1.7-bo'lim : Murakkab raqamlar
7. Ko'rsatilgan amalni bajaring va javobingizni standart shaklda yozing.
\[\frac{{7 - i}}{{2 + 10i}}\]
Maslahat: Esda tutingki, standart shakl maxrajda \(i\) ga ruxsat bermaydi.
Yechim
Standart shakl \(i\) ning maxrajda bo'lishiga yo'l qo'ymagani uchun biz pay va maxrajni maxrajning konjugatiga, ya'ni \(2 - 10i\)ga ko'paytirishimiz kerak bo'ladi.
2-qadam
Konjugatga ko'paytirish,
\[\frac{{7 - i}}{{2 + 10i}}\,\,\frac{{2 - 10i}}{{2 - 10i}} = \frac{{\left( {7 - i} \right)\left( {2 - 10i} \right)}}{{\left( {2 + 10i} \right)\left( {2 - 10i} \right)}}\]3-qadam
Endi biz qilishimiz kerak bo'lgan yagona narsa surat va maxrajdagi ko'paytirishni bajarish va natijani standart shaklga qo'yishdir.
\[\frac{{7 - i}}{{2 + 10i}} = \frac{{14 - 72i + 10{i^2}}}{{4 - 100{i^2}}} = \frac{{4 - 72i}}{{104}} = \frac{4}{{104}} - \frac{{72}}{{104}}i = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{{26}} - \frac{9}{{13}}i}}\]