ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰcomplex ᠲᠣᠭ᠎ ‍ᠠ᠋ ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ᠎ ‍᠋ᠠ᠋ ‍ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ‍ᠢ ᠥᠷᠭᠡᠵᠢᠭᠦᠯᠵᠦ᠂ 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 ᠪᠥᠭᠡᠳ i2 = −1 ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ‍ᠦᠨ ᠰᠢᠢᠳᠦᠯ ‍ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ‍ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃


ᠳᠦᠷᠰᠦᠯᠡᠯ Засварлах

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ ‍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᠃


ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ‍ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ‍ᠤᠨArgand ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ‍ᠡ᠋ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷvector ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ᠎ ‍ᠠ᠋ ᠪᠠᠷ (a, b) ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋᠃

ᠡᠭᠦᠨ ᠳ᠋ᠦ:
Re ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ‍ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂
Im ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ
i ᠢᠨᠦ i2 = −1 ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃



ᠦᠢᠯᠡᠳᠦᠯ Засварлах

ᠨᠡᠮᠡᠬᠦ Засварлах

ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎‍ᠠ᠋ ‍ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ‍ᠦᠳ ‍ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ‍ᠦᠳ ‍ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ᠎ ‍ᠡ᠋᠃[1]



ᠬᠣᠶᠠᠷ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ‍ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃



ᠬᠠᠰᠤᠬᠤ Засварлах

ᠨᠢᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎‍ᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ‍ᠦᠳ ‍ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃[1]



ᠦᠷᠡᠵᠢᠬᠦ Засварлах

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎‍ᠠ᠋ ‍ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ‍ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ‍ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃[2]



2 + i ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) 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 Засварлах

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ‍ᠳᠤ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ‍ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭ᠎‍ᠠ᠋ ‍ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ‍ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ‍ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ‍ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃



ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ Засварлах

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎‍ᠠ᠋ ‍ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲquadrate ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ‍ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨ᠎‍ᠡ᠋᠃[3]



ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ Засварлах

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ a + bi (b ≠ 0) ‍ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄ ᠪᠠ ᠪᠣᠯᠤᠮᠤᠢ᠃ sgn b ᠭᠡᠭᠴᠢ ᠢᠨᠦ ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨ᠎ᠠ᠃

ᠡᠭᠦᠨ ‍ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ‍ᠢᠢᠨ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ a + bi ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃


ᠬᠡᠷᠡᠭᠯᠡᠭᠡ Засварлах

ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ Засварлах


ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ‍ᠤᠨMandelbrot ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃



ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ‍ᠤᠨMandelbrot ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ‍ᠤᠨfractal ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ‍ᠡ᠋ ᠶᠤᠮ᠃

ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ z = 0 ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼiteration ᠬᠢᠬᠦ ᠳ᠋ᠦ fc(z)=z2+c ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ‍ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃

ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ‍ᠤᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ‍ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ‍ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ‍ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ‍ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ‍ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃[4]

ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ‍ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ‍ᠤᠨ ᠵᠠᠭᠠᠭ ‍ᠢ ᠳᠠᠪᠠᠬᠤ ‍ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ‍ᠠ᠋ ‍ᠢᠢᠨJulia ᠹᠷᠠᠻᠲᠠᠯ ‍ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ‍ᠢᠢᠨFatou ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃[5]


ᠳᠦᠷᠰᠦ ᠠᠮᠢᠯᠠᠭᠤᠯᠤᠨ ᠵᠦᠯᠢᠶ ‍ᠠ᠋ ‍ᠢᠢᠨ Julia ᠣᠯᠠᠨᠯᠢᠭ ‍ᠢ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠢᠨᠦ᠃


ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ Засварлах

ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌quantum mechanics ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨ᠎ᠡ ᠪᠣᠽᠣᠨboson (ᠵᠢᠱᠢᠶ᠎ᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨphoton ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤfermion (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃

ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬ᠎ᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃

ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃

ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ Schrödinger ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᠻ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃

ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ impuls ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢfunction ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ᠎ᠠ ᠲᠠᠢ ᠶᠤᠮ᠃

ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃

1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷWalter Heitler ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨFritz London ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨvalency ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃

ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃

ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄







ᠻᠢᠨᠧᠲᠢᠭkinetic ᠧᠨᠧᠷᠭᠢenergy






ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯpotencial ᠧᠨᠧᠷᠭᠢ





ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ


ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷ᠎ᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃

ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶ᠎ᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ‍‍ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯmolecule ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲcovalent ᠬᠣᠯᠪᠣᠭ᠎ᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃[6]


ᠤᠳᠤᠷᠢᠳᠤᠯᠭ᠎ᠠ ‍ᠢᠢᠨ ᠣᠨᠤᠯ Засварлах


ᠤᠳᠤᠷᠢᠳᠤᠯᠭ᠎ᠠ ‍ᠢᠢᠨ ᠲᠣᠭᠲᠠᠯᠴᠠᠭᠠᠨ ‍ᠤ ᠡᠩ ‍ᠦᠨ ᠪᠦᠳᠦᠭᠦᠪᠴᠢ᠃

X(s) ᠣᠷᠤᠯᠲᠠ ‍ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂

Y(s) ᠭᠠᠷᠤᠯᠲᠠ ‍ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂

G(s) ᠤᠷᠤᠭᠰᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠬᠤ ᠹᠦᠨ᠍ᠻᠼfunction

H(s) ᠡᠷᠭᠢᠬᠦ ᠬᠣᠯᠪᠤᠭ᠎ᠠ



ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭ᠎ᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶ᠎ᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃

ᠤᠳᠤᠷᠢᠳᠤᠯᠭ᠎ᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃

ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬ᠎ᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨ᠎ᠠ᠃

ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭ᠎ᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤtechnology ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨ᠎ᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃[7]

ᠤᠳᠤᠷᠢᠳᠤᠯᠭ᠎ᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨLaplace ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢsystem ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃

ᠳᠠᠷᠠᠭ᠎ᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨ᠎ᠡ᠃

ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠮᠥᠷ ᠦᠨ (root locus) ᠠᠷᠭ᠎ᠠ ᠪᠠᠷ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ ‍ᠤᠤ᠂ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ ‍ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ᠃

ᠥᠭᠡᠷ᠎ᠡ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠡᠭ ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨ᠎ᠠ ‍ᠤᠤ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠭ᠎ᠠ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ ‍ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ ᠶᠤᠮ᠃

ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠱᠤᠭᠤᠮᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠨᠢ᠄


- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠂

- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠂

- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢ ᠦᠭᠡᠶ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠃


ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ Засварлах

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ - complex number - комплексное число

ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица

ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ᠎ᠠ - imaginary number - чисто мнимое число

ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть

ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ᠎ᠠ - real number - действительное число

ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ - complex plane - комлексная плоскость

ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось

ᠣᠯᠠᠨᠯᠢᠭ - set - множество

ᠡᠶᠡᠷᠭᠦ - positive - положительный

ᠰᠥᠷᠭᠦ - negative - отрицательный

ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение

ᠲᠣᠮᠢᠶ ‍ᠠ᠋ - formula - формула

ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот

ᠮᠠᠭᠠᠳᠯᠠᠯ - probability - вероятность

ᠬᠡᠮ - degree - градус

ᠤᠳᠤᠷᠢᠳᠤᠯᠭ᠎ᠠ ‍ᠢᠢᠨ ᠣᠨᠤᠯ - control theory - теория управления

ᠠᠲ᠋ᠣᠮ - atom - атом

ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ - elementary particle - элементарная частица

ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ - wave function - волновая функция

ᠳᠠᠪᠲᠠᠮᠵᠢ - frequency - частота

ᠹᠷᠠᠻᠲ᠋ᠠᠯ ‍- fractal - фрактал

ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ - quantum mechanics - квантовая механика

ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᠻ - classical physics - классическая физика


ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ Засварлах

  1. 1.0 1.1 ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17
  2. 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.
  3. Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01
  4. ᠷᠡᠨᠴᠡᠨ ‍ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ‍ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ‍ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020
  5. ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02
  6. Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755
  7. Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf