- A k s i o m a 4.1. - Çdo numri real a
i përket një dhe vetëm një pikë A të boshtit numerik x' x dhe anasjelltas, çdo pike A të boshtit numerik x' x i përket një dhe vetëm një numër real a .
- Në bazë të kësaj aksiome konkludohet se elementet e bashkësisë
plotësisht e mbulojnë boshtin numerik x'x, e shi për këtë bashkësia quhet kontinuumi numerik ose kontinuumi aritmetik, kurse bashkësia e pikave të boshtit x' x quhet kontinuumi linear ose kontinuumi gjeometrik.
4.I. VLERA ABSOLUTE E NUMRIT REAL
- P ë r k u f i z i m i 4.1.1. - Vlera absolute (moduli) e numrit real a
![{\displaystyle \scriptstyle \in }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a4e4b18f88a540b881dab023807160f159f8a9) përcaktohet me relacionin :
.
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(6)
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- Nga ky përkufizim drejtpërsëdrejti rrjedhin këto relacione:
-
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(a1) a![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) ![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) -a ,
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(a2) a-b![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) ![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) b-a ,
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(a3) a![](//upload.wikimedia.org/wikibooks/sq/5/52/Mavog%C3%ABlbarabart.PNG) a ,
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(a4) - a![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) a,
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(a5) a b![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) ![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) a b ,
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(a6) ![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) ![{\displaystyle \textstyle \mathrm {\frac {a}{b}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef955aea745dd00c37cfddf6f42031f6370d0265) ![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) ![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) ![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) ![{\displaystyle \textstyle \mathrm {\frac {a}{b}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef955aea745dd00c37cfddf6f42031f6370d0265) , b 0.
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- që vlejnë për çdo dy numra realë a, b.
- Në rastin e përgjithshëm me relacionin
x![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) r, ku 0 r![{\displaystyle \scriptstyle \in }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a4e4b18f88a540b881dab023807160f159f8a9)
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(7)
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- shprehet jobarazimi i dyfishtë:
-r x r,
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(7a)
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- ndërkaq me relacionin
x-a![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) r, ku a![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) r![{\displaystyle \scriptstyle \in }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a4e4b18f88a540b881dab023807160f159f8a9)
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(8)
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- jobarazimi i dyfishtë:
- Vetitë e vlerës absolute të numrave realë:
- T e o r e m a 4.1.1. - Vlera absolute e shumës së dy numrave realë a, b nuk është më e madhe se shuma e vlerave absolute të tyre, pra:
( a,b![{\displaystyle \scriptstyle \in }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a4e4b18f88a540b881dab023807160f159f8a9) ) a+b![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) ![](//upload.wikimedia.org/wikibooks/sq/5/52/Mavog%C3%ABlbarabart.PNG) a + b .
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(9)
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- V ë r t e t i m: Dallojmë këto dy raste: (a) kur a+b
0 dhe (b) kur a+b<0.
- (a) Nëse a+b
0, atëherë në bazë të relacioneve (6) dhe (a3) del:
a+b![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) a+b![](//upload.wikimedia.org/wikibooks/sq/5/52/Mavog%C3%ABlbarabart.PNG) a + b![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) ![{\displaystyle \scriptstyle {\Rightarrow }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5320e9341c58a8f7d0e74df1fb5234f97a19d3) a+b![{\displaystyle \scriptstyle \mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbe6d41da20053a02be3cc4822039754fe0215) ![](//upload.wikimedia.org/wikibooks/sq/5/52/Mavog%C3%ABlbarabart.PNG) a + b ;
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