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1.2.7. Ìîäóëü ä³éñíîãî ÷èñëà<br />
Ìîäóëåì ä³éñíîãî ÷èñëà x íàçèâàºòüñÿ ÷èñëî x, ÿêùî<br />
x ≥ 0, ³ ïðîòèëåæíå éîìó ÷èñëî –x, ÿêùî x < 0. Ìîäóëü ÷èñëà<br />
õ ïîçíà÷àºòüñÿ ñèìâîëîì x ³ ÷èòàºòüñÿ “ìîäóëü ÷èñëà õ”.<br />
Îòæå, çà îçíà÷åííÿì<br />
⎧x, x ≥ 0<br />
x = ⎨<br />
⎩ − x, x < 0<br />
.<br />
Ç ãåîìåòðè÷íî¿ òî÷êè çîðó ìîäóëü ÷èñëà x îçíà÷ຠâ³äñòàíü<br />
òî÷êè ÷èñëîâî¿ îñ³ ç àáñöèñîþ õ äî òî÷êè â³äë³êó 0.<br />
Îñíîâí³ âëàñòèâîñò³ ìîäóëÿ ä³éñíîãî ÷èñëà òàê³:<br />
1. x ≤ x .<br />
2. −x ≤ x .<br />
3. x < δ ⇔ − δ < x < δ .<br />
Ñë³ä ñêàçàòè, ùî ïîíÿòòÿ ìîäóëÿ ä³éñíîãî ÷èñëà ³ éîãî<br />
îñíîâí³ âëàñòèâîñò³ âèâ÷àþòüñÿ â êóðñ³ ìàòåìàòèêè ñåðåäíüî¿<br />
øêîëè. Äëÿ âèâ÷åííÿ êóðñó âèùî¿ ìàòåìàòèêè öüîãî<br />
íåäîñòàòíüî. Ðîçãëÿíåìî ³íø³ âëàñòèâîñò³ ìîäóëÿ ÷èñëà.<br />
Äëÿ öüîãî ñôîðìóëþºìî ³ äîâåäåìî òàê³ òåîðåìè.<br />
Òåîðåìà 1.2.2. Ìîäóëü ñóìè äâîõ ÷èñåë õ ³ ó íå ïåðåâèùóº<br />
ñóìè ìîäóë³â öèõ ÷èñåë, òîáòî x + y ≤ x + y .<br />
Äîâåäåííÿ. Ðîçãëÿíåìî äâà âèïàäêè:<br />
1) x+ y≥0 ⇒ x+ y = ( x+ y)<br />
≤ x + y ;<br />
2) x+ y< 0 ⇒ x+ y =− ( x+ y)<br />
≤ x + y .<br />
Ïðè äîâåäåíí³ òåîðåìè ìè ñêîðèñòóâàëèñÿ ïåðøèìè äâîìà<br />
îñíîâíèìè âëàñòèâîñòÿìè ìîäóëÿ ä³éñíîãî ÷èñëà.<br />
Òåîðåìà 1.2.3. Ìîäóëü ð³çíèö³ äâîõ ÷èñåë õ ³ ó á³ëüøå<br />
àáî äîð³âíþº ð³çíèö³ ìîäóë³â öèõ ÷èñåë, òîáòî<br />
x−y ≥ x − y .<br />
Äîâåäåííÿ. Çàïèøåìî î÷åâèäíó òîòîæí³ñòü:<br />
x = ( x− y)<br />
+ y .<br />
Íà îñíîâ³ òåîðåìè 1.2.2 ìàºìî<br />
x = ( x− y)<br />
+ y ≤ x− y + y ,<br />
çâ³äê³ëÿ ³ âèïëèâຠíåð³âí³ñòü, ÿêó òðåáà áóëî äîâåñòè.<br />
Òåîðåìà 1.2.4. Ìîäóëü äîáóòêó äâîõ ÷èñåë õ ³ ó äîð³âíþº<br />
äîáóòêó ìîäóë³â öèõ ÷èñåë, òîáòî<br />
x⋅ y = x ⋅ y .<br />
Öþ òåîðåìó ðåêîìåíäóºìî ÷èòà÷àì äîâåñòè ñàìîñò³éíî.<br />
Äîâåäåííÿ òåîðåìè 1.2.4 ïðîñòå, àëå ïîòðåáóº ðîçãëÿäàííÿ<br />
âñ³õ ìîæëèâèõ âèïàäê³â ³ âèêîðèñòàííÿ îçíà÷åííÿ ìîäóëÿ<br />
÷èñëà.<br />
Òåîðåìà 1.2.5. Ìîäóëü ñòåïåíÿ ÷èñëà õ äîð³âíþº ñòåïåíþ<br />
ìîäóëÿ öüîãî ÷èñëà, òîáòî<br />
n<br />
x<br />
=<br />
Äîâåäåííÿ ö³º¿ òåîðåìè âèïëèâຠ³ç òåîðåìè 1.2.4.<br />
Òåîðåìà 1.2.6. Ìîäóëü ÷àñòêè äâîõ ÷èñåë õ ³ ó (y ≠ 0)<br />
äîð³âíþº ÷àñòö³ ìîäóë³â öèõ ÷èñåë, òîáòî<br />
x<br />
x x<br />
= , y ≠ 0<br />
y y .<br />
x<br />
Äîâåäåííÿ. Çîáðàçèìî ÷èñëî õ òàê: x = y, y ≠ 0. Òîä³ çà<br />
y<br />
òåîðåìîþ 1.2.4<br />
x<br />
x = y ,<br />
y<br />
çâ³äêè îòðèìàºìî ð³âí³ñòü, ÿêó òðåáà áóëî äîâåñòè.<br />
n<br />
.<br />
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