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Partea Intreaga, Partea Fractionara, Modulul Unui Numar Real

The document discusses three topics: 1. The integer part of a real number, defined as the greatest integer less than the number. Properties include that the integer part plus the fractional part equals the original number. 2. The fractional part of a real number, defined as the number minus its integer part. Properties include that it is between 0 and 1. 3. The absolute value of a real number, defined as the number if positive and its opposite if negative. Properties include that it is always positive and equals the absolute value of the opposite number.

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Marta Kita
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1K views2 pages

Partea Intreaga, Partea Fractionara, Modulul Unui Numar Real

The document discusses three topics: 1. The integer part of a real number, defined as the greatest integer less than the number. Properties include that the integer part plus the fractional part equals the original number. 2. The fractional part of a real number, defined as the number minus its integer part. Properties include that it is between 0 and 1. 3. The absolute value of a real number, defined as the number if positive and its opposite if negative. Properties include that it is always positive and equals the absolute value of the opposite number.

Uploaded by

Marta Kita
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOC, PDF, TXT or read online on Scribd
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PARTEA INTREAGA, PARTEA FRACTIONARA,

MODULUL UNUI NUMAR REAL

1. Partea intreaga a unui numar real


a) Definitie
[x]=k, k  Z si k < x < k + 1

b) Proprietati
Daca x, y  R, atunci:
 x − 1 < [x] < x
 [x + y] > [x] + [y]
 [x + y] = [x] + y,  y  Z
 [x] + [−x] = 0 daca x  Z si [x] + [-x] = −1 daca x  R − Z
 1  2  3  n  1
 [x] +  x  n  +  x  n  +  x  n  +…+  x  n  = [nx],  n  N, n  0

2. Partea fractionara a unui numar real


a) Definitie
Daca x  R, partea fractionara a lui x este numarul notat {x} = x − [x]

b) Proprietati
 0 < {x} < 1,  x  R
 {x} = 0  x  Z

3. Modulul unui numar real


a) Definitie
|x| = x daca x > 0 si |x| = −x daca x < 0

b) Proprietati
 |x| > 0 ,  x  R
 |x| = 0  x = 0
 |x| = |−x| ,  x  R
 |x| = |y|  x = y sau x = −y
 |x · y| > |x| · |y|,  x, y  R
x x
 y = y ,  x, y  R, y  0

 [x + y] < [x] + [y] ,  x, y  R


MODULUL UNUI NUMAR REAL

|x| > 0 , x  R
a) DEFINITIE
|x| = 0  x = 0
|x| = x daca x > 0 si |x| = −x daca x < 0
|x| = |−x| , x  R
|x| = |y|  x = y sau x = −y
b)· Py|ROPRIETATI
|x > |x| · |y|, x, y  R
= , x, y  R, y  0
|x + y| < |x| + |y| , x, y  R

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