ref:
3.1.13
Permalink:
http://dlmf.nist.gov/1.2.E19
Encodings:
TeX, pMML, png

If r is a nonzero real number, then the weighted mean M(r) of n nonnegative numbers a_{1},a_{2},\dots,a_{n}, and n positive numbers p_{1},p_{2},\dots,p_{n} with

is defined by

1.2.21M(r)=(p_{1}a_{1}^{r}+p_{2}a_{2}^{r}+\dots+p_{n}a_{n}^{r})^{{1/r}},
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Defines:
M: weighted mean
Symbols:
n: nonnegative integer and p_{j}; positive numbers
Permalink:
http://dlmf.nist.gov/1.2.E21
Encodings:
TeX, pMML, png

with the exception

1.2.22M(r)=0,r<0 and a_{1}a_{2}\dots a_{n}=0.
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Symbols:
M: weighted mean and n: nonnegative integer
A&S Ref:
3.1.15
Permalink:
http://dlmf.nist.gov/1.2.E22
Encodings:
TeX, pMML, png
1.2.23\lim_{{r\to\infty}}M(r)=\max(a_{1},a_{2},\dots,a_{n}),
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Symbols:
M: weighted mean and n: nonnegative integer
A&S Ref:
3.1.16
Permalink:
http://dlmf.nist.gov/1.2.E23
Encodings:
TeX, pMML, png
1.2.24\lim_{{r\to-\infty}}M(r)=\min(a_{1},a_{2},\dots,a_{n}).
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Symbols:
M: weighted mean and n: nonnegative integer
A&S Ref:
3.1.17
Permalink:
http://dlmf.nist.gov/1.2.E24
Encodings:
TeX, pMML, png

For p_{j}=1/n, j=1,2,\dots,n,

1.2.25
M(1)=A,
M(-1)=H,
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Symbols:
A: arithmetic mean, H: harmonic mean and M: weighted mean
A&S Ref:
3.1.19 3.1.20
Permalink:
http://dlmf.nist.gov/1.2.E25
Encodings:
TeX, TeX, pMML, pMML, png, png

and

1.2.26\lim_{{r\to 0}}M(r)=G.
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Symbols:
G: geometric mean and M: weighted mean
A&S Ref:
3.1.18
Permalink:
http://dlmf.nist.gov/1.2.E26
Encodings:
TeX, pMML, png

The last two equations require a_{j}>0 for all j.

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