831:
859:
714:
65:
122:
1276:
2998:
1302:
of that function). However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing
3218:
1326:
the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. In fact, the rules for additive and multiplicative inverses are both examples of applying a
2903:
2622:
546:. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and — in the case of applying a function — monotonic functions are limited to
3406:
3654:
3050:
2872:
2089:. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example,
3484:
3962:
4259:
3757:
987:
In other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved. More generally, this applies for an
342:
rather than below the < and >. Later in 1734, ≦ and ≧, known as "less than (greater-than) over equal to" or "less than (greater than) or equal to with double horizontal bars", first appeared in
2776:
2697:
3843:
4132:
346:'s work . After that, mathematicians simplified Bouguer's symbol to "less than (greater than) or equal to with one horizontal bar" (≤), or "less than (greater than) or slanted equal to" (⩽).
4586:
4536:
2016:
1958:
1908:
3030:
4851:
is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them. The complexity of this algorithm is
4041:
4315:
3533:
3298:
3313:
4474:
3552:
413:
380:
5491:
5454:
5421:
2532:
2213:
When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance, to solve the inequality 4
4389:
2993:{\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,}
2475:
145:
is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the
3426:
4334:
often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names:
830:
3862:
1298:, by its definition, may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the
858:
4151:
2786:
3669:
5365:
5656:
5250:
5026:
454:
In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several
3776:
3213:{\displaystyle \left(\sum _{i=1}^{n}u_{i}v_{i}\right)^{2}\leq \left(\sum _{i=1}^{n}u_{i}^{2}\right)\left(\sum _{i=1}^{n}v_{i}^{2}\right).}
86:
4848:
4056:
5724:
5632:
5589:
5475:
5438:
5405:
5377:
5162:
5109:
2229:
in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding
108:
5347:
334:
In the 17th and 18th centuries, personal notations or typewriting signs were used to signal inequalities. For example, In 1670,
3977:
439:
sometimes is considered a form of strict inequality. It does not say that one is greater than the other; it does not even require
5531:
4841:
50:
2707:
5275:
2632:
5215:
4855:
in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.
4359:
2883:
5683:
4955:
Halmaghi, Elena; Liljedahl, Peter. "Inequalities in the
History of Mathematics: From Peculiarities to a Hard Discipline".
4898:
2371:
5564:
4374:
4274:
5719:
5678:
4852:
3495:
79:
73:
5646:
4545:
4495:
3261:
1975:
1917:
1867:
713:
4399:
3003:
2276:
Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the
2036:
1745:
2326:> ... . Mixed chained notation is used more often with compatible relations, like <, =, ≤. For instance,
4429:
4379:
4364:
4344:
818:
In other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an
4384:
90:
2432:
2375:
4369:
2374:. In contrast, in programming languages that provide an ordering on the type of comparison results, such as
4414:
819:
512:
4991:
4888:
4878:
4424:
4404:
4394:
3244:
2395:
1541:
1475:
1295:
1045:
891:
5714:
5125:
4903:
4869:
4739:
1581:
543:
4419:
4354:
4339:
121:
4893:
4409:
4349:
4326:
2411:
2367:
1299:
237:
In contrast to strict inequalities, there are two types of inequality relations that are not strict:
4908:
4837:
4485:
4439:
2897:
2423:
2277:
1791:
1654:
1623:
1572:
1479:
1048:
is opposite of that between the original numbers. More specifically, for any non-zero real numbers
4457:
27:
This article is about relations "greater than" and "less than". For the relation "not equal", see
5580:. North-Holland Mathematical Library. Vol. 67 (first ed.). Amsterdam, the Netherlands:
5485:
5448:
5415:
5313:
4970:
4434:
1600:
1510:
1471:
1291:
550:
455:
130:
5673:
389:
2617:{\displaystyle H={\frac {n}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\cdots +{\frac {1}{a_{n}}}}}}
356:
5652:
5642:
5628:
5595:
5585:
5471:
5434:
5401:
5373:
5271:
5246:
5240:
5158:
5150:
5105:
5022:
1761:
1749:
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1396:
562:
39:
5016:
4710:
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if
507:
This implies that the lesser value can be neglected with little effect on the accuracy of an
5611:
5508:
5305:
2780:
1006:
43:
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5690:
5615:
5603:
4864:
3041:
2701:
1460:
1061:
1057:
126:
5332:
4931:
165:
There are several different notations used to represent different kinds of inequalities:
5648:
The Cauchy-Schwarz Master Class: An
Introduction to the Art of Mathematical Inequalities
5179:
4957:
Proceedings of the 2012 annual meeting of the canadian mathematics education study group
5699:
5520:
5464:
4481:
4452:
2626:
887:
343:
35:
3401:{\displaystyle {\frac {x^{p}-1}{p}}\geq \ln(x)\geq {\frac {1-{\frac {1}{x^{p}}}}{p}}.}
2280:
of the inequalities between adjacent terms. For example, the defining condition of a
5708:
4589:
4539:
4331:
3033:
2526:
2021:
1963:
1808:
1781:
988:
508:
5270:. Prentice Hall Software Series (2nd ed.). Englewood Cliffs/NJ: Prentice Hall.
3649:{\displaystyle \left(x+y\right)^{z}+\left(x+z\right)^{y}+\left(y+z\right)^{x}>2.}
651:
of the premises is a strict inequality, then the conclusion is a strict inequality:
5694:
2281:
5204:
5499:
Grinshpan, A. Z. (2005), "General inequalities, consequences, and applications",
5099:
2480:
There are many inequalities between means. For example, for any positive numbers
518:
In all of the cases above, any two symbols mirroring each other are symmetrical;
5074:
4653:
3037:
2439:
1795:
1757:
1699:
1665:
758:
754:
603:
566:
448:
335:
146:
138:
1446:(this is true because the natural logarithm is a strictly increasing function.)
382:
the symbol for "greater than" bisected by a slash, "not". The same is true for
5568:
5552:
5512:
4883:
1661:
Some types of partial orders are specified by adding further axioms, such as:
435:
28:
5599:
5333:"Solution of One Conjecture on Inequalities with Power-Exponential Functions"
5155:
Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics
5049:
5543:
2867:{\displaystyle Q={\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\cdots +a_{n}^{2}}{n}}}}
1585:. Those are the very basic axioms that every kind of order has to satisfy.
17:
5581:
4477:
750:
3243:
are real positive numbers or variable expressions. They often appear in
2081:", from which, by the transitivity property above, it also follows that
5317:
4873:
2128:
This notation can be generalized to any number of terms: for instance,
1044:
If both numbers are positive, then the inequality relation between the
34:"Less than" and "Greater than" redirect here. For the symbol <, see
2402:
is called sharp if, for every valid universally quantified inequality
2253:≥ −1 respectively, which can be combined into the final solution −1 ≤
3479:{\displaystyle x^{x}\geq \left({\frac {1}{e}}\right)^{\frac {1}{e}}.}
5309:
2378:, even homogeneous chains may have a completely different meaning.
3047:
with the standard inner product, the Cauchy–Schwarz inequality is
1274:
857:
829:
3957:{\displaystyle x^{x}y^{y}z^{z}\geq \left(xyz\right)^{(x+y+z)/3}.}
5400:. Cambridge Mathematical Library, Cambridge University Press.
4254:{\displaystyle a^{2a}+b^{2b}+c^{2c}\geq a^{2b}+b^{2c}+c^{2a}.}
1275:
58:
5340:
Australian
Journal of Mathematical Analysis and Applications
4932:"Inequality Definition (Illustrated Mathematics Dictionary)"
2043:
can be defined as the only ordered field with that quality.
1588:
A strict partial order is a relation < that satisfies:
3752:{\displaystyle {\frac {e^{b}-e^{a}}{b-a}}>e^{(a+b)/2}.}
3036:. Examples of inner products include the real and complex
2888:
The Cauchy–Schwarz inequality states that for all vectors
2522:
where they represent the following means of the sequence:
602:
The transitive property of inequality states that for any
4592:, it would have to satisfy the following two properties:
5286:, p.167: Quote: "a<b<c is parsed as (a<b)<c"
2771:{\displaystyle A={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}}
5104:. Springer Science & Business Media. pp. 2–3.
2692:{\displaystyle G={\sqrt{a_{1}\cdot a_{2}\cdots a_{n}}}}
2180:− 1. By transitivity, this condition is equivalent to
5266:
Brian W. Kernighan and Dennis M. Ritchie (Apr 1988).
4674:(in which case the first property above implies that
4548:
4498:
4460:
4277:
4154:
4059:
3980:
3865:
3779:
3672:
3555:
3498:
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3053:
3006:
2906:
2789:
2710:
2635:
2535:
1978:
1920:
1870:
392:
359:
5018:
4911:, used in programming languages to denote inequality
4872:, for the use of similar ‹ and › signs as
3838:{\displaystyle x^{p}+y^{p}>\left(x+y\right)^{p}.}
5463:
4580:
4530:
4468:
4309:
4253:
4126:
4035:
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2866:
2770:
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2010:
1952:
1902:
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374:
4800:It can easily be proven that for this definition
5149:Simovici, Dan A. & Djeraba, Chabane (2008).
4127:{\displaystyle a^{ea}+b^{eb}\geq a^{eb}+b^{ea}.}
149:by their size. The main types of inequality are
5396:Hardy, G., Littlewood J. E., Pólya, G. (1999).
3412:→ 0, the upper and lower bounds converge to ln(
1498:, it must satisfy the three following clauses:
1338:Raising both sides of an inequality to a power
230:is strictly less than or strictly greater than
5466:Inequalities: With Applications to Engineering
5462:Drachman, Byron C., Cloud, Michael J. (1998).
5101:Inequalities: With Applications to Engineering
5098:Drachman, Bryon C.; Cloud, Michael J. (2006).
4707:, which means (−1 + 1) > 0; contradiction.
4581:{\displaystyle (\mathbb {C} ,+,\times ,\leq )}
4531:{\displaystyle (\mathbb {C} ,+,\times ,\leq )}
4488:, but it is impossible to define any relation
3231:is an inequality containing terms of the form
2011:{\displaystyle (\mathbb {C} ,+,\times ,\leq )}
1953:{\displaystyle (\mathbb {R} ,+,\times ,\leq )}
1903:{\displaystyle (\mathbb {Q} ,+,\times ,\leq )}
757:from both sides of an inequality. So, for any
56:Mathematical relation expressed with < or ≤
3025:{\displaystyle \langle \cdot ,\cdot \rangle }
8:
5490:: CS1 maint: multiple names: authors list (
5453:: CS1 maint: multiple names: authors list (
5420:: CS1 maint: multiple names: authors list (
5242:Why Beauty Is Truth: The History of Symmetry
4390:Inequality of arithmetic and geometric means
4036:{\displaystyle a^{a}+b^{b}\geq a^{b}+b^{a}.}
3019:
3007:
2984:
2968:
2962:
2946:
2928:
2912:
2476:Inequality of arithmetic and geometric means
49:"≪" redirects here. Not to be confused with
2394:and still be valid in general. Formally, a
542:Inequalities are governed by the following
5370:Understanding and Using Linear Programming
4553:
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2649:
2642:
2634:
2603:
2594:
2577:
2568:
2557:
2548:
2542:
2534:
2430:also holds. For instance, the inequality
1983:
1982:
1977:
1925:
1924:
1919:
1875:
1874:
1869:
391:
358:
109:Learn how and when to remove this message
5296:Laub, M.; Ilani, Ishai (1990). "E3116".
5015:Polyanin, A.D.; Manzhirov, A.V. (2006).
712:
120:
72:This article includes a list of general
5429:Beckenbach, E. F., Bellman, R. (1975).
4920:
1579:A set with a partial order is called a
561:The relations ≤ and ≥ are each other's
5483:
5446:
5413:
4971:"Earliest Uses of Symbols of Relation"
1451:Formal definitions and generalizations
4977:. University of St Andrews, Scotland.
1399:on both sides of an inequality, when
1123:
133:are defined by a set of inequalities.
7:
4986:
4984:
4926:
4924:
4849:cylindrical algebraic decomposition
2225:+ 2, it is not possible to isolate
1970:cannot be defined in order to make
1331:monotonically decreasing function.
993:
5700:AoPS Wiki entry about Inequalities
1334:A few examples of this rule are:
1114:All of the cases for the signs of
78:it lacks sufficient corresponding
25:
5298:The American Mathematical Monthly
4310:{\displaystyle a^{b}+b^{a}>1.}
2450:is sharp, whereas the inequality
2366:. This notation exists in a few
2028:and would therefore be positive.
1271:Applying a function to both sides
1009:states that for any real numbers
894:state that for any real numbers,
5537:from the original on 2022-10-09.
5353:from the original on 2022-10-09.
4447:Complex numbers and inequalities
3528:{\displaystyle x^{x^{x}}\geq x.}
2980:
2972:
2958:
2950:
2924:
2916:
2031:Besides being an ordered field,
63:
5501:Advances in Applied Mathematics
5431:An Introduction to Inequalities
5221:from the original on 2022-10-09
222:. These relations are known as
51:Absolute continuity of measures
5651:. Cambridge University Press.
5559:. Kungliga Tekniska högskolan.
5553:"Mathematical Problem Solving"
5546:. Online e-book in PDF format.
5544:"Introduction to Inequalities"
4575:
4549:
4525:
4499:
3938:
3920:
3733:
3721:
3658:For any real distinct numbers
3357:
3351:
3293:{\displaystyle e^{x}\geq 1+x.}
2933:
2908:
2024:, because −1 is the square of
2005:
1979:
1947:
1921:
1897:
1871:
886:The properties that deal with
1:
4899:List of triangle inequalities
1306:If the inequality is strict (
538:Properties on the number line
338:used a single horizontal bar
5245:. Hachette UK. p. 106.
4469:{\displaystyle \mathbb {C} }
2386:An inequality is said to be
1407:are positive real numbers:
991:. For more information, see
322:(or, equivalently, at least
5679:Encyclopedia of Mathematics
4842:Fourier–Motzkin elimination
1354:are positive real numbers:
826:Multiplication and division
353:can also be represented by
276:(or, equivalently, at most
38:. For the symbol >, see
5741:
5625:Multicriteria Optimization
5623:Ehrgott, Matthias (2005).
5268:The C Programming Language
4324:
2881:
2473:
2470:Inequalities between means
2037:Least-upper-bound property
1779:
1746:Least-upper-bound property
408:{\displaystyle a\nless b.}
48:
33:
26:
5578:Mathematical Inequalities
5576:Pachpatte, B. G. (2005).
5513:10.1016/j.aam.2004.05.001
5021:. CRC Press. p. 29.
4360:Cauchy–Schwarz inequality
2884:Cauchy–Schwarz inequality
2878:Cauchy–Schwarz inequality
375:{\displaystyle a\ngtr b,}
5725:Mathematical terminology
5542:Arthur Lohwater (1982).
5521:"'Quickie' inequalities"
5151:"Partially Ordered Sets"
1806:, +, ×, ≤) is called an
709:Addition and subtraction
317:greater than or equal to
234:. Equality is excluded.
5557:The Old Problem Seminar
5551:Harold Shapiro (2005).
5180:"Partially Ordered Set"
4832:Systems of inequalities
4476:with its operations of
4400:Kolmogorov's inequality
4321:Well-known inequalities
1122:can also be written in
1046:multiplicative inverses
565:, meaning that for any
513:ultrarelativistic limit
93:more precise citations.
5203:Feldman, Joel (2014).
4889:Interval (mathematics)
4879:Inclusion (set theory)
4582:
4532:
4470:
4430:Samuelson's inequality
4380:Hoeffding's inequality
4365:Chebyshev's inequality
4345:Bernoulli's inequality
4311:
4255:
4128:
4037:
3958:
3839:
3753:
3650:
3529:
3480:
3402:
3294:
3245:mathematical olympiads
3214:
3186:
3140:
3080:
3026:
2994:
2868:
2772:
2693:
2618:
2396:universally quantified
2012:
1954:
1904:
1287:
1040:Multiplicative inverse
883:
855:
742:
409:
376:
280:, or not greater than
134:
5691:Graph of Inequalities
5239:Stewart, Ian (2007).
5184:mathworld.wolfram.com
5126:"ProvingInequalities"
5079:mathworld.wolfram.com
5054:mathworld.wolfram.com
4904:Partially ordered set
4870:Bracket (mathematics)
4840:can be simplified by
4740:lexicographical order
4583:
4533:
4471:
4375:Cramér–Rao inequality
4370:Chernoff's inequality
4312:
4256:
4129:
4038:
3959:
3840:
3754:
3651:
3530:
3481:
3403:
3295:
3215:
3166:
3120:
3060:
3027:
2995:
2869:
2773:
2694:
2619:
2368:programming languages
2013:
1955:
1905:
1582:partially ordered set
1278:
1005:The property for the
861:
833:
716:
534:are equivalent, etc.
511:(such as the case of
410:
377:
271:less than or equal to
124:
5433:. Random House Inc.
5372:. Berlin: Springer.
5331:Manyama, S. (2010).
5284:Relational Operators
4909:Relational operators
4894:List of inequalities
4742:definition is used:
4546:
4496:
4458:
4415:Nesbitt's inequality
4410:Minkowski inequality
4327:List of inequalities
4275:
4152:
4057:
3978:
3863:
3777:
3670:
3553:
3496:
3427:
3314:
3262:
3051:
3004:
2904:
2787:
2708:
2633:
2533:
1976:
1918:
1868:
1184:< 0, then 0 >
390:
357:
42:. For the song, see
5627:. Springer-Berlin.
5519:Murray S. Klamkin.
5470:. Springer-Verlag.
5178:Weisstein, Eric W.
5073:Weisstein, Eric W.
5048:Weisstein, Eric W.
4996:www.learnalberta.ca
4838:linear inequalities
4440:Triangle inequality
4425:Poincaré inequality
4405:Markov's inequality
4395:Jensen's inequality
4385:Hölder's inequality
3201:
3155:
2898:inner product space
2856:
2832:
2814:
2278:logical conjunction
1482:. That is, for all
551:monotonic functions
456:orders of magnitude
447:to be member of an
326:, or not less than
224:strict inequalities
5720:Elementary algebra
5643:Steele, J. Michael
4936:www.mathsisfun.com
4853:doubly exponential
4738:"). Sometimes the
4688:; this means that
4681:). In either case
4578:
4528:
4466:
4435:Sobolev inequality
4420:Pedoe's inequality
4355:Boole's inequality
4340:Azuma's inequality
4307:
4251:
4124:
4033:
3954:
3835:
3769:> 0 and 0 <
3749:
3646:
3525:
3476:
3398:
3290:
3223:Power inequalities
3210:
3187:
3141:
3022:
2990:
2864:
2842:
2818:
2800:
2768:
2689:
2614:
2382:Sharp inequalities
2008:
1950:
1900:
1748:: Every non-empty
1288:
884:
856:
745:A common constant
743:
405:
372:
135:
131:linear programming
5658:978-0-521-54677-5
5282:Here: Sect.A.7.9
5252:978-0-4650-0875-9
5028:978-1-4200-1051-0
4656:, for any number
4350:Bell's inequality
3711:
3470:
3456:
3393:
3387:
3340:
2862:
2861:
2766:
2687:
2612:
2609:
2583:
2563:
2105:is equivalent to
1397:natural logarithm
1342:> 0 (equiv., −
498:much greater than
119:
118:
111:
40:greater-than sign
16:(Redirected from
5732:
5687:
5662:
5638:
5619:
5572:
5567:. Archived from
5560:
5547:
5538:
5536:
5525:
5515:
5495:
5489:
5481:
5469:
5458:
5452:
5444:
5425:
5419:
5411:
5384:
5383:
5364:Gärtner, Bernd;
5361:
5355:
5354:
5352:
5337:
5328:
5322:
5321:
5293:
5287:
5281:
5263:
5257:
5256:
5236:
5230:
5229:
5227:
5226:
5220:
5209:
5200:
5194:
5193:
5191:
5190:
5175:
5169:
5168:
5146:
5140:
5139:
5137:
5136:
5122:
5116:
5115:
5095:
5089:
5088:
5086:
5085:
5070:
5064:
5063:
5061:
5060:
5045:
5039:
5038:
5036:
5035:
5012:
5006:
5005:
5003:
5002:
4988:
4979:
4978:
4967:
4961:
4960:
4952:
4946:
4945:
4943:
4942:
4928:
4827:
4809:
4794:
4782:
4768:
4754:
4737:
4719:
4706:
4702:
4698:
4694:
4687:
4680:
4673:
4666:
4647:
4640:
4633:
4623:
4605:
4587:
4585:
4584:
4579:
4556:
4537:
4535:
4534:
4529:
4506:
4491:
4475:
4473:
4472:
4467:
4465:
4316:
4314:
4313:
4308:
4300:
4299:
4287:
4286:
4260:
4258:
4257:
4252:
4247:
4246:
4231:
4230:
4215:
4214:
4199:
4198:
4183:
4182:
4167:
4166:
4133:
4131:
4130:
4125:
4120:
4119:
4104:
4103:
4088:
4087:
4072:
4071:
4042:
4040:
4039:
4034:
4029:
4028:
4016:
4015:
4003:
4002:
3990:
3989:
3963:
3961:
3960:
3955:
3950:
3949:
3945:
3918:
3914:
3895:
3894:
3885:
3884:
3875:
3874:
3844:
3842:
3841:
3836:
3831:
3830:
3825:
3821:
3802:
3801:
3789:
3788:
3758:
3756:
3755:
3750:
3745:
3744:
3740:
3712:
3710:
3699:
3698:
3697:
3685:
3684:
3674:
3655:
3653:
3652:
3647:
3639:
3638:
3633:
3629:
3610:
3609:
3604:
3600:
3581:
3580:
3575:
3571:
3534:
3532:
3531:
3526:
3515:
3514:
3513:
3512:
3485:
3483:
3482:
3477:
3472:
3471:
3463:
3461:
3457:
3449:
3439:
3438:
3408:In the limit of
3407:
3405:
3404:
3399:
3394:
3389:
3388:
3386:
3385:
3373:
3364:
3341:
3336:
3329:
3328:
3318:
3299:
3297:
3296:
3291:
3274:
3273:
3229:power inequality
3219:
3217:
3216:
3211:
3206:
3202:
3200:
3195:
3185:
3180:
3160:
3156:
3154:
3149:
3139:
3134:
3111:
3110:
3105:
3101:
3100:
3099:
3090:
3089:
3079:
3074:
3031:
3029:
3028:
3023:
2999:
2997:
2996:
2991:
2983:
2975:
2961:
2953:
2942:
2941:
2936:
2927:
2919:
2911:
2900:it is true that
2873:
2871:
2870:
2865:
2863:
2857:
2855:
2850:
2831:
2826:
2813:
2808:
2798:
2797:
2777:
2775:
2774:
2769:
2767:
2762:
2761:
2760:
2742:
2741:
2729:
2728:
2718:
2698:
2696:
2695:
2690:
2688:
2686:
2681:
2680:
2679:
2667:
2666:
2654:
2653:
2643:
2623:
2621:
2620:
2615:
2613:
2611:
2610:
2608:
2607:
2595:
2584:
2582:
2581:
2569:
2564:
2562:
2561:
2549:
2543:
2521:
2465:
2449:
2429:
2417:
2390:if it cannot be
2272:
2270:
2269:
2266:
2263:
2248:
2246:
2245:
2242:
2239:
2047:Chained notation
2019:
2017:
2015:
2014:
2009:
1986:
1969:
1961:
1959:
1957:
1956:
1951:
1928:
1911:
1909:
1907:
1906:
1901:
1878:
1812:if and only if:
1266:
1264:
1263:
1258:
1255:
1248:
1246:
1245:
1240:
1237:
1219:
1217:
1216:
1211:
1208:
1201:
1199:
1198:
1193:
1190:
1172:
1170:
1169:
1164:
1161:
1154:
1152:
1151:
1146:
1143:
1124:chained notation
1110:
1108:
1107:
1102:
1099:
1092:
1090:
1089:
1084:
1081:
1007:additive inverse
1001:Additive inverse
994:§ Ordered fields
822:under addition.
429:is not equal to
414:
412:
411:
406:
381:
379:
378:
373:
351:not greater than
218:is not equal to
214:In either case,
127:feasible regions
114:
107:
103:
100:
94:
89:this article by
80:inline citations
67:
66:
59:
44:Less Than (song)
21:
5740:
5739:
5735:
5734:
5733:
5731:
5730:
5729:
5705:
5704:
5672:
5669:
5659:
5641:
5635:
5622:
5592:
5575:
5563:
5550:
5541:
5534:
5528:Math Strategies
5523:
5518:
5498:
5482:
5478:
5461:
5445:
5441:
5428:
5412:
5408:
5395:
5392:
5387:
5380:
5363:
5362:
5358:
5350:
5335:
5330:
5329:
5325:
5310:10.2307/2324012
5295:
5294:
5290:
5278:
5265:
5264:
5260:
5253:
5238:
5237:
5233:
5224:
5222:
5218:
5207:
5202:
5201:
5197:
5188:
5186:
5177:
5176:
5172:
5165:
5148:
5147:
5143:
5134:
5132:
5130:www.cs.yale.edu
5124:
5123:
5119:
5112:
5097:
5096:
5092:
5083:
5081:
5072:
5071:
5067:
5058:
5056:
5047:
5046:
5042:
5033:
5031:
5029:
5014:
5013:
5009:
5000:
4998:
4990:
4989:
4982:
4969:
4968:
4964:
4954:
4953:
4949:
4940:
4938:
4930:
4929:
4922:
4918:
4865:Binary relation
4861:
4834:
4811:
4801:
4784:
4772:
4758:
4746:
4721:
4711:
4704:
4700:
4696:
4689:
4682:
4675:
4668:
4661:
4652:Because ≤ is a
4642:
4635:
4628:
4607:
4597:
4544:
4543:
4494:
4493:
4489:
4456:
4455:
4453:complex numbers
4449:
4444:
4329:
4323:
4291:
4278:
4273:
4272:
4235:
4219:
4203:
4187:
4171:
4155:
4150:
4149:
4108:
4092:
4076:
4060:
4055:
4054:
4020:
4007:
3994:
3981:
3976:
3975:
3904:
3900:
3899:
3886:
3876:
3866:
3861:
3860:
3811:
3807:
3806:
3793:
3780:
3775:
3774:
3716:
3700:
3689:
3676:
3675:
3668:
3667:
3619:
3615:
3614:
3590:
3586:
3585:
3561:
3557:
3556:
3551:
3550:
3504:
3499:
3494:
3493:
3444:
3443:
3430:
3425:
3424:
3377:
3365:
3320:
3319:
3312:
3311:
3265:
3260:
3259:
3225:
3165:
3161:
3119:
3115:
3091:
3081:
3059:
3055:
3054:
3049:
3048:
3042:Euclidean space
3002:
3001:
2931:
2902:
2901:
2886:
2880:
2799:
2785:
2784:
2752:
2733:
2720:
2719:
2706:
2705:
2702:Arithmetic mean
2671:
2658:
2645:
2644:
2631:
2630:
2599:
2573:
2553:
2547:
2531:
2530:
2504:
2502:
2493:
2486:
2478:
2472:
2451:
2431:
2419:
2407:
2384:
2325:
2318:
2311:
2304:
2297:
2290:
2267:
2264:
2261:
2260:
2258:
2243:
2240:
2237:
2236:
2234:
2197:
2188:
2171:
2161:
2151:
2142:
2135:
2049:
1974:
1973:
1971:
1967:
1916:
1915:
1913:
1866:
1865:
1863:
1784:
1778:
1461:binary relation
1455:A (non-strict)
1453:
1445:
1426:
1392:
1373:
1273:
1268:
1259:
1256:
1253:
1252:
1250:
1241:
1238:
1235:
1234:
1232:
1221:
1212:
1209:
1206:
1205:
1203:
1194:
1191:
1188:
1187:
1185:
1174:
1165:
1162:
1159:
1158:
1156:
1147:
1144:
1141:
1140:
1138:
1112:
1103:
1100:
1097:
1096:
1094:
1085:
1082:
1079:
1078:
1076:
1042:
1037:
1003:
985:
946:
828:
816:
711:
706:
679:
645:
600:
595:
594:are equivalent.
559:
540:
388:
387:
355:
354:
226:, meaning that
163:
115:
104:
98:
95:
85:Please help to
84:
68:
64:
57:
54:
47:
32:
23:
22:
15:
12:
11:
5:
5738:
5736:
5728:
5727:
5722:
5717:
5707:
5706:
5703:
5702:
5697:
5688:
5668:
5667:External links
5665:
5664:
5663:
5657:
5639:
5633:
5620:
5590:
5573:
5571:on 2008-02-03.
5561:
5548:
5539:
5516:
5496:
5476:
5459:
5439:
5426:
5406:
5391:
5388:
5386:
5385:
5378:
5366:Matoušek, Jiří
5356:
5323:
5288:
5276:
5258:
5251:
5231:
5195:
5170:
5163:
5141:
5117:
5110:
5090:
5075:"Much Greater"
5065:
5040:
5027:
5007:
4980:
4962:
4947:
4919:
4917:
4914:
4913:
4912:
4906:
4901:
4896:
4891:
4886:
4881:
4876:
4867:
4860:
4857:
4833:
4830:
4798:
4797:
4796:
4795:
4770:
4650:
4649:
4625:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4555:
4551:
4527:
4524:
4521:
4518:
4515:
4512:
4509:
4505:
4501:
4482:multiplication
4464:
4448:
4445:
4443:
4442:
4437:
4432:
4427:
4422:
4417:
4412:
4407:
4402:
4397:
4392:
4387:
4382:
4377:
4372:
4367:
4362:
4357:
4352:
4347:
4342:
4336:
4332:Mathematicians
4322:
4319:
4318:
4317:
4306:
4303:
4298:
4294:
4290:
4285:
4281:
4261:
4250:
4245:
4242:
4238:
4234:
4229:
4226:
4222:
4218:
4213:
4210:
4206:
4202:
4197:
4194:
4190:
4186:
4181:
4178:
4174:
4170:
4165:
4162:
4158:
4134:
4123:
4118:
4115:
4111:
4107:
4102:
4099:
4095:
4091:
4086:
4083:
4079:
4075:
4070:
4067:
4063:
4043:
4032:
4027:
4023:
4019:
4014:
4010:
4006:
4001:
3997:
3993:
3988:
3984:
3964:
3953:
3948:
3944:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3917:
3913:
3910:
3907:
3903:
3898:
3893:
3889:
3883:
3879:
3873:
3869:
3845:
3834:
3829:
3824:
3820:
3817:
3814:
3810:
3805:
3800:
3796:
3792:
3787:
3783:
3759:
3748:
3743:
3739:
3735:
3732:
3729:
3726:
3723:
3719:
3715:
3709:
3706:
3703:
3696:
3692:
3688:
3683:
3679:
3656:
3645:
3642:
3637:
3632:
3628:
3625:
3622:
3618:
3613:
3608:
3603:
3599:
3596:
3593:
3589:
3584:
3579:
3574:
3570:
3567:
3564:
3560:
3535:
3524:
3521:
3518:
3511:
3507:
3502:
3486:
3475:
3469:
3466:
3460:
3455:
3452:
3447:
3442:
3437:
3433:
3417:
3397:
3392:
3384:
3380:
3376:
3371:
3368:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3339:
3335:
3332:
3327:
3323:
3300:
3289:
3286:
3283:
3280:
3277:
3272:
3268:
3254:For any real
3224:
3221:
3209:
3205:
3199:
3194:
3190:
3184:
3179:
3176:
3173:
3169:
3164:
3159:
3153:
3148:
3144:
3138:
3133:
3130:
3127:
3123:
3118:
3114:
3109:
3104:
3098:
3094:
3088:
3084:
3078:
3073:
3070:
3067:
3063:
3058:
3021:
3018:
3015:
3012:
3009:
2989:
2986:
2982:
2978:
2974:
2970:
2967:
2964:
2960:
2956:
2952:
2948:
2945:
2940:
2935:
2930:
2926:
2922:
2918:
2914:
2910:
2879:
2876:
2875:
2874:
2860:
2854:
2849:
2845:
2841:
2838:
2835:
2830:
2825:
2821:
2817:
2812:
2807:
2803:
2795:
2792:
2781:Quadratic mean
2778:
2765:
2759:
2755:
2751:
2748:
2745:
2740:
2736:
2732:
2727:
2723:
2716:
2713:
2699:
2685:
2678:
2674:
2670:
2665:
2661:
2657:
2652:
2648:
2641:
2638:
2627:Geometric mean
2624:
2606:
2602:
2598:
2593:
2590:
2587:
2580:
2576:
2572:
2567:
2560:
2556:
2552:
2546:
2541:
2538:
2498:
2491:
2484:
2471:
2468:
2466:is not sharp.
2383:
2380:
2323:
2316:
2309:
2302:
2295:
2288:
2284:is written as
2193:
2184:
2166:
2157:
2147:
2140:
2133:
2048:
2045:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1985:
1981:
1964:ordered fields
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1927:
1923:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1877:
1873:
1860:
1859:
1840:
1780:Main article:
1777:
1776:Ordered fields
1774:
1773:
1772:
1767:(supremum) in
1743:
1697:
1659:
1658:
1627:
1604:
1577:
1576:
1545:
1514:
1452:
1449:
1448:
1447:
1427:
1408:
1393:
1374:
1355:
1346:< 0), when
1272:
1269:
1222:
1175:
1128:
1126:, as follows:
1066:
1056:that are both
1041:
1038:
1019:
1002:
999:
947:
908:
888:multiplication
827:
824:
774:
710:
707:
680:
653:
619:
599:
596:
578:
558:
555:
539:
536:
505:
504:
482:
476:much less than
404:
401:
398:
395:
371:
368:
365:
362:
344:Pierre Bouguer
332:
331:
285:
212:
211:
189:
162:
159:
117:
116:
71:
69:
62:
55:
36:less-than sign
24:
14:
13:
10:
9:
6:
4:
3:
2:
5737:
5726:
5723:
5721:
5718:
5716:
5713:
5712:
5710:
5701:
5698:
5696:
5692:
5689:
5685:
5681:
5680:
5675:
5671:
5670:
5666:
5660:
5654:
5650:
5649:
5644:
5640:
5636:
5634:3-540-21398-8
5630:
5626:
5621:
5617:
5613:
5609:
5605:
5601:
5597:
5593:
5591:0-444-51795-2
5587:
5583:
5579:
5574:
5570:
5566:
5562:
5558:
5554:
5549:
5545:
5540:
5533:
5529:
5522:
5517:
5514:
5510:
5507:(1): 71–100,
5506:
5502:
5497:
5493:
5487:
5479:
5477:0-387-98404-6
5473:
5468:
5467:
5460:
5456:
5450:
5442:
5440:0-394-01559-2
5436:
5432:
5427:
5423:
5417:
5409:
5407:0-521-05206-8
5403:
5399:
5394:
5393:
5389:
5381:
5379:3-540-30697-8
5375:
5371:
5367:
5360:
5357:
5349:
5345:
5341:
5334:
5327:
5324:
5319:
5315:
5311:
5307:
5303:
5299:
5292:
5289:
5285:
5279:
5273:
5269:
5262:
5259:
5254:
5248:
5244:
5243:
5235:
5232:
5217:
5213:
5206:
5199:
5196:
5185:
5181:
5174:
5171:
5166:
5164:9781848002012
5160:
5156:
5152:
5145:
5142:
5131:
5127:
5121:
5118:
5113:
5111:0-3872-2626-5
5107:
5103:
5102:
5094:
5091:
5080:
5076:
5069:
5066:
5055:
5051:
5044:
5041:
5030:
5024:
5020:
5019:
5011:
5008:
4997:
4993:
4987:
4985:
4981:
4976:
4972:
4966:
4963:
4958:
4951:
4948:
4937:
4933:
4927:
4925:
4921:
4915:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4871:
4868:
4866:
4863:
4862:
4858:
4856:
4854:
4850:
4845:
4843:
4839:
4831:
4829:
4826:
4822:
4818:
4814:
4808:
4804:
4792:
4788:
4780:
4776:
4771:
4766:
4762:
4757:
4756:
4753:
4749:
4745:
4744:
4743:
4741:
4736:
4732:
4728:
4724:
4718:
4714:
4708:
4692:
4686:
4679:
4671:
4665:
4659:
4655:
4646:
4639:
4632:
4626:
4622:
4618:
4614:
4610:
4604:
4600:
4595:
4594:
4593:
4591:
4590:ordered field
4572:
4569:
4566:
4563:
4560:
4557:
4541:
4540:ordered field
4522:
4519:
4516:
4513:
4510:
4507:
4487:
4483:
4479:
4454:
4446:
4441:
4438:
4436:
4433:
4431:
4428:
4426:
4423:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4398:
4396:
4393:
4391:
4388:
4386:
4383:
4381:
4378:
4376:
4373:
4371:
4368:
4366:
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4346:
4343:
4341:
4338:
4337:
4335:
4333:
4328:
4320:
4304:
4301:
4296:
4292:
4288:
4283:
4279:
4271:> 0, then
4270:
4266:
4262:
4248:
4243:
4240:
4236:
4232:
4227:
4224:
4220:
4216:
4211:
4208:
4204:
4200:
4195:
4192:
4188:
4184:
4179:
4176:
4172:
4168:
4163:
4160:
4156:
4148:> 0, then
4147:
4143:
4139:
4135:
4121:
4116:
4113:
4109:
4105:
4100:
4097:
4093:
4089:
4084:
4081:
4077:
4073:
4068:
4065:
4061:
4053:> 0, then
4052:
4048:
4044:
4030:
4025:
4021:
4017:
4012:
4008:
4004:
3999:
3995:
3991:
3986:
3982:
3974:> 0, then
3973:
3969:
3965:
3951:
3946:
3942:
3935:
3932:
3929:
3926:
3923:
3915:
3911:
3908:
3905:
3901:
3896:
3891:
3887:
3881:
3877:
3871:
3867:
3859:> 0, then
3858:
3854:
3850:
3846:
3832:
3827:
3822:
3818:
3815:
3812:
3808:
3803:
3798:
3794:
3790:
3785:
3781:
3773:< 1, then
3772:
3768:
3764:
3760:
3746:
3741:
3737:
3730:
3727:
3724:
3717:
3713:
3707:
3704:
3701:
3694:
3690:
3686:
3681:
3677:
3665:
3661:
3657:
3643:
3640:
3635:
3630:
3626:
3623:
3620:
3616:
3611:
3606:
3601:
3597:
3594:
3591:
3587:
3582:
3577:
3572:
3568:
3565:
3562:
3558:
3549:> 0, then
3548:
3544:
3540:
3536:
3522:
3519:
3516:
3509:
3505:
3500:
3492:> 0, then
3491:
3487:
3473:
3467:
3464:
3458:
3453:
3450:
3445:
3440:
3435:
3431:
3423:> 0, then
3422:
3418:
3415:
3411:
3395:
3390:
3382:
3378:
3374:
3369:
3366:
3360:
3354:
3348:
3345:
3342:
3337:
3333:
3330:
3325:
3321:
3310:> 0, then
3309:
3305:
3301:
3287:
3284:
3281:
3278:
3275:
3270:
3266:
3257:
3253:
3252:
3251:
3248:
3246:
3242:
3238:
3234:
3230:
3222:
3220:
3207:
3203:
3197:
3192:
3188:
3182:
3177:
3174:
3171:
3167:
3162:
3157:
3151:
3146:
3142:
3136:
3131:
3128:
3125:
3121:
3116:
3112:
3107:
3102:
3096:
3092:
3086:
3082:
3076:
3071:
3068:
3065:
3061:
3056:
3046:
3043:
3039:
3035:
3034:inner product
3016:
3013:
3010:
2987:
2976:
2965:
2954:
2943:
2938:
2920:
2899:
2895:
2891:
2885:
2877:
2858:
2852:
2847:
2843:
2839:
2836:
2833:
2828:
2823:
2819:
2815:
2810:
2805:
2801:
2793:
2790:
2782:
2779:
2763:
2757:
2753:
2749:
2746:
2743:
2738:
2734:
2730:
2725:
2721:
2714:
2711:
2703:
2700:
2683:
2676:
2672:
2668:
2663:
2659:
2655:
2650:
2646:
2639:
2636:
2628:
2625:
2604:
2600:
2596:
2591:
2588:
2585:
2578:
2574:
2570:
2565:
2558:
2554:
2550:
2544:
2539:
2536:
2528:
2527:Harmonic mean
2525:
2524:
2523:
2519:
2515:
2511:
2507:
2501:
2497:
2490:
2483:
2477:
2469:
2467:
2463:
2459:
2455:
2447:
2443:
2442:
2437:
2434:
2428:
2425:
2422:
2416:
2413:
2410:
2405:
2401:
2397:
2393:
2389:
2381:
2379:
2377:
2373:
2369:
2365:
2361:
2357:
2353:
2349:
2345:
2341:
2337:
2333:
2329:
2322:
2315:
2308:
2301:
2294:
2287:
2283:
2279:
2274:
2256:
2252:
2232:
2228:
2224:
2220:
2216:
2211:
2209:
2205:
2201:
2196:
2192:
2187:
2183:
2179:
2176:= 1, 2, ...,
2175:
2169:
2165:
2160:
2156:
2152:
2150:
2146:
2139:
2132:
2126:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2088:
2084:
2080:
2076:
2072:
2068:
2064:
2063:
2059:
2055:
2051:The notation
2046:
2044:
2042:
2038:
2035:also has the
2034:
2029:
2027:
2023:
2022:ordered field
2002:
1999:
1996:
1993:
1990:
1987:
1965:
1944:
1941:
1938:
1935:
1932:
1929:
1894:
1891:
1888:
1885:
1882:
1879:
1857:
1853:
1849:
1845:
1841:
1838:
1834:
1830:
1826:
1822:
1818:
1815:
1814:
1813:
1811:
1810:
1809:ordered field
1805:
1801:
1797:
1793:
1790:, +, ×) is a
1789:
1783:
1782:Ordered field
1775:
1770:
1766:
1764:
1759:
1755:
1751:
1747:
1744:
1741:
1737:
1733:
1729:
1725:
1722:, there is a
1721:
1717:
1713:
1709:
1705:
1701:
1698:
1695:
1691:
1687:
1683:
1679:
1675:
1671:
1667:
1664:
1663:
1662:
1656:
1652:
1648:
1644:
1640:
1636:
1632:
1628:
1625:
1621:
1617:
1613:
1609:
1605:
1602:
1601:irreflexivity
1598:
1594:
1591:
1590:
1589:
1586:
1584:
1583:
1574:
1570:
1566:
1562:
1558:
1554:
1550:
1546:
1543:
1539:
1535:
1531:
1527:
1523:
1519:
1515:
1512:
1508:
1504:
1501:
1500:
1499:
1497:
1493:
1489:
1485:
1481:
1477:
1476:antisymmetric
1473:
1469:
1466:
1462:
1458:
1457:partial order
1450:
1443:
1439:
1435:
1431:
1424:
1420:
1416:
1412:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1371:
1367:
1363:
1359:
1353:
1349:
1345:
1341:
1337:
1336:
1335:
1332:
1330:
1325:
1321:
1317:
1313:
1309:
1304:
1301:
1297:
1293:
1292:monotonically
1286:
1282:
1279:The graph of
1277:
1270:
1262:
1244:
1230:
1226:
1215:
1197:
1183:
1179:
1168:
1150:
1136:
1132:
1127:
1125:
1121:
1117:
1106:
1088:
1074:
1070:
1065:
1063:
1059:
1055:
1051:
1047:
1039:
1035:
1031:
1027:
1023:
1018:
1016:
1012:
1008:
1000:
998:
996:
995:
990:
989:ordered field
983:
979:
975:
971:
967:
963:
960:< 0, then
959:
955:
951:
944:
940:
936:
932:
928:
924:
921:> 0, then
920:
916:
912:
907:
905:
902:and non-zero
901:
897:
893:
889:
881:
877:
874:< 0, then
873:
869:
865:
860:
853:
849:
846:> 0, then
845:
841:
837:
832:
825:
823:
821:
820:ordered group
814:
810:
806:
802:
798:
794:
790:
786:
782:
778:
773:
771:
767:
763:
760:
756:
752:
748:
740:
736:
732:
728:
724:
720:
715:
708:
704:
700:
696:
692:
688:
684:
677:
673:
669:
665:
661:
657:
652:
650:
643:
639:
635:
631:
627:
623:
618:
616:
612:
608:
605:
597:
593:
589:
585:
581:
577:
575:
571:
568:
564:
556:
554:
552:
549:
545:
537:
535:
533:
529:
525:
521:
516:
515:in physics).
514:
510:
509:approximation
502:
499:
495:
491:
487:
484:The notation
483:
480:
477:
473:
469:
465:
462:The notation
461:
460:
459:
457:
452:
450:
446:
442:
438:
437:
432:
428:
424:
420:
417:The notation
415:
402:
399:
396:
393:
385:
384:not less than
369:
366:
363:
360:
352:
349:The relation
347:
345:
341:
337:
329:
325:
321:
318:
314:
310:
306:
302:
298:
294:
290:
287:The notation
286:
283:
279:
275:
272:
268:
264:
260:
256:
252:
248:
244:
241:The notation
240:
239:
238:
235:
233:
229:
225:
221:
217:
209:
206:
202:
198:
194:
191:The notation
190:
187:
184:
180:
176:
172:
169:The notation
168:
167:
166:
160:
158:
156:
152:
148:
144:
140:
132:
128:
123:
113:
110:
102:
92:
88:
82:
81:
75:
70:
61:
60:
52:
45:
41:
37:
30:
19:
5715:Inequalities
5695:Ed Pegg, Jr.
5677:
5674:"Inequality"
5647:
5624:
5577:
5569:the original
5556:
5527:
5504:
5500:
5465:
5430:
5398:Inequalities
5397:
5369:
5359:
5343:
5339:
5326:
5304:(1): 65–67.
5301:
5297:
5291:
5283:
5267:
5261:
5241:
5234:
5223:. Retrieved
5211:
5198:
5187:. Retrieved
5183:
5173:
5157:. Springer.
5154:
5144:
5133:. Retrieved
5129:
5120:
5100:
5093:
5082:. Retrieved
5078:
5068:
5057:. Retrieved
5053:
5043:
5032:. Retrieved
5017:
5010:
4999:. Retrieved
4995:
4992:"Inequality"
4974:
4965:
4956:
4950:
4939:. Retrieved
4935:
4846:
4835:
4824:
4820:
4816:
4812:
4806:
4802:
4799:
4790:
4786:
4778:
4774:
4764:
4760:
4751:
4747:
4734:
4730:
4726:
4722:
4716:
4712:
4709:
4690:
4684:
4677:
4669:
4663:
4657:
4651:
4644:
4637:
4630:
4620:
4616:
4612:
4608:
4602:
4598:
4450:
4330:
4268:
4264:
4145:
4141:
4137:
4050:
4046:
3971:
3967:
3856:
3852:
3848:
3770:
3766:
3762:
3663:
3659:
3546:
3542:
3538:
3489:
3420:
3413:
3409:
3307:
3303:
3255:
3249:
3240:
3236:
3232:
3228:
3226:
3044:
2893:
2889:
2887:
2517:
2513:
2509:
2505:
2499:
2495:
2488:
2481:
2479:
2461:
2457:
2453:
2445:
2440:
2435:
2426:
2420:
2418:holds, then
2414:
2408:
2403:
2399:
2391:
2387:
2385:
2363:
2359:
2355:
2351:
2347:
2343:
2339:
2335:
2331:
2327:
2320:
2313:
2306:
2299:
2292:
2285:
2282:zigzag poset
2275:
2254:
2250:
2230:
2226:
2222:
2218:
2214:
2212:
2207:
2203:
2199:
2198:for any 1 ≤
2194:
2190:
2185:
2181:
2177:
2173:
2167:
2163:
2158:
2154:
2148:
2144:
2137:
2130:
2129:
2127:
2122:
2118:
2114:
2110:
2106:
2102:
2098:
2094:
2090:
2086:
2082:
2078:
2074:
2070:
2066:
2065:stands for "
2061:
2057:
2053:
2052:
2050:
2040:
2032:
2030:
2025:
1861:
1855:
1851:
1850:implies 0 ≤
1847:
1843:
1836:
1832:
1828:
1824:
1820:
1816:
1807:
1803:
1799:
1787:
1785:
1768:
1762:
1753:
1739:
1735:
1731:
1727:
1723:
1719:
1715:
1711:
1707:
1703:
1693:
1689:
1685:
1681:
1677:
1673:
1669:
1668:: For every
1660:
1655:transitivity
1650:
1646:
1642:
1638:
1634:
1630:
1619:
1615:
1611:
1607:
1596:
1592:
1587:
1580:
1578:
1573:transitivity
1568:
1564:
1560:
1556:
1552:
1548:
1542:antisymmetry
1537:
1533:
1529:
1525:
1521:
1517:
1506:
1502:
1495:
1491:
1487:
1483:
1467:
1456:
1454:
1441:
1437:
1433:
1429:
1422:
1418:
1414:
1410:
1404:
1400:
1388:
1384:
1380:
1376:
1369:
1365:
1361:
1357:
1351:
1347:
1343:
1339:
1333:
1328:
1323:
1319:
1315:
1311:
1307:
1305:
1289:
1284:
1280:
1260:
1249:< 0 <
1242:
1228:
1227:< 0 <
1224:
1213:
1195:
1181:
1177:
1166:
1148:
1134:
1130:
1119:
1115:
1113:
1104:
1086:
1072:
1068:
1053:
1049:
1043:
1033:
1029:
1025:
1021:
1014:
1010:
1004:
992:
986:
981:
977:
973:
969:
965:
961:
957:
953:
949:
942:
938:
934:
930:
926:
922:
918:
914:
910:
903:
899:
895:
885:
879:
875:
871:
867:
863:
851:
847:
843:
839:
835:
817:
812:
808:
804:
800:
796:
792:
788:
784:
780:
776:
769:
765:
761:
759:real numbers
746:
744:
738:
734:
730:
726:
722:
718:
702:
698:
694:
690:
686:
682:
675:
671:
667:
663:
659:
655:
648:
646:
641:
637:
633:
629:
625:
621:
614:
610:
606:
604:real numbers
601:
598:Transitivity
591:
587:
583:
579:
573:
569:
567:real numbers
560:
547:
541:
531:
527:
523:
519:
517:
506:
500:
497:
493:
489:
485:
478:
475:
471:
467:
463:
453:
444:
440:
434:
430:
426:
422:
418:
416:
383:
350:
348:
339:
333:
327:
323:
319:
316:
312:
308:
304:
300:
296:
292:
288:
281:
277:
273:
270:
266:
262:
258:
254:
250:
246:
242:
236:
231:
227:
223:
219:
215:
213:
207:
205:greater than
204:
200:
196:
192:
185:
182:
178:
174:
170:
164:
155:greater than
154:
150:
142:
136:
105:
96:
77:
18:Greater than
5565:"3rd USAMO"
5212:math.ubc.ca
5050:"Much Less"
4836:Systems of
4654:total order
4538:becomes an
4451:The set of
3306:> 0 and
3247:exercises.
3038:dot product
2398:inequality
2342:means that
2153:means that
2039:. In fact,
1796:total order
1794:and ≤ is a
1765:upper bound
1758:upper bound
1700:Dense order
1666:Total order
1511:reflexivity
1395:Taking the
1294:increasing
492:means that
470:means that
449:ordered set
425:means that
336:John Wallis
311:means that
265:means that
199:means that
177:means that
147:number line
139:mathematics
91:introducing
5709:Categories
5616:1091.26008
5277:0131103628
5225:2019-12-03
5189:2019-12-03
5135:2019-12-03
5084:2019-12-03
5059:2019-12-03
5034:2021-11-19
5001:2019-12-03
4941:2019-12-03
4916:References
4884:Inequation
4763:) < Re(
4542:. To make
4325:See also:
3250:Examples:
2882:See also:
2474:See also:
1730:such that
1714:for which
1702:: For all
1480:transitive
1440:) < ln(
1303:function.
1129:If 0 <
755:subtracted
544:properties
436:inequation
143:inequality
74:references
29:Inequation
5684:EMS Press
5600:0924-6509
5486:cite book
5449:cite book
5416:cite book
4701:−1 > 0
4660:, either
4573:≤
4567:×
4523:≤
4517:×
4201:≥
4090:≥
4005:≥
3897:≥
3705:−
3687:−
3517:≥
3441:≥
3370:−
3361:≥
3349:
3343:≥
3331:−
3276:≥
3168:∑
3122:∑
3113:≤
3062:∑
3020:⟩
3017:⋅
3011:⋅
3008:⟨
2985:⟩
2969:⟨
2966:⋅
2963:⟩
2947:⟨
2944:≤
2929:⟩
2913:⟨
2837:⋯
2747:⋯
2669:⋯
2656:⋅
2589:⋯
2003:≤
1997:×
1945:≤
1939:×
1895:≤
1889:×
1624:asymmetry
1472:reflexive
1470:which is
1463:≤ over a
1060:(or both
397:≮
364:≯
183:less than
151:less than
5645:(2004).
5582:Elsevier
5532:Archived
5368:(2006).
5348:Archived
5346:(2): 1.
5216:Archived
5205:"Fields"
4975:MacTutor
4874:brackets
4859:See also
4810:implies
4705:1 > 0
4697:1 > 0
4492:so that
4478:addition
3235:, where
2783: :
2704: :
2629: :
2529: :
2503:we have
2370:such as
2143:≤ ... ≤
1846:and 0 ≤
1823:implies
1802:, then (
1756:with an
1329:strictly
1296:function
1062:negative
1058:positive
1028:, then −
892:division
563:converse
557:Converse
548:strictly
161:Notation
99:May 2017
5686:, 2001
5608:2147066
5390:Sources
5318:2324012
4789:) ≤ Im(
4777:) = Re(
4720:, then
4641:, then
4606:, then
3032:is the
2494:, ...,
2392:relaxed
2271:
2259:
2247:
2235:
2221:+ 1 ≤ 3
2018:
1972:
1960:
1914:
1910:
1864:
1645:, then
1618:<⃥͏
1614:, then
1595:<⃥͏
1563:, then
1532:, then
1428:0 <
1421:) ≤ ln(
1409:0 <
1265:
1251:
1247:
1233:
1231:, then
1218:
1204:
1200:
1186:
1173:> 0.
1171:
1157:
1153:
1139:
1137:, then
1109:
1095:
1091:
1077:
1075:, then
783:, then
749:may be
725:, then
697:, then
670:, then
636:, then
433:; this
87:improve
5655:
5631:
5614:
5606:
5598:
5588:
5474:
5437:
5404:
5376:
5316:
5274:
5249:
5161:
5108:
5025:
4693:> 0
3000:where
2896:of an
2372:Python
2358:, and
2217:< 2
1966:, but
1760:has a
1750:subset
1490:, and
1478:, and
1364:⇔ 0 ≤
1300:domain
753:to or
649:either
76:, but
5535:(PDF)
5524:(PDF)
5351:(PDF)
5336:(PDF)
5314:JSTOR
5219:(PDF)
5208:(PDF)
4755:, if
4699:; so
4676:0 ≤ −
4486:field
4484:is a
3040:; In
2406:, if
2388:sharp
2346:<
2330:<
2319:<
2312:>
2305:<
2298:>
2291:<
2257:<
2233:<
2117:<
2113:<
2101:<
2093:<
2085:<
2077:<
2069:<
2060:<
2056:<
1862:Both
1792:field
1763:least
1738:<
1734:<
1718:<
1649:<
1641:<
1633:<
1610:<
1459:is a
1436:⇔ ln(
1432:<
1417:⇔ ln(
1318:>
1310:<
1283:= ln
878:>
866:<
850:<
838:<
751:added
733:<
721:<
701:<
685:<
674:<
666:<
530:>
522:<
340:above
195:>
173:<
141:, an
5653:ISBN
5629:ISBN
5596:ISSN
5586:ISBN
5492:link
5472:ISBN
5455:link
5435:ISBN
5422:link
5402:ISBN
5374:ISBN
5272:ISBN
5247:ISBN
5159:ISBN
5106:ISBN
5023:ISBN
4847:The
4783:and
4769:, or
4703:and
4695:and
4683:0 ≤
4662:0 ≤
4643:0 ≤
4636:0 ≤
4634:and
4629:0 ≤
4480:and
4302:>
3804:>
3714:>
3662:and
3641:>
3239:and
2892:and
2464:≥ −1
2249:and
2172:for
2073:and
1962:are
1912:and
1842:0 ≤
1786:If (
1706:and
1672:and
1637:and
1555:and
1524:and
1403:and
1391:≥ 0.
1375:0 ≤
1356:0 ≤
1350:and
1290:Any
1118:and
1052:and
1013:and
968:and
956:and
929:and
917:and
890:and
870:and
842:and
799:and
689:and
662:and
628:and
586:and
572:and
526:and
458:.
443:and
303:or
153:and
125:The
5693:by
5612:Zbl
5509:doi
5306:doi
4785:Im(
4773:Re(
4759:Re(
4672:≤ 0
4667:or
4627:if
4596:if
4588:an
4263:If
4136:If
4045:If
3966:If
3847:If
3761:If
3537:If
3488:If
3419:If
3302:If
2448:≥ 0
2020:an
1798:on
1752:of
1726:in
1710:in
1688:or
1676:in
1629:if
1606:if
1547:if
1516:if
1494:in
1465:set
1324:and
1223:If
1176:If
1067:If
1064:):
1032:≥ −
1020:If
948:If
909:If
862:If
834:If
775:If
717:If
681:If
654:If
647:If
620:If
496:is
474:is
315:is
295:or
269:is
257:or
249:or
203:is
181:is
137:In
129:of
5711::
5682:,
5676:,
5610:.
5604:MR
5602:.
5594:.
5584:.
5555:.
5530:.
5526:.
5505:34
5503:,
5488:}}
5484:{{
5451:}}
5447:{{
5418:}}
5414:{{
5342:.
5338:.
5312:.
5302:97
5300:.
5214:.
5210:.
5182:.
5153:.
5128:.
5077:.
5052:.
4994:.
4983:^
4973:.
4934:.
4923:^
4844:.
4828:.
4823:+
4819:≤
4815:+
4805:≤
4750:≤
4733:+
4729:≤
4725:+
4715:≤
4645:ab
4619:+
4615:≤
4611:+
4601:≤
4305:1.
4267:,
4144:,
4140:,
4049:,
3970:,
3855:,
3851:,
3765:,
3666:,
3644:2.
3545:,
3541:,
3416:).
3346:ln
3258:,
3227:A
2516:≤
2512:≤
2508:≤
2487:,
2460:.
2456:∈
2444:.
2438:∈
2362:≤
2354:=
2350:,
2338:≤
2334:=
2273:.
2210:.
2206:≤
2202:≤
2189:≤
2170:+1
2162:≤
2136:≤
2125:.
2121:−
2109:−
2097:+
1854:×
1835:+
1831:≤
1827:+
1819:≤
1692:≤
1684:≤
1680:,
1567:≤
1559:≤
1551:≤
1536:=
1528:≤
1520:≤
1505:≤
1486:,
1474:,
1444:).
1425:).
1413:≤
1387:≥
1383:⇔
1379:≤
1368:≤
1360:≤
1322:)
1314:,
1202:≥
1180:≤
1155:≥
1133:≤
1093:≥
1071:≤
1024:≤
1017::
997:.
976:≥
966:bc
964:≥
962:ac
952:≤
937:≤
927:bc
925:≤
923:ac
913:≤
906::
898:,
880:ay
876:ax
852:ay
848:ax
811:−
807:≤
803:−
795:+
791:≤
787:+
779:≤
772::
768:,
764:,
737:+
729:+
693:≤
658:≤
640:≤
632:≤
624:≤
617::
613:,
609:,
590:≥
582:≤
576::
553:.
488:≫
466:≪
451:.
421:≠
386:,
330:).
307:≧
299:⩾
291:≥
284:).
261:≦
253:⩽
245:≤
157:.
5661:.
5637:.
5618:.
5511::
5494:)
5480:.
5457:)
5443:.
5424:)
5410:.
5382:.
5344:7
5320:.
5308::
5280:.
5255:.
5228:.
5192:.
5167:.
5138:.
5114:.
5087:.
5062:.
5037:.
5004:.
4959:.
4944:.
4825:c
4821:b
4817:c
4813:a
4807:b
4803:a
4793:)
4791:b
4787:a
4781:)
4779:b
4775:a
4767:)
4765:b
4761:a
4752:b
4748:a
4735:c
4731:b
4727:c
4723:a
4717:b
4713:a
4691:i
4685:a
4678:a
4670:a
4664:a
4658:a
4648:.
4638:b
4631:a
4624:;
4621:c
4617:b
4613:c
4609:a
4603:b
4599:a
4576:)
4570:,
4564:,
4561:+
4558:,
4554:C
4550:(
4526:)
4520:,
4514:,
4511:+
4508:,
4504:C
4500:(
4490:≤
4463:C
4297:a
4293:b
4289:+
4284:b
4280:a
4269:b
4265:a
4249:.
4244:a
4241:2
4237:c
4233:+
4228:c
4225:2
4221:b
4217:+
4212:b
4209:2
4205:a
4196:c
4193:2
4189:c
4185:+
4180:b
4177:2
4173:b
4169:+
4164:a
4161:2
4157:a
4146:c
4142:b
4138:a
4122:.
4117:a
4114:e
4110:b
4106:+
4101:b
4098:e
4094:a
4085:b
4082:e
4078:b
4074:+
4069:a
4066:e
4062:a
4051:b
4047:a
4031:.
4026:a
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4018:+
4013:b
4009:a
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3996:b
3992:+
3987:a
3983:a
3972:b
3968:a
3952:.
3947:3
3943:/
3939:)
3936:z
3933:+
3930:y
3927:+
3924:x
3921:(
3916:)
3912:z
3909:y
3906:x
3902:(
3892:z
3888:z
3882:y
3878:y
3872:x
3868:x
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3833:.
3828:p
3823:)
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3396:.
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3326:p
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3304:x
3288:.
3285:x
3282:+
3279:1
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3267:e
3256:x
3241:b
3237:a
3233:a
3208:.
3204:)
3198:2
3193:i
3189:v
3183:n
3178:1
3175:=
3172:i
3163:(
3158:)
3152:2
3147:i
3143:u
3137:n
3132:1
3129:=
3126:i
3117:(
3108:2
3103:)
3097:i
3093:v
3087:i
3083:u
3077:n
3072:1
3069:=
3066:i
3057:(
3045:R
3014:,
2988:,
2981:v
2977:,
2973:v
2959:u
2955:,
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2806:1
2802:a
2794:=
2791:Q
2764:n
2758:n
2754:a
2750:+
2744:+
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2726:1
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2715:=
2712:A
2684:n
2677:n
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2647:a
2640:=
2637:G
2605:n
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2452:∀
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2436:a
2433:∀
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2424:⇔
2421:ψ
2415:φ
2412:⇒
2409:ψ
2404:ψ
2400:φ
2376:C
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2006:)
2000:,
1994:,
1991:+
1988:,
1984:C
1980:(
1968:≤
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1942:,
1936:,
1933:+
1930:,
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882:.
872:a
868:y
864:x
854:.
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836:x
815:.
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781:b
777:a
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