ЛЕКЦІЇ ² ВПРАВИ

Äëÿ ñïðÿæåíèõ êîìïëåêñíèõ ÷èñåë ìîäóë³ ð³âí³, à àðãóìåíòè
ïðîòèëåæí³ arg z =− arg z .
13.1.3. ij¿ íàä êîìïëåêñíèìè ÷èñëàìè
1. Ñóìîþ äâîõ êîìïëåêñíèõ ÷èñåë z 1 =a 1 +ib 1 ³ z 2 =a 2 +ib 2
íàçèâàºòüñÿ êîìïëåêñíå ÷èñëî z, ÿêå îáóìîâëåíå ð³âí³ñòþ
z=z 1 +z 2 = (a 1 +b 1 i) + (a 2 +b 2 i) = (a 1 +a 2 ) + (b 1 +b 2 ) i. (13.1.2)
2. гçíèöåþ äâîõ êîìïëåêñíèõ ÷èñåë z 1 = a 1 + b 1 i ³
z 2 =a 2 +b 2 i íàçèâàºòüñÿ òàêå êîìïëåêñíå ÷èñëî z, ÿêå, áóäó-
÷è ñêëàäåíå ç z 2 , äຠâ ñóì³ êîìïëåêñíå ÷èñëî z 1 :
z=z 1 –z 2 = (a 1 +b 1 i)–(a 2 +b 2 i) = (a 1 –a 2 ) + (b 1 –b 2 )i. (13.1.3)
³äçíà÷èìî, ùî ìîäóëü ð³çíèö³ äâîõ êîìïëåêñíèõ ÷èñåë
äîð³âíþº â³äñòàí³ ì³æ òî÷êàìè, ùî çîáðàæóþòü ö³ ÷èñëà íà
ïëîùèí³ êîìïëåêñíî¿ çì³ííî¿
z − z = ( a − a ) + ( b −b
) .
2 2
1 2 1 2 1 2
3. Äîáóòêîì êîìïëåêñíèõ ÷èñåë z 1 =a 1 +b 1 i ³ z 2 =a 2 +b 2 i
íàçèâàºòüñÿ òàêå êîìïëåêñíå ÷èñëî z, ÿêå âèõîäèòü, ÿêùî
ïåðåìíîæèòè ö³ ÷èñëà ÿê äâî÷ëåíè çà ïðàâèëàìè àëãåáðè,
âðàõîâóþ÷è ò³ëüêè, ùî i 2 =−1.
Íà ï³äñòàâ³ öüîãî ïðàâèëà îäåðæóºìî
z = z 1 z 2 =(a 1 + b 1 i)(a 2 + b 2 i)=a 1 a 2 + b 1 a 2 i + a 1 b 2 i + b 1 b 2 i 2 =
= (a 1 a 2 –b 1 b 2 )+(b 1 a 2 +a 1 b 2 )i. (13.1.4)
ßêùî êîìïëåêñí³ ÷èñëà çàäàí³ â òðèãîíîìåòðè÷í³é ôîðì³,
òî íåâàæêî äîâåñòè, ùî
z=z 1 z 2 =r 1 (cos ϕ 1 + i sin ϕ 1 ) ⋅ r 2 (cos ϕ 2 + i sin ϕ 2 ) =
=r 1 r 2 (cos ϕ 1 cos ϕ 2 + i sin ϕ 1 cos ϕ 2 + i cos ϕ 1 sin ϕ 2 + i 2 sin ϕ 1 sin ϕ 2 )=
=r 1 r 2 [(cos ϕ 1 cos ϕ 2 −sin ϕ 1 sin ϕ 2 )+i (sin ϕ 1 cosϕ 2 + cosϕ 1 sinϕ 2 )]=
=r 1 r 2 [cos(ϕ 1 +ϕ 2 )+i sin(ϕ 1 +ϕ 2 )], (13.1.5)
òîáòî äîáóòîê äâîõ êîìïëåêñíèõ ÷èñåë º òàêå êîìïëåêñíå
÷èñëî, ìîäóëü ÿêîãî äîð³âíþº äîáóòêó ìîäóë³â ñï³âìíîæíèê³â,
à àðãóìåíò äîð³âíþº ñóì³ àðãóìåíò³â ñï³âìíîæíèê³â.
Çàóâàæåííÿ. Äîáóòîê ñïðÿæåíèõ êîìïëåêñíèõ ÷èñåë
z=a+bi ³ z=a− bi íà ï³äñòàâ³ ð³âíîñò³ (13.1.4) çíàõîäèòüñÿ
òàêèì ÷èíîì:
2 2 2 2
zz a b z z
= + = = ,
òîáòî äîáóòîê ñïðÿæåíèõ êîìïëåêñíèõ ÷èñåë äîð³âíþº êâàäðàòó
ìîäóëÿ êîæíîãî ç íèõ.
4. ijëåííÿ êîìïëåêñíèõ ÷èñåë âèçíà÷àºòüñÿ ÿê ä³ÿ, îáåðíåíà
ìíîæåííþ.
2 2
Íåõàé z 1 =a 1 +b 1 i, z 2 =a 2 +b 2 i, z2 = a2 + b2 ≠ 0 . Òîä³
z 1 /z 2 =z º òàêå êîìïëåêñíå ÷èñëî, ùî z 1 =z 2 z. ßêùî
a
a
+ bi
= x+
yi
+ bi
,
1 1
2 2
òî a 1 +b 1 i=(a 2 +b 2 i)(x +yi) = (a 2 x–b 2 y) + (a 2 y+b 2 x)i, äå õ, ó
âèçíà÷àþòüñÿ ç ñèñòåìè ð³âíÿíü
⎧ a1 = a2x −b2y aa
1 2+ bb
1 2
ab
2 1−ab
1 2
⎨
⇒ x = , y =
.
2 2 2 2
⎩b1 = b2x − a2y a2 + b2 a2 + b2
Îñòàòî÷íî îòðèìàºìî
aa + bb ab − ab
z = +
i.
1 2 1 2 2 1 1 2
2 2 2 2
a2 + b2 a2 + b2
(13.1.6)
Ïðàêòè÷íî ä³ëåííÿ êîìïëåêñíèõ ÷èñåë âèêîíóºòüñÿ òàêèì
÷èíîì: ùîá ðîçä³ëèòè z 1 =a 1 +b 1 i íà z 2 =a 2 +b 2 i, ïîìíîæèìî
ä³ëåíå ³ ä³ëüíèê íà êîìïëåêñíå ÷èñëî, ñïðÿæåíå ä³ëüíèêó
(òîáòî íà a 2 − b 2 i), â ðåçóëüòàò³ îòðèìàºìî ôîðìóëó
(13.1.6).
ßêùî êîìïëåêñí³ ÷èñëà çàäàí³ â òðèãîíîìåòðè÷í³é ôîðì³
z 1 = r 1 (cos ϕ 1 + i sin ϕ 1 ), z 2 =r 2 (cos ϕ 2 + i sin ϕ 2 ),
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