A. J. Lohwater's Russian-English Dictionary of the Mathematical Sciences

a. The number "one" is essentially an adjective, and as such agrees in gender, number, and case with its noun. The plural form, одни, is used when the following noun occurs only in the plural:

Если одни часы колеблются с периодом T, ... → If one clock vibrates with period T, ...

When the number "one" is part of a compound number, both the number "one" and the noun remain in the singular:¹

Он написал пятьдесят одну статью. → He wrote fifty-one articles.

The word один may also express the meanings "certain," "alone," and "a":

Одни математики согласились с Лобачевским. → Certain mathematicians agree with Lobachevsky.

одна теорема Лобачевского → a theorem of Lobachevsky

b. When the numbers два, две, три, and четыре (and compounds ending in them) are used in the nominative and inanimate accusative (which looks like the nominative), the accompanying noun appears in the genitive singular:

два примера: (gen. sing. of пример) → two examples

разбить на 2 интеграла (gen. sing. of интеграл) → to split into two integrals

Пусть i, j,

k — три вектора единичной длины, ... → Let i, j, k be three vectors of unit length, ...

If the noun following the number is preceded by an adjective, it may be in either the genitive plural or the nominative plural (although the noun is genitive singular in both instances). Generally, the genitive plural is used with masculine and neuter nouns, while the nominative plural is used with feminine nouns:

три сложных примера → three complex examples (masculine noun)

три сложные книги → three complex books (feminine noun)

Вводя три единичных вектора i₁, i₂, i₃, ... → If we introduce the three unit vectors i₁, i₂

, i₃, ...

If the item following the number is a substantivized adjective, that is, an adjective acting as a noun, the word is treated as an adjective under the above conditions:

две переменных → two variables

две кривые → two curves

c. When the numbers пять (five) and up through девятнадцать (nineteen), and any compound numbers ending in five, six, seven, eight, nine, or zero are in the nominative or inanimate accusative cases, both the noun and its adjective appear in the genitive plural:

тридцать шесть русских книг → thirty-six Russian books

d. When a number is used in a case other than the nominative or inanimate accusative, the number and the noun and adjectives following that number are in the plural of that particular case.² In other words, the number no longer determines the case of the adjective and noun following it; rather, it agrees in case with the following noun. In compound numbers, all components decline:

из двадцати новых книг → from twenty new books

из двадцати двух новых книг → from twenty-two new books

Если угол между двумя зеркалами равен α, ... → If the angle between two mirrors is equal to α, ...

V является функцией 12 величин. → V is a function of twelve variables.

e. Nouns following the numbers тысяча, миллион, and

миллиард are usually in the genitive plural, regardless of the case in which the number is being used. These words are declined as nouns.

2. Indefinite numerals. When the actual number is unspecified, the noun following the number is usually in the genitive plural:

Пусть уравнение L = 0 имеет n различных корней. → Let the equation L = 0 have n distinct roots.

Similarly, the expressions of quantity много (much, many), мало (little, few), несколько (several, some, a few) are indefinite numerals. When the noun following them can be counted, it appears in the genitive plural; when the noun can be measured but not counted, it is in the genitive singular:

Приведём несколько примеров. → We cite a few examples.

Имеется много различных вариантов. → Many different variants are possible.

мало света, много воды → a little light, a lot of water

The above are sentences in which the indefinite numerals are in the nominative or accusative case. These expressions may themselves occur in the other cases as well; the nouns following them occur also in that particular case, and generally in the plural:

много новых книг → many new books (nominative plus genitive plural)

из многих новых книг → from many new books (all in genitive plural)

со многими новыми книгами → with many new books (all in instrumental plural

)

Footnotes

1.	Unless the noun occurs only in the plural, like the Russian word for "clock," in which case they will be plural. ↑
2.	If the number is a compound ending in "one," the noun is in the singular. ↑

Some Mathematical Conventions

tg = tan

ctg = cotangent

sh = sinh

ch = cosh

Ареа- before a hyperbolic function means the inverse of the function

i = 1, n stands for i = 1, ..., n.