This article is about relations "greater than" and "less than". For the relation "not equal", seeInequation."Less than" redirects here; not to be confused withLess Than (song)."≪" redirects here; not to be confused withAbsolute continuity of measures.
Inmathematics, aninequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions.[1] It is used most often to compare two numbers on thenumber line by their size. The main types of inequality areless than andgreater than (denoted by< and>, respectively theless-than andgreater-than signs).
There are several different notations used to represent different kinds of inequalities:
The notationa <b means thata isless thanb.
The notationa >b means thata isgreater thanb.
In either case,a is not equal tob. These relations are known asstrict inequalities,[1] meaning thata is strictly less than or strictly greater thanb. Equality is excluded.
In contrast to strict inequalities, there are two types of inequality relations that are not strict:
The notationa ≤b ora ⩽b ora ≦b means thata isless than or equal tob (or, equivalently, at mostb, or not greater thanb).
The notationa ≥b ora ⩾b ora ≧b means thata isgreater than or equal tob (or, equivalently, at leastb, or not less thanb).
In the 17th and 18th centuries, personal notations or typewriting signs were used to signal inequalities.[2] For example, In 1670,John Wallis used a single horizontal barabove 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 inPierre Bouguer's work .[3] 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" (⩽).
The relationnot greater than can also be represented by the symbol for "greater than" bisected by a slash, "not". The same is true fornot less than,
The notationa ≠b means thata is not equal tob; thisinequation sometimes is considered a form of strict inequality.[4] It does not say that one is greater than the other; it does not even requirea andb to be member of anordered set.
In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another,[5] normally by severalorders of magnitude.
The notationa ≪b means thata ismuch less thanb.[6]
The notationa ≫b means thata ismuch greater thanb.[7]
This implies that the lesser value can be neglected with little effect on the accuracy of anapproximation (such as the case ofultrarelativistic limit in physics).
In all of the cases above, any two symbols mirroring each other are symmetrical;a <b andb >a are equivalent, etc.
Inequalities are governed by the followingproperties. 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 tostrictlymonotonic functions.
Ifx <y anda > 0, thenax <ay.Ifx <y anda < 0, thenax >ay.
The properties that deal withmultiplication anddivision state that for any real numbers,a,b and non-zeroc:
Ifa ≤b andc > 0, thenac ≤bc anda/c ≤b/c.
Ifa ≤b andc < 0, thenac ≥bc anda/c ≥b/c.
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 anordered field. For more information, see§ Ordered fields.
If both numbers are positive, then the inequality relation between themultiplicative inverses is opposite of that between the original numbers. More specifically, for any non-zero real numbersa andb that are bothpositive (or bothnegative):
Ifa ≤b, then1/a ≥1/b.
All of the cases for the signs ofa andb can also be written inchained notation, as follows:
Anymonotonically increasingfunction, by its definition,[9] may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in thedomain 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 function.
If the inequality is strict (a <b,a >b)and 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 astrictly monotonically decreasing function.
A few examples of this rule are:
Raising both sides of an inequality to a powern > 0 (equiv., −n < 0), whena andb are positive real numbers:
0 ≤a ≤b ⇔ 0 ≤an ≤bn.
0 ≤a ≤b ⇔a−n ≥b−n ≥ 0.
Taking thenatural logarithm on both sides of an inequality, whena andb are positive real numbers:
0 <a ≤b ⇔ ln(a) ≤ ln(b).
0 <a <b ⇔ ln(a) < ln(b).
(this is true because the natural logarithm is a strictly increasing function.)
Both and areordered fields, but≤ cannot be defined in order to make anordered field,[12] because −1 is the square ofi and would therefore be positive.
Besides being an ordered field,R also has theLeast-upper-bound property. In fact,R can be defined as the only ordered field with that quality.[13]
The notationa <b <c stands for "a <b andb <c", from which, by the transitivity property above, it also follows thata <c. 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,a <b +e <c is equivalent toa −e <b <c −e.
This notation can be generalized to any number of terms: for instance,a1 ≤a2 ≤ ... ≤an means thatai ≤ai+1 fori = 1, 2, ...,n − 1. By transitivity, this condition is equivalent toai ≤aj for any 1 ≤i ≤j ≤n.
When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolatex in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yieldingx <1/2 andx ≥ −1 respectively, which can be combined into the final solution −1 ≤x <1/2.
Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is thelogical conjunction of the inequalities between adjacent terms. For example, the defining condition of azigzag poset is written asa1 <a2 >a3 <a4 >a5 <a6 > ... . Mixed chained notation is used more often with compatible relations, like <, =, ≤. For instance,a <b =c ≤d means thata <b,b =c, andc ≤d. This notation exists in a fewprogramming languages such asPython. In contrast, in programming languages that provide an ordering on the type of comparison results, such asC, even homogeneous chains may have a completely different meaning.[14]
An inequality is said to besharp if it cannot berelaxed and still be valid in general. Formally, auniversally quantified inequalityφ is called sharp if, for every valid universally quantified inequalityψ, ifψ⇒φ holds, thenψ⇔φ also holds. For instance, the inequality∀a ∈R.a2 ≥ 0 is sharp, whereas the inequality∀a ∈R.a2 ≥ −1 is not sharp.[citation needed]
The Cauchy–Schwarz inequality states that for all vectorsu andv of aninner product space it is true thatwhere is theinner product. Examples of inner products include the real and complexdot product; InEuclidean spaceRn with the standard inner product, the Cauchy–Schwarz inequality is
Apower inequality is an inequality containing terms of the formab, wherea andb are real positive numbers or variable expressions. They often appear inmathematical olympiads exercises.
Examples:
For any realx,
Ifx > 0 andp > 0, then In the limit ofp → 0, the upper and lower bounds converge to ln(x).
Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names:
Because ≤ is atotal order, for any numbera, either0 ≤a ora ≤ 0 (in which case the first property above implies that0 ≤ −a). In either case0 ≤a2; this means thati2 > 0 and12 > 0; so−1 > 0 and1 > 0, which means (−1 + 1) > 0; contradiction.
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "ifa ≤b, thena +c ≤b +c"). Sometimes thelexicographical order definition is used:
a ≤b, if
Re(a) < Re(b), or
Re(a) = Re(b) andIm(a) ≤ Im(b)
It can easily be proven that for this definitiona ≤b impliesa +c ≤b +c.
Thecylindrical algebraic decomposition 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 isdoubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.
^Halmaghi, Elena; Liljedahl, Peter. "Inequalities in the History of Mathematics: From Peculiarities to a Hard Discipline".Proceedings of the 2012 Annual Meeting of the Canadian Mathematics Education Study Group.
^Simovici, Dan A. & Djeraba, Chabane (2008)."Partially Ordered Sets".Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics. Springer.ISBN9781848002012.
^Brian W. Kernighan and Dennis M. Ritchie (Apr 1988).The C Programming Language. Prentice Hall Software Series (2nd ed.). Englewood Cliffs/NJ: Prentice Hall.ISBN0131103628. Here: Sect.A.7.9Relational Operators, p.167: Quote: "a<b<c is parsed as (a<b)<c"
Hardy, G., Littlewood J. E., Pólya, G. (1999).Inequalities. Cambridge Mathematical Library, Cambridge University Press.ISBN0-521-05206-8.{{cite book}}: CS1 maint: multiple names: authors list (link)
Beckenbach, E. F., Bellman, R. (1975).An Introduction to Inequalities. Random House Inc.ISBN0-394-01559-2.{{cite book}}: CS1 maint: multiple names: authors list (link)
Grinshpan, A. Z. (2005), "General inequalities, consequences, and applications",Advances in Applied Mathematics,34 (1):71–100,doi:10.1016/j.aam.2004.05.001
