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30-838: [REDACTED] Look up below in Wiktionary, the free dictionary. Below may refer to: Earth Ground (disambiguation) Soil Floor Bottom (disambiguation) Less than Temperatures below freezing Hell or underworld People with the surname [ edit ] Ernst von Below (1863–1955), German World War I general Fred Below (1926–1988), American blues drummer Fritz von Below (1853–1918), German World War I general Gerd-Paul von Below (1892-1953), German World War II general Otto von Below (1857–1944), German World War I general Nicolaus von Below (1907–1983), German adjutant of Adolf Hitler Other uses [ edit ] Below (album) ,

60-399: A ≯ b , {\displaystyle a\ngtr b,} the symbol for "greater than" bisected by a slash, "not". The same is true for not less than , a ≮ b . {\displaystyle a\nless b.} The notation a ≠ b means that a is not equal to b ; this inequation sometimes is considered a form of strict inequality. It does not say that one

90-731: A 1 , a 2 , ..., a n we have where they represent the following means of the sequence: The Cauchy–Schwarz inequality states that for all vectors u and v of an inner product space it is true that | ⟨ u , v ⟩ | 2 ≤ ⟨ u , u ⟩ ⋅ ⟨ v , v ⟩ , {\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle }

120-426: A and b : The transitive property of inequality states that for any real numbers a , b , c : If either of the premises is a strict inequality, then the conclusion is a strict inequality: A common constant c may be added to or subtracted from both sides of an inequality. So, for any real numbers a , b , c : In other words, the inequality relation is preserved under addition (or subtraction) and

150-423: A is not equal to b . These relations are known as strict inequalities , meaning that a is strictly less than or strictly greater than b . Equality is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: In the 17th and 18th centuries, personal notations or typewriting signs were used to signal inequalities. For example, In 1670, John Wallis used

180-512: A ≤ b , then a + c ≤ b + c "). Sometimes the lexicographical order definition is used: It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c . Systems of linear inequalities can be simplified by Fourier–Motzkin elimination . The cylindrical 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

210-415: A 2021 album by Beartooth Below (film) , a 2002 film by David Twohy Below (video game) , a video game by Capybara Games Below Records , a record label Below Par Records , a record label The Great Below , a song by Nine Inch Nails See also [ edit ] Belov Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

240-415: A 2021 album by Beartooth Below (film) , a 2002 film by David Twohy Below (video game) , a video game by Capybara Games Below Records , a record label Below Par Records , a record label The Great Below , a song by Nine Inch Nails See also [ edit ] Belov Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

270-565: A single horizontal bar above 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 Pierre Bouguer '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" (⩽). The relation not greater than can also be represented by

300-488: Is a binary relation ≤ over a set P which is reflexive , antisymmetric , and transitive . That is, for all a , b , and c in P , it must satisfy the three following clauses: A set with a partial order is called a partially ordered set . Those are the very basic axioms that every kind of order has to satisfy. A strict partial order is a relation < that satisfies: Some types of partial orders are specified by adding further axioms, such as: If ( F , +, ×)

330-673: Is a field and ≤ is a total order on F , then ( F , +, ×, ≤) is called an ordered field if and only if: Both ⁠ ( Q , + , × , ≤ ) {\displaystyle (\mathbb {Q} ,+,\times ,\leq )} ⁠ and ⁠ ( R , + , × , ≤ ) {\displaystyle (\mathbb {R} ,+,\times ,\leq )} ⁠ are ordered fields , but ≤ cannot be defined in order to make ⁠ ( C , + , × , ≤ ) {\displaystyle (\mathbb {C} ,+,\times ,\leq )} ⁠ an ordered field , because −1

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360-420: Is a field , but it is impossible to define any relation ≤ so that ( C , + , × , ≤ ) {\displaystyle (\mathbb {C} ,+,\times ,\leq )} becomes an ordered field . To make ( C , + , × , ≤ ) {\displaystyle (\mathbb {C} ,+,\times ,\leq )} an ordered field , it would have to satisfy

390-508: Is an inequality containing terms of the form a , where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises. Examples: 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: The set of complex numbers C {\displaystyle \mathbb {C} } with its operations of addition and multiplication

420-524: Is equivalent to a i ≤ a j 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 4 x < 2 x + 1 ≤ 3 x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < ⁠ 1 / 2 ⁠ and x ≥ −1 respectively, which can be combined into

450-447: Is greater than the other; it does not even require a and b to be member of an ordered set . In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude . This implies that the lesser value can be neglected with little effect on the accuracy of an approximation (such as the case of ultrarelativistic limit in physics). In all of

480-471: Is said to be sharp if it cannot be relaxed and still be valid in general. Formally, a universally quantified inequality φ is called sharp if, for every valid universally quantified inequality ψ , if ψ ⇒ φ holds, then ψ ⇔ φ also holds. For instance, the inequality ∀ a ∈ R . a ≥ 0 is sharp, whereas the inequality ∀ a ∈ R . a ≥ −1 is not sharp. There are many inequalities between means. For example, for any positive numbers

510-410: 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 a strictly monotonically decreasing function. A few examples of this rule are: A (non-strict) partial order

540-781: Is the inner product . Examples of inner products include the real and complex dot product ; In Euclidean space R with the standard inner product, the Cauchy–Schwarz inequality is ( ∑ i = 1 n u i v i ) 2 ≤ ( ∑ i = 1 n u i 2 ) ( ∑ i = 1 n v i 2 ) . {\displaystyle {\biggl (}\sum _{i=1}^{n}u_{i}v_{i}{\biggr )}^{2}\leq {\biggl (}\sum _{i=1}^{n}u_{i}^{2}{\biggr )}{\biggl (}\sum _{i=1}^{n}v_{i}^{2}{\biggr )}.} A power inequality

570-484: Is the square of i and would therefore be positive. Besides being an ordered field, R also has the Least-upper-bound property . In fact, R can be defined as the only ordered field with that quality. The notation a < b < c stands for " a < b and b < c ", from which, by the transitivity property above, it also follows that a < c . By the above laws, one can add or subtract

600-420: Is used more often with compatible relations, like <, =, ≤. For instance, a < b = c ≤ d means that a < b , b = c , and c ≤ d . This notation exists in a few programming languages such as Python . In contrast, in programming languages that provide an ordering on the type of comparison results, such as C , even homogeneous chains may have a completely different meaning. An inequality

630-556: The additive inverse states that for any real numbers a and b : If both numbers are positive, then the inequality relation between the multiplicative inverses is opposite of that between the original numbers. More specifically, for any non-zero real numbers a and b that are both positive (or both negative ): All of the cases for the signs of a and b can also be written in chained notation , as follows: Any monotonically increasing function , by its definition, may be applied to both sides of an inequality without breaking

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660-534: The cases above, any two symbols mirroring each other are symmetrical; a < b and b > a are equivalent, etc. Inequalities are governed by the following properties . 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 strictly monotonic functions . The relations ≤ and ≥ are each other's converse , meaning that for any real numbers

690-431: The final solution −1 ≤ x < ⁠ 1 / 2 ⁠ . Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For example, the defining condition of a zigzag poset is written as a 1 < a 2 > a 3 < a 4 > a 5 < a 6 > ... . Mixed chained notation

720-421: The following two properties: Because ≤ is a total order , for any number a , either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ − a ). In either case 0 ≤ a ; this means that i > 0 and 1 > 0 ; so −1 > 0 and 1 > 0 , which means (−1 + 1) > 0; contradiction. However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if

750-763: The 💕 [REDACTED] Look up below in Wiktionary, the free dictionary. Below may refer to: Earth Ground (disambiguation) Soil Floor Bottom (disambiguation) Less than Temperatures below freezing Hell or underworld People with the surname [ edit ] Ernst von Below (1863–1955), German World War I general Fred Below (1926–1988), American blues drummer Fritz von Below (1853–1918), German World War I general Gerd-Paul von Below (1892-1953), German World War II general Otto von Below (1857–1944), German World War I general Nicolaus von Below (1907–1983), German adjutant of Adolf Hitler Other uses [ edit ] Below (album) ,

780-404: The inequality relation (provided that both expressions are in the domain 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

810-466: The real numbers are an ordered group under addition. The properties that deal with multiplication and division state that for any real numbers, a , b and non-zero 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 an ordered field . For more information, see § Ordered fields . The property for

840-467: 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 to a − e < b < c − e . This notation can be generalized to any number of terms: for instance, a 1 ≤ a 2 ≤ ... ≤ a n means that a i ≤ a i +1 for i = 1, 2, ..., n − 1. By transitivity, this condition

870-503: The title Below . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Below&oldid=1059821480 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages below From Misplaced Pages,

900-597: The title Below . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Below&oldid=1059821480 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Less than There are several different notations used to represent different kinds of inequalities: In either case,

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