Integer

This article is about numbers traditionally known as "integers". For computer representations, see integer (computer science). For the concept in algebraic number theory, see integral element.

Algebraic structure → Group theory
Group theory

Basic notions

Group homomorphisms

List of group theory topics

Glossary of group theory

Finite groups

Classification of finite simple groups

Lagrange's theorem

Frobenius group

Schur multiplier

Symmetric group S_n

Integers (Z)
Lattice

Modular groups

PSL(2,Z)
SL(2,Z)

Topological / Lie groups

General linear GL(n)

Special linear SL(n)

Orthogonal O(n)

Euclidean E(n)

Special orthogonal SO(n)

Unitary U(n)

Special unitary SU(n)

Symplectic Sp(n)

Infinite dimensional Lie group

O(∞)
SU(∞)
Sp(∞)

Algebraic groups

Elliptic curve

Linear algebraic group

Abelian variety

An integer (from the Latin integer meaning "whole")^{[note 1]} is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 ¹⁄₂, and √2 are not.

The set of integers consists of zero (0), the natural numbers (1, 2, 3, …), also called whole numbers or counting numbers,^[1] and their additive inverses (the negative integers, i.e. −1, −2, −3, …). This is often denoted by a boldface Z (" $Z$ ") or blackboard bold $\mathbb{Z}$ (Unicode U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "numbers").^[2]^[3] ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite.

The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the (rational) integers are the algebraic integers that are also rational numbers.

Algebraic properties

Integers can be thought of as discrete, equally spaced points on an infinitely long number line. In the above, non-negative integers are shown in purple and negative integers in red.

Like the natural numbers, $Z$ is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, 0, $Z$ (unlike the natural numbers) is also closed under subtraction. The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of rings, characterizes the ring $Z$ .

$Z$ is not closed under division, since the quotient of two integers (e.g. 1 divided by 2), need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative).

The following lists some of the basic properties of addition and multiplication for any integers $a$ , $b$ and $c$ .

Properties of addition and multiplication on integers
	Addition	Multiplication
Closure:	$a + b$ is an integer	$a \times b$ is an integer
Associativity:	$a + (b + c) = (a + b) + c$	$a \times (b \times c) = (a \times b) \times c$
Commutativity:	$a + b = b + a$	$a \times b = b \times a$
Existence of an identity element:	$a + 0 = a$	$a \times 1 = a$
Existence of inverse elements:	$a + (- a) = 0$	An inverse element usually does not exist at all.
Distributivity:	$a \times (b + c) = (a \times b) + (a \times c)$ and $(a + b) \times c = (a \times c) + (b \times c)$
No zero divisors: (*)		If $a \times b = 0$ , then $a = 0$ or $b = 0$ (or both)

In the language of abstract algebra, the first five properties listed above for addition say that $Z$ under addition is an abelian group. As a group under addition, $Z$ is a cyclic group, since every non-zero integer can be written as a finite sum 1 + 1 + … + 1 or (−1) + (−1) + … + (−1). In fact, $Z$ under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to $Z$ .

The first four properties listed above for multiplication say that $Z$ under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse; e.g. there is no integer $x$ such that $2 x = 1,$ because the left hand side is even, while the right hand side is odd. This means that $Z$ under multiplication is not a group.

All the rules from the above property table, except for the last, taken together say that $Z$ together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in $Z$ for all values of variables, which are true in any unital commutative ring. Note that certain non-zero integers map to zero in certain rings.

At last, the property (*) says that the commutative ring $Z$ is an integral domain. In fact, $Z$ provides the motivation for defining such a structure.

The lack of multiplicative inverses, which is equivalent to the fact that $Z$ is not closed under division, means that $Z$ is not a field. The smallest field with the usual operations containing the integers is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes $Z$ as its subring.

Moreover, $Z$ is a principal ring.^[4]

Although ordinary division is not defined on $Z$ , the division "with remainder" is defined on them. It is called Euclidean division and possesses the following important property: that is, given two integers $a$ and $b$ with $b \neq 0$ , there exist unique integers $q$ and $r$ such that $a = q \times b + r$ and $0 \leq r < | b |$ , where $| b |$ denotes the absolute value of $b$ . The integer $q$ is called the quotient and $r$ is called the remainder of the division of $a$ by $b$ . The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions.

Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. This implies that $Z$ is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.

Order-theoretic properties

$Z$ is a totally ordered set without upper or lower bound. The ordering of $Z$ is given by: $:\dots -3 < -2 < -1 < 0 < 1 < 2 < 3 < \dots$ An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive.

The ordering of integers is compatible with the algebraic operations in the following way:

if $a < b$ and $c < d$ , then $a + c < b + d$
if $a < b$ and $0 < c$ , then $ac < bc$ .

It follows that $Z$ together with the above ordering is an ordered ring.

The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered.^[5] This is equivalent to the statement that any Noetherian valuation ring is either a field or a discrete valuation ring.

Construction

Representation of equivalence classes for the numbers −5 to 5

Red points represent ordered pairs of natural numbers. Linked red points are equivalence classes representing the blue integers at the end of the line.

In elementary school teaching, integers are often intuitively defined as the (positive) natural numbers, zero, and the negations of the natural numbers. However, this style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that these operations obey the laws of arithmetic.^[6] Therefore, in modern set-theoretic mathematics a more abstract construction,^[7] which allows one to define the arithmetical operations without any case distinction, is often used instead.^[8] The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers $(a, b)$ .^[9]

The intuition is that $(a, b)$ stands for the result of subtracting $b$ from $a$ .^[9] To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation $~$ on these pairs with the following rule:

(a,b) \sim (c,d) \,\!

precisely when

a + d = b + c. \,\!

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;^[9] denoting by $[(a, b)]$ the equivalence class having $(a, b)$ as a member, one has:

[(a,b)] + [(c,d)] := [(a+c,b+d)].\,

[(a,b)]\cdot[(c,d)] := [(ac+bd,ad+bc)].\,

The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:

-[(a,b)] := [(b,a)].\,

Hence subtraction can be defined as the addition of the additive inverse:

[(a,b)] - [(c,d)] := [(a+d,b+c)].\,

The standard ordering on the integers is given by:

[(a,b)] < [(c,d)]\,

iff

a+d < b+c.\,

It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.

Every equivalence class has a unique member that is of the form $(n,0)$ or $(0, n)$ (or both at once). The natural number $n$ is identified with the class $[(n,0)]$ (in other words the natural numbers are embedded into the integers by map sending $n$ to $[(n,0)]$ ), and the class $[(0, n)]$ is denoted $- n$ (this covers all remaining classes, and gives the class $[(0,0)]$ a second time since $-0 = 0.$

Thus, $[(a, b)]$ is denoted by

\begin{cases} a - b, & \mbox{if } a \ge b \\ -(b - a), & \mbox{if } a < b. \end{cases}

If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity.

This notation recovers the familiar representation of the integers as ${\dots, -2, -1, 0, 1, 2, \dots$ }.

Some examples are:

\begin{align} 0 &= [(0,0)] &= [(1,1)] &= \cdots & &= [(k,k)] \\ 1 &= [(1,0)] &= [(2,1)] &= \cdots & &= [(k+1,k)] \\ -1 &= [(0,1)] &= [(1,2)] &= \cdots & &= [(k,k+1)] \\ 2 &= [(2,0)] &= [(3,1)] &= \cdots & &= [(k+2,k)] \\ -2 &= [(0,2)] &= [(1,3)] &= \cdots & &= [(k,k+2)]. \end{align}

Computer science

Main article: Integer (computer science)

An integer is often a primitive data type in computer languages. However, integer data types can only represent a subset of all integers, since practical computers are of finite capacity. Also, in the common two's complement representation, the inherent definition of sign distinguishes between "negative" and "non-negative" rather than "negative, positive, and 0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.).

Variable-length representations of integers, such as bignums, can store any integer that fits in the computer's memory. Other integer data types are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10).

Cardinality

The cardinality of the set of integers is equal to $ℵ 0$ (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from $Z$ to $N$ . If $N = {0, 1, 2, \dots}$ then consider the function:

f(x) = \begin{cases} 2|x|, & \mbox{if } x \leq 0\\ 2x-1, & \mbox{if } x > 0. \end{cases}

${\dots (-4,8) (-3,6) (-2,4) (-1,2) (0,0) (1,1) (2,3) (3,5) \dots}$

If $N = {1, 2, 3, ...}$ then consider the function:

g(x) = \begin{cases} 2|x|, & \mbox{if } x < 0 \\ 2x+1, & \mbox{if } x \ge 0. \end{cases}

${\dots (-4,8) (-3,6) (-2,4) (-1,2) (0,1) (1,3) (2,5) (3,7) \dots}$

If the domain is restricted to $Z$ then each and every member of $Z$ has one and only one corresponding member of N and by the definition of cardinal equality the two sets have equal cardinality.

Notes

↑ Integer 's first, literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch"). "Entire" derives from the same origin via French (see: Evans, Nick (1995). "A-Quantifiers and Scope". In Bach, Emmon W. Quantification in Natural Languages. Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. p. 262. ISBN 0-7923-3352-7. )

References

↑ Weisstein, Eric W., "Counting Number", and "Whole Number", MathWorld.
↑ Miller, Jeff (2010-08-29). "Earliest Uses of Symbols of Number Theory". Retrieved 2010-09-20.
↑ Peter Jephson Cameron (1998). Introduction to Algebra. Oxford University Press. p. 4. ISBN 978-0-19-850195-4.
↑ Serge, Lang (1993), Algebra (3rd ed.), Addison-Wesley, pp. 86–87, ISBN 0-201-55540-9
↑ Warner, Seth (2012), Modern Algebra, Dover Books on Mathematics, Courier Corporation, Theorem 20.14, p. 185, ISBN 9780486137094 .
↑ Mendelson, Elliott (2008), Number Systems and the Foundations of Analysis, Dover Books on Mathematics, Courier Dover Publications, p. 86, ISBN 9780486457925 .
↑ Ivorra Castillo: Álgebra
↑ Frobisher, Len (1999), Learning to Teach Number: A Handbook for Students and Teachers in the Primary School, The Stanley Thornes Teaching Primary Maths Series, Nelson Thornes, p. 126, ISBN 9780748735150 .
1 2 3 Campbell, Howard E. (1970). The structure of arithmetic. Appleton-Century-Crofts. p. 83. ISBN 0-390-16895-5.

Sources

Bell, E.T., Men of Mathematics. New York: Simon & Schuster, 1986. (Hardcover; ISBN 0-671-46400-0)/(Paperback; ISBN 0-671-62818-6)
Herstein, I.N., Topics in Algebra, Wiley; 2 edition (June 20, 1975), ISBN 0-471-01090-1.
Mac Lane, Saunders, and Garrett Birkhoff; Algebra, American Mathematical Society; 3rd edition (April 1999). ISBN 0-8218-1646-2.
Weisstein, Eric W., "Integer", MathWorld.

External links

Look up integer in Wiktionary, the free dictionary.

Hazewinkel, Michiel, ed. (2001), "Integer", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
The Positive Integers — divisor tables and numeral representation tools
On-Line Encyclopedia of Integer Sequences cf OEIS

This article incorporates material from Integer on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Number systems

Countable sets	Natural numbers (ℕ) Integers (ℤ) Rational numbers (ℚ) Constructible numbers Algebraic numbers (𝔸) Periods Computable numbers Definable real numbers Arithmetical numbers Gaussian integers

Real numbers and extensions	Real numbers (ℝ) Complex numbers (ℂ) Quaternions (ℍ) Octonions (𝕆) Sedenions (𝕊) Cayley–Dickson construction Dual numbers Split-complex numbers Bicomplex numbers Hypercomplex numbers Superreal numbers Irrational numbers Transcendental numbers Hyperreal numbers Levi-Civita field Surreal numbers

Other systems	Cardinal numbers Ordinal numbers p-adic numbers Supernatural numbers

Classification List

Rational numbers

Integer Dedekind cut Dyadic rational Half-integer Superparticular ratio

Integers

0s

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

100s

100 101 102 103 104 105 106 107 108 109

110 111 112 113 114 115 116 117 118 119

120 121 122 123 124 125 126 127 128 129

130 131 132 133 134 135 136 137 138 139

140 141 142 143 144 145 146 147 148 149

150 151 152 153 154 155 156 157 158 159

160 161 162 163 164 165 166 167 168 169

170 171 172 173 174 175 176 177 178 179

180 181 182 183 184 185 186 187 188 189

190 191 192 193 194 195 196 197 198 199

200s

200 201 202 203 204 205 206 207 208 209

210 211 212 213 214 215 216 217 218 219

220 221 222 223 224 225 226 227 228 229

230 231 232 233 234 235 236 237 238 239

240 241 242 243 244 245 246 247 248 249

250 251 252 253 254 255 256 257 258 259

260 261 262 263 264 265 266 267 268 269

270 271 272 273 274 275 276 277 278 279

280 281 282 283 284 285 286 287 288 289

290 291 292 293 294 295 296 297 298 299

300s

300 301 302 303 304 305 306 307 308 309

310 311 312 313 314 315 316 317 318 319

320 321 322 323 324 325 326 327 328 329

330 331 332 333 334 335 336 337 338 339

340 341 342 343 344 345 346 347 348 349

350 351 352 353 354 355 356 357 358 359

360 361 362 363 364 365 366 367 368 369

370 371 372 373 374 375 376 377 378 379

380 381 382 383 384 385 386 387 388 389

390 391 392 393 394 395 396 397 398 399

400s

400 401 402 403 404 405 406 407 408 409

410 411 412 413 414 415 416 417 418 419

420 421 422 423 424 425 426 427 428 429

430 431 432 433 434 435 436 437 438 439

440 441 442 443 444 445 446 447 448 449

450 451 452 453 454 455 456 457 458 459

460 461 462 463 464 465 466 467 468 469

470 471 472 473 474 475 476 477 478 479

480 481 482 483 484 485 486 487 488 489

490 491 492 493 494 495 496 497 498 499

500s

500 501 502 503 504 505 506 507 508 509

510 511 512 513 514 515 516 517 518 519

520 521 522 523 524 525 526 527 528 529

530 531 532 533 534 535 536 537 538 539

540 541 542 543 544 545 546 547 548 549

550 551 552 553 554 555 556 557 558 559

560 561 562 563 564 565 566 567 568 569

570 571 572 573 574 575 576 577 578 579

580 581 582 583 584 585 586 587 588 589

590 591 592 593 594 595 596 597 598 599

600s

600 601 602 603 604 605 606 607 608 609

610 611 612 613 614 615 616 617 618 619

620 621 622 623 624 625 626 627 628 629

630 631 632 633 634 635 636 637 638 639

640 641 642 643 644 645 646 647 648 649

650 651 652 653 654 655 656 657 658 659

660 661 662 663 664 665 666 667 668 669

670 671 672 673 674 675 676 677 678 679

680 681 682 683 684 685 686 687 688 689

690 691 692 693 694 695 696 697 698 699

700s

700 701 702 703 704 705 706 707 708 709

710 711 712 713 714 715 716 717 718 719

720 721 722 723 724 725 726 727 728 729

730 731 732 733 734 735 736 737 738 739

740 741 742 743 744 745 746 747 748 749

750 751 752 753 754 755 756 757 758 759

760 761 762 763 764 765 766 767 768 769

770 771 772 773 774 775 776 777 778 779

780 781 782 783 784 785 786 787 788 789

790 791 792 793 794 795 796 797 798 799

800s

800 801 802 803 804 805 806 807 808 809

810 811 812 813 814 815 816 817 818 819

820 821 822 823 824 825 826 827 828 829

830 831 832 833 834 835 836 837 838 839

840 841 842 843 844 845 846 847 848 849

850 851 852 853 854 855 856 857 858 859

860 861 862 863 864 865 866 867 868 869

870 871 872 873 874 875 876 877 878 879

880 881 882 883 884 885 886 887 888 889

890 891 892 893 894 895 896 897 898 899

900s

900 901 902 903 904 905 906 907 908 909

910 911 912 913 914 915 916 917 918 919

920 921 922 923 924 925 926 927 928 929

930 931 932 933 934 935 936 937 938 939

940 941 942 943 944 945 946 947 948 949

950 951 952 953 954 955 956 957 958 959

960 961 962 963 964 965 966 967 968 969

970 971 972 973 974 975 976 977 978 979

980 981 982 983 984 985 986 987 988 989

990 991 992 993 994 995 996 997 998 999

Authority control	GND: 4134668-3 NDL: 00570428

This article is issued from Wikipedia - version of the Tuesday, February 02, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.