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Corollary: Properties of the Absolute Value edit contribute as guest [id:619] The absolute value has the following properties:
\(|-x|=|x|\) for all \(x\in\mathbb R\).
\(|xy|=|x||y|\) for all \(x,y\in\mathbb R\).
\(|\frac xy|=\frac{|x|}{|y|}\) for all \(x,y\in\mathbb R\), \(y\neq 0\).
Please note that another, very important property of the absolute value is the triangle inequality \(|x+y|\le |x|+|y|\) for all \(x,y\in\mathbb R\), which is proven here .
Further Reading
[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983
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1081: absolute value of rational numbers 890: addition of integers 842: addition of natural numbers 892: algebraic structure of integers together with addition and multiplication 841: algebraic structure of natural numbers together with addition 877: algebraic structure of natural numbers together with multiplication 668: axiom of associativity 672: axiom of commutativity 682: axiom of distributivity 669: axiom of existence of an identity 670: axiom of existence of inverse elements 771: bijective function 571: binary relations 837: cancellation property 748: cartesian product 1072: cauchy sequence 553: commutative (abelian) group 706: commutative monoid 880: commutative ring 1103: commutative semigroup 888: construction of fields from integral domains 839: construction of groups from commutative and cancellative semigroups 148: convergent sequences and limits 827: cosets 844: definition of integers 1033: definition of rational numbers 1105: definition of real numbers 574: equivalence relation, equivalent classes, partitions, representative elements, quotient sets 191: factor groups 274: factor rings 557: field 671: group 679: group homomorphism 836: groupoid (magma) 401: homomorphism 1062: ideal 661: identity, neutral element, left identity, right identity 769: injective function 575: irreflexive, asymmetric and antisymmetric relation 838: isomorphic semigroups 614: metric (distance) 617: metric space 659: monoid 891: multiplication of integers 876: multiplication of natural numbers 273: normal subgroups 849: open ball, neighborhood 1075: order relation for integers 697: order relation for natural numbers 1076: order relation for rational numbers 1107: order relation for real numbers 747: ordered pair, n-tuple 723: ordinal number 504: peano axioms 576: preorder (quasiorder), partial and total order, poset and chain 1308: properties of binary relations between two sets 829: properties of cosets 721: properties of transitive sets 572: reflexive, symmetric and transitive relation 683: ring 885: ring homomorphism 660: semigroup 874: sequence 664: set of natural numbers (peano) 550: set, set element, empty set 718: set-theoretic definitions of natural numbers 887: subfield 554: subgroup 552: subset, superset, union, intersection, set difference, set complement, power set 774: successor of oridinal 770: surjective function 1090: the absolute value makes the set of rational numbers a metric space. 1030: the distributivity law for natural numbers 657: the proving principle of complete induction (variant 1) 1101: the unit ring of all rational cauchy sequences 592: total maps (functions) 720: transitive set 821: unit and unit group, zero divisor and integral domain 1427: zermelo-fraenkel axioms 879: zero ring
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