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11: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

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12: Bibliography I

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IEEE Standard for Floating-Point Arithmetic. The Institute of Electrical and Electronics Engineers, Inc..

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Notes:

IEEE Std 754-2008, Revision of IEEE Std 754-1985

External Links:

ISBN 978-0-7381-5753-5, Document

Referenced by:

§3.1(i).

Encodings:

BibTeX

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IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:

ISBN 978-0-7381-9720-3, Document

Referenced by:

§3.1(ii), §3.1(ii), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:

ISBN 978-1-5044-4602-0, Document

Referenced by:

§3.1(ii), §3.1(ii), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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IEEE International Standard for Information Technology—Microprocessor Systems—Floating-Point arithmetic: IEEE Std 754-2019. The Institute of Electrical and Electronics Engineers, Inc..

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Notes:

IEEE Std ISO/IEC/IEEE 60559, Revision of IEEE Std 754-1985

External Links:

ISBN 978-2-88912-566-1, Document

Referenced by:

Figure 3.1.1, Figure 3.1.1, §3.1(i), §3.1(i), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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13: Daniel W. Lozier

… ►His research interests have centered on numerical analysis, special functions, computer arithmetic, and mathematical software construction and testing. …

14: 27.15 Chinese Remainder Theorem

… ►Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers $a_1(mod{m}_1)$, $a_2(mod{m}_2)$, $a_3(mod{m}_3)$, and $a_4(mod{m}_4)$, where each $a_j$ has no more than five digits. …

15: Bibliography Y

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Software for interval arithmetic: A reasonably portable package. ACM Trans. Math. Software 5 (1), pp. 50–63.

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External Links:

ISSN 0098-3500, Document

Referenced by:

§4.48(ii).

Encodings:

BibTeX

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16: 1.7 Inequalities

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§1.7(iii) Means

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1.7.7 $$H\le G\le A,$$

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Symbols:

$A$: arithmetic mean, $G$: geometric mean and $H$: harmonic mean

A&S Ref:

3.2.1

Permalink:

http://dlmf.nist.gov/1.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §1.7(iii), §1.7 and Ch.1

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17: 23.22 Methods of Computation

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In the general case, given by $cd\ne 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^3-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_1$, $e_2$, $e_3$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^3-27d^2>0$, then $e_1$, $e_2$, $e_3$ can be identified via (23.5.1), and $k^2$, $k_{}^{\prime }{}_{}{}^2$ obtained from (23.6.16).

If , or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha ,\gamma ]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha ,\gamma )$. In consequence, $k^2=(\beta -\gamma )/(\alpha -\gamma )$, $k_{}^{\prime }{}_{}{}^2=(\alpha -\beta )/(\alpha -\gamma )$ satisfy $\mathrm{\Im }{k}^2\ge 0\ge \mathrm{\Im }k_{}^{\prime }{}_{}{}^2$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^2|$, $|k_{}^{\prime }{}_{}{}^2|\le 1$.

Finally, on taking the principal square roots of $k^2$ and $k_{}^{\prime }{}_{}{}^2$ we obtain values for $k$ and $k^{\prime }$ that lie in the 1st and 4th quadrants, respectively, and $2{\omega }_1$, $2{\omega }_3$ are given by

23.22.1 $$2{\omega }_{1}M(1,k^{\prime })=-2\mathrm{i}{\omega }_{3}M(1,k)=\frac{\pi }{3}\sqrt{\frac{c(2+k^{2}k_{}^{\prime }{}_{}{}^2)(k_{}^{\prime }{}_{}{}^2-k^2)}{d(1-k^{2}k_{}^{\prime }{}_{}{}^2)}},$$

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

Proof sketch:

Combine (23.6.16), (23.6.17), (22.20.6), (23.3.5)–(23.3.7), and use analytical continuation.

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 2 possible choices of the square root.

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18: 27.11 Asymptotic Formulas: Partial Sums

… ►where $\left(h,k\right)=1$, $k>0$. ►Letting $x\to \mathrm{\infty }$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h(modk)$ if $h,k$ are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. … ►The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$, then the number of primes $p\le x$ with $p\equiv h(modk)$ is asymptotic to $x/(\varphi \left(k\right)\ln x)$ as $x\to \mathrm{\infty }$.

19: 1.2 Elementary Algebra

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Arithmetic Progression

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§1.2(iv) Means

►The arithmetic mean of $n$ numbers $a_1,a_2,\mathrm{\dots },a_n$ is ►

1.2.17 $$A=\frac{a_1+a_2+\mathrm{\cdots }+a_n}{n}.$$

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Defines:

$A$: arithmetic mean (locally)

Symbols:

$n$: nonnegative integer

A&S Ref:

3.1.11

Permalink:

http://dlmf.nist.gov/1.2.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(iv), §1.2 and Ch.1

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$M(1)=A,$

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20: 27.2 Functions

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§27.2(i) Definitions

►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer $n>1$ can be represented uniquely as a product of prime powers, …