"ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ"-ны өөр хувилбарууд

11:11, 4 Долоодугаар сар 2021-ий байдлаарх засвар

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

ᠥᠭᠡᠷ ‍ᠡ᠋ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠻᠸᠠᠲᠷᠠᠲ᠋ 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}}

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᠃

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

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

ᠵᠠᠷᠢᠮ ᠦᠢᠯᠡᠳᠦᠯ

ᠦᠷᠵᠢᠭᠦᠯᠬᠦ ᠬᠤᠪᠢᠶᠠᠬᠤ

ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠷᠴᠡ ᠪᠤᠶᠤ i ᠲᠣᠭᠠᠨ ᠳ᠋ᠦ ᠦᠷᠵᠢᠬᠦᠯᠦᠭᠰᠡᠨ ‍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 ᠭᠠᠷ 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

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 ᠲᠣᠳᠤᠷᠬᠠᠶᠢᠯᠠᠬᠤ ‍ᠢᠢᠨ ᠲᠤᠯᠠᠳᠠ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠦᠵᠡᠭᠦᠯᠦᠭᠴᠢ ‍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

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

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

k = 0 ᠨᠥᠭᠦᠴᠡᠯ ᠳ᠋ᠦ ᠦᠨᠳᠢᠰᠦᠨ ᠤᠳᠬ ‍ᠠ᠋ ᠨᠢ e^−π/2 ᠪᠤᠶᠤ ᠣᠢᠢᠷᠠᠯᠴᠠᠭ᠎ ‍ᠠ᠋ ᠪᠠᠷ 0.207879576 ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋᠃

ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ

ᠦᠵᠡᠭᠦᠯᠦᠭᠴᠢ - exponent - экспонента

ᠦᠯᠡᠳᠡᠭᠳᠡᠯ - remainder - остаток

ᠣᠯᠠᠨᠯᠢᠭ

ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ᠎ ‍ᠠ᠋

ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ

↑ 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᠃

[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᠃

[1]

[2]

[3]

[4]

@@ 93-р мөр: / 93-р мөр: @@
 ᠢᠢᠨ ᠬᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ‍ᠢᠢ ᠠᠯᠢᠮᠠᠳ ᠪᠦᠬᠦᠯᠢ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠥᠭᠦᠯᠪᠡᠯ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>1</i></span>᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span>᠂ <span style="writing-mode: horizontal-tb;"><i>-1</i></span>᠂ <span style="writing-mode: horizontal-tb;"><i>-i</i></span> - ᠡᠳᠡᠭᠡᠷ ‍ᠦᠨ ᠨᠢᠭᠡ ᠢᠮᠠᠭᠲᠠ ᠪᠠᠢᠢᠬᠤ ᠠᠵᠤᠭᠤ᠃
+===<span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>i</i></span> 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᠃
+<span style="writing-mode: horizontal-tb;">''k'' = 0</span> ᠨᠥᠭᠦᠴᠡᠯ ᠳ᠋ᠦ ᠦᠨᠳᠢᠰᠦᠨ 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᠃
 ==ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==
@@ 98-р мөр: / 112-р мөр: @@
 ᠦᠯᠡᠳᠡᠭᠳᠡᠯ - remainder - остаток
+ᠣᠯᠠᠨᠯᠢᠭ
+ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ᠎ ‍ᠠ᠋