"ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ"-ны өөр хувилбарууд

13:55, 7 Долоодугаар сар 2021-ий байдлаарх одоогийн засвар

ᠦᠨᠳᠦᠰᠦᠨ ᠨᠡᠢᠢᠲᠡᠯᠡᠯ ‍ᠢ ᠤᠩᠰᠢᠬᠤ ᠳ᠋ᠤ ᠰᠧᠷᠸᠧᠷ ‍ᠦᠨ^server ᠬᠦᠴᠦᠨ ᠴᠢᠳᠠᠯ ᠡᠴᠡ ᠰᠢᠯᠲᠠᠭᠠᠯᠠᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠺ ᠲᠣᠮᠢᠶᠠᠨ ᠨᠤᠭᠤᠳ ᠡᠪᠳᠡᠷᠡᠵᠦ ᠬᠠᠷᠠᠭᠳᠠᠵᠤ ᠪᠠᠢᠢᠪᠠᠯ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠬᠣᠯᠪᠤᠭᠠᠰᠤ ᠪᠠᠷ ᠣᠷᠤᠵᠤ ᠤᠩᠰᠢᠨ᠎ᠠ ‍ᠤᠤ᠄ Хэлэлцүүлэг:ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ

ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠪᠣᠯ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ -1 ᠭᠠᠷᠳᠠᠭ ᠲᠣᠭ᠎ ‍ᠠ᠋ ᠶᠤᠮ᠃

ᠥᠭᠡᠷ ‍ᠡ᠋ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠻᠸᠠᠲᠷᠠᠲ᠋^quadratic ᠵᠡᠷᠭᠡ ᠢᠨᠦ -1 ᠤᠳᠬ ‍ᠠ᠋ ᠠᠪᠳᠠᠭ (i² = -1) ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠭᠡᠨ᠎ ‍ᠡ᠋᠃

ᠪᠠᠰᠠ -1 ‍ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠭᠡᠵᠦ ᠬᠡᠯᠡᠵᠦ ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋:

i={\sqrt {-1}}

ᠨᠥᠭᠦᠭᠡᠲᠡᠭᠦᠷ᠂ ${\sqrt {-1}}$ ᠪᠤᠶᠤ ᠬᠠᠰᠠᠬᠤ ᠨᠢᠭᠡ ‍ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ‍ᠤᠨ ᠰᠢᠢᠳᠦᠯ ᠢᠨᠦ $i$ ᠲᠡᠳᠦᠢ ᠦᠭᠡᠢ ᠪᠠᠰᠠ $-i$ ᠪᠠᠢᠢᠬᠤ ᠶᠤᠮ᠃^[1]

ᠠᠩᠭᠯᠢ ᠪᠠᠷ imaginary unit, ᠣᠷᠤᠰ ‍ᠢᠢᠠᠷ мнимая единица ᠬᠡᠮᠡᠬᠦ ᠡᠨᠡ ᠤᠬᠠᠭᠳᠠᠬᠤᠨ ᠠᠨᠤ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠨᠢᠭᠡᠴᠡ᠂ ᠲᠥᠰᠦᠭᠡᠯᠡᠭᠰᠡᠨ ᠨᠢᠭᠡᠴᠡ᠂ ᠬᠣᠭᠤᠰᠤᠨ ᠨᠢᠭᠡᠴᠡ᠂ ᠬᠡᠢᠢᠰᠪᠦᠷᠢ ᠨᠢᠭᠡᠴᠡ ᠭᠡᠰᠡᠨ ᠤᠳᠬ ‍ᠠ᠋ ᠲᠠᠢ᠃

ᠬᠠᠯᠢᠮᠠᠭ ᠲᠦᠮᠡᠨ imaginary number ᠬᠡᠮᠡᠬᠦᠢ ‍ᠢᠢ ухалдаг тойг (ᠤᠬᠠᠭᠠᠯᠠᠳᠠᠭ ᠲᠣᠭ᠎ᠠ) ᠬᠡᠮᠡᠨ ᠪᠠᠭᠤᠯᠭᠠᠭᠰᠠᠨ ᠪᠠᠢᠢᠬᠤ ᠲᠤᠯᠠ imaginary unit ᠭᠡᠳᠡᠭ ‍ᠢ ᠪᠠᠰᠠ "ᠤᠬᠠᠭᠠᠯᠠᠳᠠᠭ ᠨᠢᠭᠡᠴᠡ" ᠭᠡᠵᠦ ᠪᠠᠭᠤᠯᠭᠠᠳᠠᠭ ᠪᠤᠢ ᠵᠠ᠃

ᠬᠠᠷᠢᠨ комплексное число ᠭᠡᠳᠡᠭ ‍ᠢ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ᠎ᠡ ᠲᠣᠮᠢᠶᠠᠨ  ᠤ ᠲᠣᠯᠢ ᠳ᠋ᠤ "ᠬᠠᠪᠰᠤᠷᠠᠭᠰᠠ ᠲᠣᠭ᠎ᠠ" (ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ^complex ᠲᠣᠭ᠎ᠠ) ᠬᠡᠮᠡᠵᠦᠬᠦᠢ᠃^[2]


	ᠢᠵᠢ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷ ‍ᠡ᠋ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠬᠢᠭᠡᠳ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ᠎ ‍ᠠ᠋ 1 ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ i᠃

ᠵᠠᠷᠢᠮ ᠦᠢᠯᠡᠳᠦᠯ[засварлах | кодоор засварлах]

ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠬᠤᠪᠢᠶᠠᠬᠤ[засварлах | кодоор засварлах]

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠪᠤᠶᠤ i ᠲᠣᠭᠠᠨ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠬᠦᠯᠦᠭᠰᠡᠨ ‍ᠢᠢᠡᠷ᠄

$i\,(a+bi)$
$=ai+bi^{2}$
$=-b+ai$

ᠭᠠᠷᠤᠮᠤᠢ᠃

ᠡᠨᠡ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ‍ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠸᠧᠺᠲ᠋ᠣᠷ ‍ᠢ^vector ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ 90°ᠡᠷᠭᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ ᠪᠤᠶ ‍ᠠ᠋᠃

ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠳ᠋ᠦ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠨᠢ i ᠲᠣᠭᠠᠨ ‍ᠤ ᠤᠷᠪᠠᠭᠤ ᠲᠣᠭᠠᠨ ᠳ᠋ᠤ ᠦᠷᠵᠢᠭᠦᠯᠬᠦ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ:

${\frac {1}{i}}={\frac {1}{i}}\cdot {\frac {i}{i}}$
$={\frac {i}{i^{2}}}={\frac {i}{-1}}$
$=-i$

ᠡᠭᠦᠨ ‍ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳ᠋ᠤ ᠬᠡᠷᠡᠭᠯᠡᠪᠡᠰᠦ:

${\frac {a+bi}{i}}=-i\,(a+bi)$
$=-ai-bi^{2}=b-ai~.$

ᠡᠨᠡ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ‍ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠸᠧᠺᠲ᠋ᠣᠷ ‍ᠢ^vector ᠨᠠᠷᠠ ᠵᠥᠪ 90°ᠡᠷᠭᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ ᠪᠤᠶ ‍ᠠ᠋᠃

ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ[засварлах | кодоор засварлах]

$i$ ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠨᠢ ᠮᠥᠴᠢᠯᠭᠡ ᠪᠡᠷ ᠳᠠᠪᠲᠠᠭᠳᠠᠬᠤ ᠰᠢᠨᠵᠢ ᠴᠢᠨᠠᠷ ᠲᠠᠢ:^[3]


i⁰ = 1	i¹ = i	i² = -1	i³ = i² • i = (-1) • i = -i
i⁴ = i³ • i = (-i) • i = -i² = 1	i⁵ = i⁴ • i = 1 • (i) = i	i⁶ = i⁴ • i² = 1 • (-1) =-1	i⁷= i⁴ • i³ = 1 • (-i) = -i
i⁸ = i⁴ • i⁴ = 1 • 1 = 1	i⁹= i⁴ • i⁴ • i = 1 • 1• i = i	i¹⁰ = (i⁴)² • i² = 1 • (-1) = -1	i¹¹ = (i⁴)² • i³ = 1 • (-i) = -i

ᠡᠭᠦᠨ ‍ᠢ ᠶᠡᠷᠦᠩᠬᠡᠢᠢᠯᠡᠭᠰᠡᠨ ᠬᠡᠯᠪᠡᠷᠢ ‍ᠪᠡᠷ ᠪᠢᠴᠢᠪᠡᠯ᠄

$i^{4n}=1$
$i^{4n+1}=i$
$i^{4n+2}=-1$
$i^{4n+3}=-i$

ᠡ᠊ᠨᠳᠡ ᠡᠴᠡ᠄ $i^{n}=i^{(n{\bmod {4}})}$

ᠡᠩ ‍ᠦᠨ ‍ᠢᠢᠡᠷ ᠠᠷᠭᠠᠴᠢᠯᠠᠪᠠᠰᠤ᠂ i ᠲᠣᠭᠠᠨ ‍ᠤ ᠵᠡᠷᠭᠡ ‍ᠢᠢ ᠲᠣᠳᠤᠷᠬᠠᠶ᠋ᠢᠯᠠᠬᠤ ‍ᠢᠢᠨ ᠲᠤᠯᠠᠳᠠ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠯᠡᠳᠬᠡᠭᠴᠢ ‍ᠢᠢ^[4] 4 ᠳ᠋ᠦ ᠬᠤᠪᠢᠶᠠᠮᠤᠢ᠃

ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ 0 ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ $1$ ᠂

ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ 1 ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ $i$ ᠂

ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ 2 ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ $-1$ ᠂

ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ 3 ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ $-i$

ᠪᠠᠢᠢᠨ᠎ᠠ ᠬᠡᠮᠡᠨ ᠲᠣᠭᠲᠠᠭᠠᠵᠤ ᠪᠣᠯᠤᠨ᠎ᠠ᠃

ᠢᠢᠨ ᠬᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ‍ᠢᠢ ᠠᠯᠢᠮᠠᠳ ᠪᠦᠬᠦᠯᠢ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠥᠭᠦᠯᠪᠡᠯ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ $1$ ᠂ $i$ ᠂ $-1$ ᠂ $-i$ - ᠡᠳᠡᠭᠡᠷ ‍ᠦᠨ ᠨᠢᠭᠡ ᠢᠮᠠᠭᠲᠠ ᠪᠠᠢᠢᠬᠤ ᠠᠵᠤᠭᠤ᠃

$i$ ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ $i$ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ[засварлах | кодоор засварлах]

$i^{i}=\left(e^{i(\pi /2+2k\pi )}\right)^{i}$
$=e^{i^{2}(\pi /2+2k\pi )}$
$=e^{-(\pi /2+2k\pi )}$

ᠡᠭᠦᠨ ᠳ᠋ᠦ k ∈ ℤ ᠪᠤᠶᠤ ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ᠃

k = 0 ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠨᠥᠭᠦᠴᠡᠯ ᠳ᠋ᠦ ᠦᠨᠳᠦᠰᠦᠨ ᠤᠳᠬ ‍ᠠ᠋ ᠨᠢ e^−π/2 ᠪᠤᠶᠤ ᠣᠢᠢᠷᠠᠯᠴᠠᠭ᠎ ‍ᠠ᠋ ᠪᠠᠷ 0.207879576 ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋᠃^[5]^[6]

ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠡᠴᠡ ᠢᠵᠠᠭᠤᠷ ᠭᠠᠷᠭᠠᠬᠤ[засварлах | кодоор засварлах]


	i ᠲᠣᠭᠠᠨ ‍ᠤ ᠺᠦ᠋ᠪ^cubic ᠢᠵᠠᠭᠤᠷ ᠨᠢ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠣᠷᠣᠢ ᠨᠤᠭᠤᠳ ᠋ᠲᠤ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠮᠤᠢ᠃

$i$ ᠲᠣᠭᠠᠨ ‍ᠤ n ᠵᠡᠷᠭᠡ ‍ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ᠨᠢ n ᠲᠣᠭᠠᠨ ‍ᠤ ᠬᠠᠷᠢᠭᠤ ᠲᠠᠢ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃

ᠬᠡᠳᠦᠨ ᠬᠠᠷᠢᠭᠤ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ ᠭᠠᠷᠬᠤ n-ᠥᠨᠴᠦᠭᠲᠦ ‍ᠢᠢᠨ ᠥᠨᠴᠦᠭ ‍ᠦᠨ ᠲᠣᠭ᠎ ‍ᠠ᠋ ᠪᠠᠷ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃

$u_{k}=\cos {\frac {{\frac {\pi }{2}}+2\pi k}{n}}+i\ \sin {\frac {{\frac {\pi }{2}}+2\pi k}{n}},$

$\quad k=0,1,...,n-1$

$i$ ᠲᠣᠭᠠᠨ ‍ᠤ
ᠺᠸᠠᠲᠷᠠᠲ
ᠢᠵᠠᠭᠤᠷ᠄
$\{{\sqrt {i}}\}=\left\{{\frac {1+i}{\sqrt {2}}};~{\frac {-1-i}{\sqrt {2}}}\right\}$

$i$ ᠲᠣᠭᠠᠨ ‍ᠤ
ᠺᠦᠪ
ᠢᠵᠠᠭᠤᠷ᠄
$\{{\sqrt[{3}]{i}}\}=\left\{-i;~{\frac {i+{\sqrt {3}}}{2}};~{\frac {i-{\sqrt {3}}}{2}}\right\}.$

$i$ ᠲᠣᠭᠠᠨ ᠡᠴᠡ $i$ ᠵᠡᠷᠭᠡ ‍ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ᠭᠠᠷᠭᠠᠬᠤ[засварлах | кодоор засварлах]

${\sqrt[{i}]{i}}=i^{{\frac {1}{i}}{\frac {i}{i}}}=i^{-i}$
$={\biggl (}e^{{i}{\frac {\pi }{2}}}{\biggr )}^{-i}$
$=e^{-i^{2}{\frac {\pi }{2}}}=e^{\frac {\pi }{2}}$

ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ[засварлах | кодоор засварлах]

ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица

ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ᠎ ‍ᠠ᠋ - imaginary number - чисто мнимое число

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ ‍ᠠ᠋ (ᠬᠠᠪᠰᠤᠷᠠᠭᠰᠠᠨ ᠲᠣᠭ᠎ ‍ᠠ᠋) - complex number - комплексное число

ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ᠎ ‍ᠠ᠋ - real number - вещественное число (действительное число)

ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ - multiplication - умножение

ᠬᠤᠪᠢᠶᠠᠬᠤ - division - деление

ᠵᠡᠷᠭᠡ - power - степень

ᠢᠵᠠᠭᠤᠷ - root - корень

ᠢᠯᠡᠳᠬᠡᠭᠴᠢ - exponent - экспонента

ᠦᠯᠡᠳᠡᠭᠳᠡᠯ - remainder - остаток

ᠣᠯᠠᠨᠯᠢᠭ - set - множество

ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ᠎ ‍ᠠ᠋ - integer - целое число

ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ[засварлах | кодоор засварлах]

↑ James Tamton. Encyclopedia of Mathematics. Facts on File Inc. New York 2005. ISBN 0-8160-5124-0
↑ ᠡ᠊᠂ ᠸᠠᠩᠳᠤᠢ᠃ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ ‍ᠡ᠋ ᠮᠣᠮᠢᠶᠠᠨ ‍ᠤ ᠲᠣᠯᠢ᠃ ᠪᠦ᠊᠂ ᠨᠠ᠊᠂ ᠮᠣ᠊᠂ ᠠ᠊᠂ ᠤ᠊᠂ ‍ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ‍ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠬᠡᠯᠡ ᠵᠣᠬᠢᠶᠠᠯ ‍ᠤᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ ᠤᠯᠤᠰ ‍ᠤᠨ ᠬᠡᠪᠯᠡᠯ ‍ᠦᠨ ᠬᠡᠷᠡᠭ ᠡᠷᠬᠢᠯᠡᠬᠦ ᠬᠣᠷᠢᠶ ‍ᠠ᠋᠃ ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ 1964᠃
↑ MathBits Notebook. Cyclic Nature of the Powers of "i ".https://mathbitsnotebook.com/Algebra2/ComplexNumbers/CPPowers.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/03
↑ ᠡ᠊᠂ ᠸᠠᠩᠳᠤᠢ᠃ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ ‍ᠡ᠋ ᠮᠣᠮᠢᠶᠠᠨ ‍ᠤ ᠲᠣᠯᠢ᠃ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠪᠣᠲᠢ᠃ ᠪᠦ᠊᠂ ᠨᠠ᠊᠂ ᠮᠣ᠊᠂ ᠠ᠊᠂ ᠤ᠊᠂ ‍ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ‍ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠬᠡᠯᠡ ᠵᠣᠬᠢᠶᠠᠯ ‍ᠤᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ ᠤᠯᠤᠰ ‍ᠤᠨ ᠬᠡᠪᠯᠡᠯ ‍ᠦᠨ ᠭᠠᠵᠠᠷ᠃ ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ 1970᠃
↑ David Wells. The Penguin Dictionary of Curious and Interesting Numbers. UK: Penguin Books 1997. ISBN=0-14-026149-4
↑ Brilliant. What is i to the power of i. https://brilliant.org/discussions/thread/what-is-i-to-the-power-of-i-T. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/017/04

[1] James Tamton. Encyclopedia of Mathematics. Facts on File Inc. New York 2005. ISBN 0-8160-5124-0

[2] ᠡ᠊᠂ ᠸᠠᠩᠳᠤᠢ᠃ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ ‍ᠡ᠋ ᠮᠣᠮᠢᠶᠠᠨ ‍ᠤ ᠲᠣᠯᠢ᠃ ᠪᠦ᠊᠂ ᠨᠠ᠊᠂ ᠮᠣ᠊᠂ ᠠ᠊᠂ ᠤ᠊᠂ ‍ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ‍ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠬᠡᠯᠡ ᠵᠣᠬᠢᠶᠠᠯ ‍ᠤᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ ᠤᠯᠤᠰ ‍ᠤᠨ ᠬᠡᠪᠯᠡᠯ ‍ᠦᠨ ᠬᠡᠷᠡᠭ ᠡᠷᠬᠢᠯᠡᠬᠦ ᠬᠣᠷᠢᠶ ‍ᠠ᠋᠃ ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ 1964᠃

[3] MathBits Notebook. Cyclic Nature of the Powers of "i ".https://mathbitsnotebook.com/Algebra2/ComplexNumbers/CPPowers.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/03

[4] ᠡ᠊᠂ ᠸᠠᠩᠳᠤᠢ᠃ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ ‍ᠡ᠋ ᠮᠣᠮᠢᠶᠠᠨ ‍ᠤ ᠲᠣᠯᠢ᠃ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠪᠣᠲᠢ᠃ ᠪᠦ᠊᠂ ᠨᠠ᠊᠂ ᠮᠣ᠊᠂ ᠠ᠊᠂ ᠤ᠊᠂ ‍ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ‍ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠬᠡᠯᠡ ᠵᠣᠬᠢᠶᠠᠯ ‍ᠤᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ ᠤᠯᠤᠰ ‍ᠤᠨ ᠬᠡᠪᠯᠡᠯ ‍ᠦᠨ ᠭᠠᠵᠠᠷ᠃ ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ 1970᠃

[5] David Wells. The Penguin Dictionary of Curious and Interesting Numbers. UK: Penguin Books 1997. ISBN=0-14-026149-4

[6] Brilliant. What is i to the power of i. https://brilliant.org/discussions/thread/what-is-i-to-the-power-of-i-T. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/017/04

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@@ 1-р мөр: / 1-р мөр: @@
 <div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'">
-ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠪᠣᠯ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;" ><i>-1</i></span> ᠭᠠᠷᠳᠠᠭ ᠲᠣᠭ᠎ ‍ᠠ᠋ ᠶᠤᠮ᠃
+<span style="color:Violet;">ᠦᠨᠳᠦᠰᠦᠨ ᠨᠡᠢᠢᠲᠡᠯᠡᠯ ‍ᠢ ᠤᠩᠰᠢᠬᠤ ᠳ᠋ᠤ ᠰᠧᠷᠸᠧᠷ ‍ᠦᠨ<sup>server</sup> ᠬᠦᠴᠦᠨ ᠴᠢᠳᠠᠯ ᠡᠴᠡ ᠰᠢᠯᠲᠠᠭᠠᠯᠠᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠺ ᠲᠣᠮᠢᠶᠠᠨ ᠨᠤᠭᠤᠳ ᠡᠪᠳᠡᠷᠡᠵᠦ ᠬᠠᠷᠠᠭᠳᠠᠵᠤ ᠪᠠᠢᠢᠪᠠᠯ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠬᠣᠯᠪᠤᠭᠠᠰᠤ ᠪᠠᠷ ᠣᠷᠤᠵᠤ ᠤᠩᠰᠢᠨ᠎ᠠ ‍ᠤᠤ᠄</span> [[Хэлэлцүүлэг:ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]]
-ᠥᠭᠡᠷ ‍ᠡ᠋ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠻᠸᠠᠲᠷᠠᠲ᠋ ᠵᠡᠷᠭᠡ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;" ><i>-1</i></span> ᠤᠳᠬ ‍ᠠ᠋ ᠠᠪᠳᠠᠭ <span style="writing-mode: horizontal-tb;"><i>(i<sup>2</sup> = -1)</i></span> ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠭᠡᠨ᠎ ‍ᠡ᠋᠃
+ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠪᠣᠯ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>-1</i></span> ᠭᠠᠷᠳᠠᠭ ᠲᠣᠭ᠎ ‍ᠠ᠋ ᠶᠤᠮ᠃
+ᠥᠭᠡᠷ ‍ᠡ᠋ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠻᠸᠠᠲᠷᠠᠲ᠋<sup>quadratic</sup> ᠵᠡᠷᠭᠡ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;" ><i>-1</i></span> ᠤᠳᠬ ‍ᠠ᠋ ᠠᠪᠳᠠᠭ <span style="writing-mode: horizontal-tb;"><i>(i<sup>2</sup> = -1)</i></span> ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠭᠡᠨ᠎ ‍ᠡ᠋᠃
 ᠪᠠᠰᠠ <span style="writing-mode: horizontal-tb;" ><i>-1</i></span> ‍ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠭᠡᠵᠦ ᠬᠡᠯᠡᠵᠦ ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋:
 :<math>i = \sqrt{-1}</math>
+ᠨᠥᠭᠦᠭᠡᠲᠡᠭᠦᠷ᠂ <small><small><math>\sqrt{-1}</math></small></small> ᠪᠤᠶᠤ ᠬᠠᠰᠠᠬᠤ ᠨᠢᠭᠡ ‍ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ‍ᠤᠨ ᠰᠢᠢᠳᠦᠯ ᠢᠨᠦ <math>i</math> ᠲᠡᠳᠦᠢ ᠦᠭᠡᠢ ᠪᠠᠰᠠ <math>-i</math> ᠪᠠᠢᠢᠬᠤ ᠶᠤᠮ᠃<ref>James Tamton. Encyclopedia of Mathematics. Facts on File Inc. New York 2005. [[index.php?title=Special:BookSources/0816051240|ISBN 0-8160-5124-0]]</ref>
 ᠠᠩᠭᠯᠢ ᠪᠠᠷ imaginary unit, ᠣᠷᠤᠰ ‍ᠢᠢᠠᠷ мнимая единица ᠬᠡᠮᠡᠬᠦ ᠡᠨᠡ ᠤᠬᠠᠭᠳᠠᠬᠤᠨ ᠠᠨᠤ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠨᠢᠭᠡᠴᠡ᠂ ᠲᠥᠰᠦᠭᠡᠯᠡᠭᠰᠡᠨ ᠨᠢᠭᠡᠴᠡ᠂ ᠬᠣᠭᠤᠰᠤᠨ ᠨᠢᠭᠡᠴᠡ᠂ ᠬᠡᠢᠢᠰᠪᠦᠷᠢ ᠨᠢᠭᠡᠴᠡ ᠭᠡᠰᠡᠨ ᠤᠳᠬ ‍ᠠ᠋ ᠲᠠᠢ᠃
-ᠬᠠᠯᠨᠮᠠᠭ ᠲᠦᠮᠡᠨ imaginary number ᠬᠡᠮᠡᠬᠦᠢ ‍ᠢᠢ ухалдаг тойг (ᠤᠬᠠᠭᠠᠯᠠᠳᠠᠭ ᠲᠣᠭ᠎ᠠ) ᠬᠡᠮᠡᠨ ᠪᠠᠭᠤᠯᠭᠠᠭᠰᠠᠨ ᠪᠠᠢᠢᠬᠤ ᠲᠤᠯᠠ imaginary unit ᠭᠡᠳᠡᠭ ‍ᠢ ᠪᠠᠰᠠ "ᠤᠬᠠᠭᠠᠯᠠᠳᠠᠭ ᠨᠢᠭᠡᠴᠡ" ᠭᠡᠵᠦ ᠪᠠᠭᠤᠯᠭᠠᠳᠠᠭ ᠪᠤᠢ ᠵᠠ᠃
+ᠬᠠᠯᠢᠮᠠᠭ ᠲᠦᠮᠡᠨ imaginary number ᠬᠡᠮᠡᠬᠦᠢ ‍ᠢᠢ ухалдаг тойг (ᠤᠬᠠᠭᠠᠯᠠᠳᠠᠭ ᠲᠣᠭ᠎ᠠ) ᠬᠡᠮᠡᠨ ᠪᠠᠭᠤᠯᠭᠠᠭᠰᠠᠨ ᠪᠠᠢᠢᠬᠤ ᠲᠤᠯᠠ imaginary unit ᠭᠡᠳᠡᠭ ‍ᠢ ᠪᠠᠰᠠ "ᠤᠬᠠᠭᠠᠯᠠᠳᠠᠭ ᠨᠢᠭᠡᠴᠡ" ᠭᠡᠵᠦ ᠪᠠᠭᠤᠯᠭᠠᠳᠠᠭ ᠪᠤᠢ ᠵᠠ᠃
-ᠨᠥᠭᠦᠭᠡᠲᠡᠭᠦᠷ᠂ <small><small><math>\sqrt{-1}</math></small></small> ᠪᠤᠶᠤ ᠬᠠᠰᠠᠬᠤ ᠨᠢᠭᠡ ‍ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ‍ᠤᠨ ᠰᠢᠢᠳᠦᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠡᠳᠦᠢ ᠦᠭᠡᠢ ᠪᠠᠰᠠ <span style="writing-mode: horizontal-tb;"><i>-i</i></span> ᠪᠠᠢᠢᠬᠤ ᠶᠤᠮ᠃<ref>James Tamton. Encyclopedia of Mathematics. Facts on File Inc. New York 2005. [[index.php?title=Special:BookSources/0816051240|ISBN 0-8160-5124-0]]</ref>
+ᠬᠠᠷᠢᠨ комплексное число ᠭᠡᠳᠡᠭ ‍ᠢ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ᠎ᠡ ᠲᠣᠮᠢᠶᠠᠨ  ᠤ ᠲᠣᠯᠢ ᠳ᠋ᠤ "ᠬᠠᠪᠰᠤᠷᠠᠭᠰᠠ ᠲᠣᠭ᠎ᠠ" (ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ᠎ᠠ) ᠬᠡᠮᠡᠵᠦᠬᠦᠢ᠃<ref>ᠡ᠊᠂ ᠸᠠᠩᠳᠤᠢ᠃ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ ‍ᠡ᠋ ᠮᠣᠮᠢᠶᠠᠨ ‍ᠤ ᠲᠣᠯᠢ᠃ ᠪᠦ᠊᠂ ᠨᠠ᠊᠂ ᠮᠣ᠊᠂ ᠠ᠊᠂ ᠤ᠊᠂ ‍ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ‍ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠬᠡᠯᠡ ᠵᠣᠬᠢᠶᠠᠯ ‍ᠤᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ ᠤᠯᠤᠰ ‍ᠤᠨ ᠬᠡᠪᠯᠡᠯ ‍ᠦᠨ ᠬᠡᠷᠡᠭ ᠡᠷᠬᠢᠯᠡᠬᠦ ᠬᠣᠷᠢᠶ ‍ᠠ᠋᠃ ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ 1964᠃</ref>
-ᠬᠠᠷᠢᠨ комплексное число ᠭᠡᠳᠡᠭ ‍ᠢ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ᠎ᠡ ᠲᠣᠮᠢᠶᠠᠨ  ᠤ ᠲᠣᠯᠢ ᠳ᠋ᠤ "ᠬᠠᠪᠰᠤᠷᠠᠭᠰᠠ ᠲᠣᠭ᠎ᠠ" (ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ᠎ᠠ) ᠬᠡᠮᠡᠵᠦᠬᠦᠢ᠃<ref>ᠡ᠊᠂ ᠸᠠᠩᠳᠤᠢ᠃ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ ‍ᠡ᠋ ᠮᠣᠮᠢᠶᠠᠨ ‍ᠤ ᠲᠣᠯᠢ᠃ ᠪᠦ᠊᠂ ᠨᠠ᠊᠂ ᠮᠣ᠊᠂ ᠠ᠊᠂ ᠤ᠊᠂ ‍ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ‍ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠬᠡᠯᠡ ᠵᠣᠬᠢᠶᠠᠯ ‍ᠤᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ ᠤᠯᠤᠰ ‍ᠤᠨ ᠬᠡᠪᠯᠡᠯ ‍ᠦᠨ ᠬᠡᠷᠡᠭ ᠡᠷᠬᠢᠯᠡᠬᠦ ᠬᠣᠷᠢᠶ ‍ᠠ᠋᠃ ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ 1964᠃</ref>
+<br>
+{| class="wikitable"
+|+
+|[[File:Complex Plane number i and 1.svg|thumb]]
+|style = "height: 150px;"|ᠢᠵᠢ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷ ‍ᠡ᠋ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠬᠢᠭᠡᠳ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ᠎ ‍ᠠ᠋ <span style="writing-mode: horizontal-tb;"><i>1</i></span> ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ <span style="writing-mode: horizontal-tb;"><i>i</i></span>᠃
+|}
+<br>
 __TOC__
 <br>
 == ᠵᠠᠷᠢᠮ ᠦᠢᠯᠡᠳᠦᠯ ==
 <br>
-=== ᠦᠷᠵᠢᠭᠦᠯᠬᠦ ᠬᠤᠪᠢᠶᠠᠬᠤ ===
+=== ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠬᠤᠪᠢᠶᠠᠬᠤ ===
-ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠷᠴᠡ ᠪᠤᠶᠤ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ᠳ᠋ᠦ ᠦᠷᠵᠢᠬᠦᠯᠦᠭᠰᠡᠨ ‍ᠢᠢᠡᠷ᠄
+[[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ]] ‍ᠢᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠪᠤᠶᠤ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠬᠦᠯᠦᠭᠰᠡᠨ ‍ᠢᠢᠡᠷ᠄
+<br><br>
-:<math>i\,(a + bi) = ai + bi^2 = -b + ai</math>
+<span style="writing-mode: horizontal-tb;">
+<br><math>i\,(a + bi)</math>
+<br><math> = ai + bi^2</math>
+<br><math> = -b + ai</math>
+</span>
 ᠭᠠᠷᠤᠮᠤᠢ᠃
@@ 30-р мөр: / 47-р мөр: @@
 ᠡᠨᠡ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ‍ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠸᠧᠺᠲ᠋ᠣᠷ ‍ᠢ<sup>vector</sup> ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ 90°ᠡᠷᠭᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ ᠪᠤᠶ ‍ᠠ᠋᠃
-ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠳ᠋ᠦ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ‍ᠤ ᠤᠷᠪᠠᠭᠤ ᠲᠣᠭᠠᠨ ᠳ᠋ᠤ ᠦᠷᠵᠢᠭᠦᠯᠬᠦ ᠲᠡᠢ ᠭᠠᠷ ᠨᠢᠭᠡᠨ:
+ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠳ᠋ᠦ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ‍ᠤ ᠤᠷᠪᠠᠭᠤ ᠲᠣᠭᠠᠨ ᠳ᠋ᠤ ᠦᠷᠵᠢᠭᠦᠯᠬᠦ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ:
+<br><br>
+<span style="writing-mode: horizontal-tb;">
+<br><math>\frac{1}{i} = \frac{1}{i} \cdot \frac{i}{i}</math>
+<br><math> = \frac{i}{i^2} = \frac{i}{-1}</math>
+<br><math> = -i</math>
+</span>
-:<math>\frac{1}{i} = \frac{1}{i} \cdot \frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i~.</math>
 ᠡᠭᠦᠨ ‍ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳ᠋ᠤ ᠬᠡᠷᠡᠭᠯᠡᠪᠡᠰᠦ:
+<br><br>
+<span style="writing-mode: horizontal-tb;">
+<br><math>\frac{a + bi}{i} = -i\,(a + bi)</math>
+<br><math> = -a i - bi^2 = b - a i~.</math>
+</span>
-:<math>\frac{a + bi}{i} = -i\,(a + bi) = -a i - bi^2 = b - a i~.</math>
 ᠡᠨᠡ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ‍ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠸᠧᠺᠲ᠋ᠣᠷ ‍ᠢ<sup>vector</sup> ᠨᠠᠷᠠ ᠵᠥᠪ 90°ᠡᠷᠭᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ ᠪᠤᠶ ‍ᠠ᠋᠃
@@ 42-р мөр: / 68-р мөр: @@
 ===ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ===
-<span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠨᠢ ᠮᠥᠴᠢᠯᠭᠡ ᠪᠡᠷ ᠳᠠᠪᠲᠠᠭᠳᠠᠬᠤ ᠰᠢᠨᠵᠢ ᠴᠢᠨᠠᠷ ᠲᠠᠢ:<ref>MathBits Notebook. Cyclic Nature of the Powers of "''i ".''https://mathbitsnotebook.com/Algebra2/ComplexNumbers/CPPowers.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/03</ref>
+<math>i</math> ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠨᠢ ᠮᠥᠴᠢᠯᠭᠡ ᠪᠡᠷ ᠳᠠᠪᠲᠠᠭᠳᠠᠬᠤ ᠰᠢᠨᠵᠢ ᠴᠢᠨᠠᠷ ᠲᠠᠢ:<ref>MathBits Notebook. Cyclic Nature of the Powers of "''i ".''https://mathbitsnotebook.com/Algebra2/ComplexNumbers/CPPowers.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/03</ref>
@@ 73-р мөр: / 99-р мөр: @@
 <br><math>i^{4n+2} = -1</math>
 <br><math>i^{4n+3} = -i</math>
+<br><br><span style="writing-mode: vertical-lr;">
+ᠡ᠊ᠨᠳᠡ ᠡᠴᠡ᠄
+<math>i^n = i^{(n \bmod 4)}</math>
+</span></span>
+ᠡᠩ ‍ᠦᠨ ‍ᠢᠢᠡᠷ ᠠᠷᠭᠠᠴᠢᠯᠠᠪᠠᠰᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ‍ᠤ ᠵᠡᠷᠭᠡ ‍ᠢᠢ ᠲᠣᠳᠤᠷᠬᠠᠶ᠋ᠢᠯᠠᠬᠤ ‍ᠢᠢᠨ ᠲᠤᠯᠠᠳᠠ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠯᠡᠳᠬᠡᠭᠴᠢ ‍ᠢᠢ<ref>ᠡ᠊᠂ ᠸᠠᠩᠳᠤᠢ᠃ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ ‍ᠡ᠋ ᠮᠣᠮᠢᠶᠠᠨ ‍ᠤ ᠲᠣᠯᠢ᠃ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠪᠣᠲᠢ᠃ ᠪᠦ᠊᠂ ᠨᠠ᠊᠂ ᠮᠣ᠊᠂ ᠠ᠊᠂ ᠤ᠊᠂ ‍ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ‍ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠬᠡᠯᠡ ᠵᠣᠬᠢᠶᠠᠯ ‍ᠤᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ ᠤᠯᠤᠰ ‍ᠤᠨ ᠬᠡᠪᠯᠡᠯ ‍ᠦᠨ ᠭᠠᠵᠠᠷ᠃ ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ 1970᠃</ref> <span style="writing-mode: horizontal-tb;">4</span> ᠳ᠋ᠦ ᠬᠤᠪᠢᠶᠠᠮᠤᠢ᠃
+ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">0</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>1</math>᠂
+ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">1</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>i</math>᠂
+ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">2</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>-1</math>᠂
+ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">3</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>-i</math>
+ᠪᠠᠢᠢᠨ᠎ᠠ ᠬᠡᠮᠡᠨ ᠲᠣᠭᠲᠠᠭᠠᠵᠤ ᠪᠣᠯᠤᠨ᠎ᠠ᠃
+ᠢᠢᠨ ᠬᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ‍ᠢᠢ ᠠᠯᠢᠮᠠᠳ ᠪᠦᠬᠦᠯᠢ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠥᠭᠦᠯᠪᠡᠯ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>1</math>᠂ <math>i</math>᠂ <math>-1</math>᠂ <math>-i</math> - ᠡᠳᠡᠭᠡᠷ ‍ᠦᠨ ᠨᠢᠭᠡ ᠢᠮᠠᠭᠲᠠ ᠪᠠᠢᠢᠬᠤ ᠠᠵᠤᠭᠤ᠃
+===<math>i</math> ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ <math>i</math> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ===
+<br>
+<span style="writing-mode: horizontal-tb;">
+<math>i^i = \left( e^{i (\pi/2 + 2k \pi)} \right)^i </math>
+<br>
+<math>= e^{i^2 (\pi/2 + 2k \pi)} </math>
+<br>
+<math>= e^{- (\pi/2 + 2k \pi)}</math>
 </span>
+ᠡᠭᠦᠨ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;">''k'' ∈ ℤ</span> ᠪᠤᠶᠤ ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ᠃
-ᠡ᠊ᠨᠳᠡ ᠡᠴᠡ᠄
+<span style="writing-mode: horizontal-tb;">''k'' = 0</span> ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠨᠥᠭᠦᠴᠡᠯ ᠳ᠋ᠦ ᠦᠨᠳᠦᠰᠦᠨ ᠤᠳᠬ ‍ᠠ᠋ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>e</i><sup>−''π''/2</sup></span> ᠪᠤᠶᠤ ᠣᠢᠢᠷᠠᠯᠴᠠᠭ᠎ ‍ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;">0.207879576</span> ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋᠃<ref>David Wells. The Penguin Dictionary of Curious and Interesting Numbers. UK: Penguin Books 1997. ISBN=0-14-026149-4</ref><ref>Brilliant. What is i to the power of i. https://brilliant.org/discussions/thread/what-is-i-to-the-power-of-i-T. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/017/04</ref>
-:<math>i^n = i^{(n \bmod 4)}</math>
-ᠡᠩ ‍ᠦᠨ ‍ᠢᠢᠡᠷ ᠠᠷᠭᠠᠴᠢᠯᠠᠪᠠᠰᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ‍ᠤ ᠵᠡᠷᠭᠡ ‍ᠢᠢ ᠲᠣᠳᠤᠷᠬᠠᠶᠢᠯᠠᠬᠤ ‍ᠢᠢᠨ ᠲᠤᠯᠠᠳᠠ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠦᠵᠡᠭᠦᠯᠦᠭᠴᠢ ‍ᠢᠢ<ref>ᠡ᠊᠂ ᠸᠠᠩᠳᠤᠢ᠃ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ ‍ᠡ᠋ ᠮᠣᠮᠢᠶᠠᠨ ‍ᠤ ᠲᠣᠯᠢ᠃ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠪᠣᠲᠢ᠃ ᠪᠦ᠊᠂ ᠨᠠ᠊᠂ ᠮᠣ᠊᠂ ᠠ᠊᠂ ᠤ᠊᠂ ‍ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ‍ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠬᠡᠯᠡ ᠵᠣᠬᠢᠶᠠᠯ ‍ᠤᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ ᠤᠯᠤᠰ ‍ᠤᠨ ᠬᠡᠪᠯᠡᠯ ‍ᠦᠨ ᠭᠠᠵᠠᠷ᠃ ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ 1970᠃</ref> <span style="writing-mode: horizontal-tb;">4</span> ᠳ᠋ᠦ ᠬᠤᠪᠢᠶᠠᠮᠤᠢ᠃
+===ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠡᠴᠡ ᠢᠵᠠᠭᠤᠷ ᠭᠠᠷᠭᠠᠬᠤ===
+<br>
+{| class="wikitable"
+|+
+|[[File:3-root of imaginary unit,.svg|thumb]]
+|style = "height: 150px;"|<span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ‍ᠤ ᠺᠦ᠋ᠪ<sup>cubic</sup> ᠢᠵᠠᠭᠤᠷ ᠨᠢ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠣᠷᠣᠢ ᠨᠤᠭᠤᠳ ᠋ᠲᠤ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠮᠤᠢ᠃
+|}
+<br>
+<math>i</math> ᠲᠣᠭᠠᠨ ‍ᠤ <span style="writing-mode: horizontal-tb;"><i>n</i></span> ᠵᠡᠷᠭᠡ ‍ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>n</i></span> ᠲᠣᠭᠠᠨ ‍ᠤ ᠬᠠᠷᠢᠭᠤ ᠲᠠᠢ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃
-ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">0</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;">1</span>᠂
+ᠬᠡᠳᠦᠨ ᠬᠠᠷᠢᠭᠤ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ ᠭᠠᠷᠬᠤ <span style="writing-mode: horizontal-tb;"><i>n</i></span>-ᠥᠨᠴᠦᠭᠲᠦ ‍ᠢᠢᠨ ᠥᠨᠴᠦᠭ ‍ᠦᠨ ᠲᠣᠭ᠎ ‍ᠠ᠋ ᠪᠠᠷ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃<br><br>
+<span style="writing-mode: horizontal-tb;">
+<br><small><math>u_k=\cos {\frac{{\frac{\pi}{2}} + 2\pi k}{n}} +i\ \sin {\frac{{\frac{\pi}{2}} + 2\pi k}{n}},</math> </small>
+<br><br><small><math>\quad k=0,1,...,n-1</math> </small>
+<br><br><span style="writing-mode: vertical-lr;"><math>i</math> ᠲᠣᠭᠠᠨ ‍ᠤ <br>ᠺᠸᠠᠲᠷᠠᠲ <br>ᠢᠵᠠᠭᠤᠷ᠄</span>
+<br><small><math>\{\sqrt i\} = \left\{\frac{1+i}\sqrt2; ~\frac{-1-i}\sqrt2 \right\}</math> </small>
+<br><br><span style="writing-mode: vertical-lr;"><math>i</math> ᠲᠣᠭᠠᠨ ‍ᠤ <br>ᠺᠦᠪ <br>ᠢᠵᠠᠭᠤᠷ᠄</span>
+<br><small><math>\{\sqrt[3]i\} = \left\{-i;~\frac{i+\sqrt3}2;~ \frac{i-\sqrt3}2\right\}.</math></small>
+</span>
-ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">1</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>i</i></span>᠂
-ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">2</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;">-1</span>᠂
-ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">3</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>-i</i></span>
+===<math>i</math> ᠲᠣᠭᠠᠨ ᠡᠴᠡ <math>i</math> ᠵᠡᠷᠭᠡ ‍ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ᠭᠠᠷᠭᠠᠬᠤ===
-ᠪᠠᠢᠢᠨ᠎ᠠ ᠬᠡᠮᠡᠨ ᠲᠣᠭᠲᠠᠭᠠᠵᠤ ᠪᠣᠯᠤᠨ᠎ᠠ᠃
+<br>
+<span style="writing-mode: horizontal-tb;">
+<br><math>\sqrt[i]{i}
+= i^{\frac{1}{i}\frac{i}{i}}
+= i^{-i}</math>
+<br><math>= \biggl(e^{{i}\frac{\pi}{2}}\biggr)^{-i}</math>
+<br><math>= e^{-i^2\frac{\pi}{2}}
+= e^{\frac{\pi}{2}}</math>
+</span>
 ==ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==
-ᠦᠵᠡᠭᠦᠯᠦᠭᠴᠢ - exponent - экспонента
+ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица
+ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ᠎ ‍ᠠ᠋ - imaginary number - чисто мнимое число
+ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ ‍ᠠ᠋ (ᠬᠠᠪᠰᠤᠷᠠᠭᠰᠠᠨ ᠲᠣᠭ᠎ ‍ᠠ᠋) - complex number - комплексное число
+ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ᠎ ‍ᠠ᠋ - real number - вещественное число (действительное число)
+ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ - multiplication - умножение
+ᠬᠤᠪᠢᠶᠠᠬᠤ - division - деление
+ᠵᠡᠷᠭᠡ - power - степень
+ᠢᠵᠠᠭᠤᠷ - root - корень
+ᠢᠯᠡᠳᠬᠡᠭᠴᠢ - exponent - экспонента
 ᠦᠯᠡᠳᠡᠭᠳᠡᠯ - remainder - остаток
+ᠣᠯᠠᠨᠯᠢᠭ - set - множество
+ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ᠎ ‍ᠠ᠋ - integer - целое число