ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ: Засвар хоорондын ялгаа
Images |
Images |
||
| 33-р мөр: | 33-р мөр: | ||
<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math> | <math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math> | ||
<br> | |||
{| class="wikitable" | |||
|+ | |||
|[[File:Komplexe_addition.svg|thumb]] | |||
|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ | |||
|} | |||
<br> | |||
22:51, 10 Есдүгээр сар 2021-ий байдлаарх засвар
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰcomplex ᠲᠣᠭ ᠠ᠋ ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ $ x^{2}+1=0 $ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ a + bi ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ a ᠪᠣᠯᠤᠨ b ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ i ᠨᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠪᠢᠭᠡᠳ i2 = −1 ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃
ᠳᠦᠷᠰᠦᠯᠡᠯ
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃
ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃
ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃
 |
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨArgand ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷvector ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ (a, b) ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃
ᠡᠭᠦᠨ ᠳ᠋ᠦ: |
ᠦᠢᠯᠡᠳᠦᠯ
ᠨᠡᠮᠡᠬᠦ
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃[1]
$ (a+bi)+(c+di)= $ $ =(a+c)+(b+d)i $
 |
ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ |
ᠬᠠᠰᠤᠬᠤ
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃[1]
$ (a+bi)-(c+di)= $ $ =(a-c)+(b-d)i $
ᠦᠷᠡᠵᠢᠬᠦ
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃[2]
$ (a+bi)(c+di)= $ $ =ac+bci-bd+adi= $ $ =(ac-bd)+(bc+ad)i $
 |
2 + i ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ 3 + i ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ √5 ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨhypotenuse ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ |
ᠬᠤᠪᠢᠶᠠᠬᠤ
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃
$ {\frac {a+bi}{c+di}}= $
$ ={\frac {\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}}= $
$ ={\frac {ac+bd}{c^{2}+d^{2}}}+\left({\frac {bc-ad}{c^{2}+d^{2}}}\right)i. $
ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲquadrate ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃[3]
$ (x+yi)^{2}=x^{2}-y^{2}+2xyi. $
ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ a + bi (b ≠ 0) ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ $ \pm (\gamma +\delta i) $ ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄ $ \gamma ={\sqrt {\frac {a+{\sqrt {a^{2}+b^{2}}}}{2}}} $ ᠪᠠ $ \delta =(\operatorname {sgn} b){\sqrt {\frac {-a+{\sqrt {a^{2}+b^{2}}}}{2}}} $ ᠪᠣᠯᠤᠮᠤᠢ᠃ sgn b ᠭᠡᠭᠴᠢ ᠢᠨᠦ $ \operatorname {sgn} b={\begin{cases}\ \ 1,&b>0\\\ \ 0,&b=0\\-1,&b<0\end{cases}} $ ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃
ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ $ \pm (\gamma +\delta i) $ ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ a + bi ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃
ᠬᠡᠷᠡᠭᠯᠡᠭᠡ
ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ
ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ
- ↑ Jump up to: 1.0 1.1 ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17
- ↑ Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И. Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — М.: Просвещение, 1998. — 288 с. — ISBN 5-09-008036-4.
- ↑ Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01