# mathematical pronunciation of 1

I have a question about pronouncing `1` in Russian for mathematics.

When `1` appears by itself on the right side of an equation (or inequality or congruence or other mathematical relation), it seems from what I've heard in lectures that it is always pronounced as `единица` instead of `один`. For example

1. x2 = 1 would be

• `икс квадрат равно единице` instead of
• `икс квадрат равно одному`,
2. and 3x ≡ 1 mod 7 would be pronounced

• `три икc сравнимо с единицей по модулю семь` instead of
• `три икс сравнимо с одним по модулю семь`.

Is this convention followed only when `1` is the only term on the right side, or are there other situations where an equation/inequality/congruence would have `1` pronounced as `единица`? To be clear, I am not asking about the usage of `единица` in the sense of an identity element in a group or a unit in a ring, but rather a setting where it is really synonymous with `один`.

• What about splitting this in 2 separate questions? Jun 14 '12 at 4:33
• @defaultlocale probably a good idea. The question is quite interesting though. Jun 14 '12 at 4:43
• So I should delete this question and ask each one separately?
– KCd
Jun 14 '12 at 4:46
• @KCd question is really interesting. You don't have to delete it. You can edit this one leaving only the first part. Then ask the second as a new question. Jun 14 '12 at 4:51
• The second question I had before has now been moved to a separate question.
– KCd
Jun 14 '12 at 5:02

The word "единица" is a noun and can be used alone, but "один" is a quantifier that needs something to quantify thus should not be used alone

Take a look at three in the expression 3x. It has "x" to quantify, this is why you say "три икс" instead of "тройка икс"

Update 1: It's time to summarize what we've been talking about below. Oh man, that was hard.

When it comes to math formulas, there are three players:

• number names (plain numbers in nominative case): "один", "два", "три"
• declined number names (in dative case: одному, двум, трем; or genitive case: трех, четырех, восьми)
• nouns representing numbers: "тройка", "единица"

There are some rules I can think of

1. You must use plain number names for numbers that immediately follow "плюс" or "минус", because it is not clear how to decline these numbers. Good example: "единица плюс пять игрек равно четыре миунс два корня из трех". Bad example (wrong one): "икс равен двойке плюс тройку минус четверкy".

1.1. However, this rule is not the case for stand alone numbers: "в некоторых условиях единица равна минус одному" (it's clear how "минус одному" should be declined).

2. You must use a plain number when the number is used as a quantifier: "четыре икс", "два Пи" in the context of the expression (with other numbers around).

2.1. You can use a declined number name on the right side to get it to agree with the relation connecting the two sides: "равно" + dative (equal to), "больше" + genitive, "меньше" + genitive. Good example: "эф от икс равно четырём икс", "длина окружности равна двум пи умноженным на радуис".

3. You should avoid using plain number names for a number that stands alone on either side of the equation. Although saying something like "один плюс один равно два" isn't a crime, the following would be better: "один плюс один равно двум" or "один плюс один равно двойке". You should consider either of the following:

3.1. Using a noun for a stand-alone number even if that number is wrapped into an operator/function. For example: "корень единицы равен единице", "единица больше либо равна косинусу трех икс", "двойка в квадрате равна четырем".

3.3. Or using a declined number name for a stand-alone number on the right side to get it agree with the predicate of the sentence ("равно", "не равно", "больше", "меньше"). For example: "один плюс два равно трем", "два икс плюс четыре больше трех".

3.4. However, you are highly discouraged from using a declined number name for a stand-alone number on the left side because there is nothing for it to agree with. (For comparison, the agreement of a declined number name on the right side is with the type of math relation, like = or <, that came before it). Bad example: "одному равно три минус два" even though somebody can find it OK, this sounds like Yoda's way of speaking.

4. You must decline a standalone number to get it to agree with the operators/functions "корень", "косинус", etc. For example: "корень из четырех равен двум". You cannot say "корень из два" or "косинус один".

5. Rules for one side of an equation do not affect rules for the other, so these sides should be considered separately. Which means you CAN use a noun on one side with number on the other if it doesn't break the above rules. For example: "в некоторых условиях единица равна минус одному", "единица плюс два равно тройке".

Update 2: I think we can generalize it even further.

1. You must always decline numbers, with a few exceptions:

• there is nothing to get these numbers agreed with
• when the nominative case is required by the construction you use: "[SOMETHING] минус [ALWAYS_NUMBER_IN_NOMINATIVE]", "логарифм икс по основанию [ALWAYS_NUMBER_IN_NOMINATIVE]" UPDATE 3: There is a catch: although "два" in "единица плюс два" seems to be in nominative, it is actually in the accusative case" that shares the same form with the nominative for inanimate nouns (and numbers). So what's above should be read: "[SOMETHING] минус [ALWAYS_NUMBER_IN_ACCUSIVE]", "логарифм икс по основанию [ALWAYS_NUMBER_IN_ACCUSIVE]"

• "равно", "равна", "равен" allows (but does not require) being followed by a nominative regardless of anything: "[SOMETHING] равно пять" is OK as well as "[SOMETHING] равно пяти". (On the other hand "меньше" or "больше" always require to be followed by a genitive for that matter)

• quantifiers are allowed (but not required) to be used in nominative regardless of anything: "косинус девять икс" is OK as well as "косинус девяти икс", but "косинус девять" is NOT
2. You can use a noun for a number that is not a quantifier.

• Hey all, if you want to continue discussing the pronunciation of exponents, consider posting another question about that or drop by Russian Language Chat. I'm going to clean up the comments here now that the answer has been updated and its original issues addressed. Jun 15 '12 at 19:45
• "эф от икс равно четверке икс" is wrong, nobody says this way. The correct variant is "эф от икс равно четыре икс" or "эф от икс равно четырём икс". And also it is in the wrong section (in this section you talk about plain number names rather than nouns) so I think it is your typo. Dec 24 '13 at 13:33
• I would say that one should be strongly discouraged from using words like "двойка", "тройка", "четверка" и т.д. in mathematical formulas in any way. The only exception is "единица". Dec 24 '13 at 13:41

It seems to me that this is all due to the case being used. If you are reading mathematical equations with a simple right part, you usually use the dative case. It means that the right part of the equation answers the question "than what?" or "than whom?"

-5 < 1 "Минус пять меньше единицы." The question here is: "Меньше чего?"

You can also use the word "один" here:

-5 < 1 "Минус пять меньше одного."

The problem here is that in my opinion it sounds like you are trying to answer a question "Than whom?" rather than "Than what?", and the sentence looks unfinished because the question "Than whom?" implies that there is some noun missing.

On the other hand, if there is a complex right side of the equation, the sentence is usually built without using the dative case:

5 < 1 + 10 "Пять меньше, чем один плюс десять" The question here is "Меньше, чем что?"

Here you can use both "один" and "единица", but the first word is shorter, so it is used more frequently.

• I'm not interested in cases specifically (which vary with the relation being =, <, etc.), but the settings where единица is used in place of один and why, regardless of how they are declined. For instance, if I had some integral equal to 2, I'd say этот интеграл равен двум. If the integral were 1, do you think it sounds fine to say этот интеграл равен одному?
– KCd
Jun 14 '12 at 14:17

In our practice there is no big difference in meaning between `один` and `единица`.

If you read `x*2=1` you can say both more short and convenient `икс на два равно один` either more draggy or pretending to be academical `икс умножить на двойку равняется единице`.

BTW, the same about `икс квадрат` instead of `икс в квадрате` or `икс в степени два` Please note you cannot say `икс в квадрат`.

• Thanks for the comment on икс в квадрат, which I corrected in the question.
– KCd
Jun 14 '12 at 14:09
• You'd really say икс на два равно один instead of икс на два равно одному? I'd think a number on the right belongс in dative. Would you pronounce 4x = 3 as четыре икс равно трём, not четыре икс равно три.
– KCd
Jun 14 '12 at 14:29
• It may be not absolutely correct but it's simpler and faster. Moreover, if we are not at math class we say "будет" instead of "равно" Jun 14 '12 at 15:22

Both words can be used interchangeably, though there may be slight semantic differences. Ultimately, it comes down to tradition, preference, and the way it sounds. Other answers here describe well where one would be preferred over another.

Big Explanatory Dictionary lists “один” as digit (glyph), number, and quantity, amongst other things, and specifically gives this arithmetical example: “Три минус два равняется одному” (3 – 2 = 1). BED defines “единицу” same way, adding meanings of single-digit place (“разряд единц”), and “unit” (mathematical example: y = 2x + C, where graph moves up C units (“C единиц”)).

Ushakov’s Explanatory Dictionary (1935–1940) defines “один” as 1) name of the number one (название числа 1); 2) quantity of 1; 9) first in a row of similar things. In contrast to the first of the above, “единица” is described as 1) “Цифра, изображающая число один” (figure representing number one).

Interestingly, Compendium of Figurative Words and Allegories (1904) gives this variant:

Единица (иноск.) существо, (отдѣльно отъ другихъ, одинаковыхъ съ нимъ).

Translation: being [creature] (separate from others, equal to him).

• your example of y = 2x + C is not the kind of thing I was asking about, since there is no 1 anywhere. I'd pronounce that as игрек равно два икс плюс це. That example is illustrating a different meaning of единица than I had in mind. Also, the example from the 1904 dictionary is not related to the question directly since it's not about the role of единица in place of один in mathematical formulas or relations.
– KCd
Jun 14 '12 at 14:27

`Единица`, while having an "unofficial" meaning of `one` as a number, comes from the same family as `двойка`, `тройка`, `четвёрка` etc - words that define digits. Usage of `единица` is unofficial: there are no good rules when to use it rather than `one`.

• Usage of “единица” in context the asker provided is just as “official” as anything else (see my answer). Keep in mind, that “единица”, “двойка” (“пара”), “тройка” and so forth are not only digits (figures, цифры), but also quantities, represented by those digits. And your insistence on authority being a native speaker only indicates preference, not a steadfast grammatical rule, or an academic convention. I had seen plenty text books where the word in question was used precisely in the context provided. Jun 14 '12 at 6:59
• I see the point about preference, however, while you are correct about quantities (“единица”, “двойка” (“пара”), “тройка” etc), their use in sentences describing expressions/formulas is a bit the same as "видишь цифру 250 в третьей строке?", when everyone would still understand, but it won't be correct. "Тройка плюс единица равна четвёрке" - have you ever heard it said outside of elementary school? Jun 14 '12 at 7:17
• Вычисление числа π/4 с помощью разложения в ряд функции arctg(x) в окрестности единицы. (Obviously, we are not talking elementary arithmetic here.) Jun 14 '12 at 7:29
• @theUg: Thanks for the example. That is the kind of situation I am interested in, not elementary arithmetic. A series expansion "at the point 1" would be в точке единица, yes? Would an expansion "at the point 2" or "in a neighborhood of 2" be в точке два and в окрестности двух?
– KCd
Jun 14 '12 at 14:08

`икс квадрат равно единице` is incorrect.

How to translate `ObjectA = 1` into Russian

There are three genders in Russian:
`masculine, feminine, neuter` == `мужской, женский и средний`.
You will have to know the genders of an ObjectA or either singular or plural form.
`Plural` == `множественное число`.

Алгоритм (algorithm):

``````If ObjectA is `мужской род` Then
ObjectA равен единице                // интеграл равен единице

else If ObjectA is `средний род` Then
ObjectA равно единице                // число равно единице

else If ObjectA is `женский род` Then
ObjectA равна единице                // функция равна единице

else If ObjectA are `множественное число` Then
ObjectA равны единице                // неизвестные равны единице

EndIf
``````

Итак, правильно будет:

Квадрат, интеграл, момент, факториал, коэффициент == `мужской род`

`икс квадрат равен единице`
`интеграл равен единице`
`момент равен единице`
`факториал равен единице`
`коэффициент равен единице`

Сумма, разность, сторона, площадь, функция == `женский род`

`сумма равна единице`
`разность равна единице`
`сторона равна единице`
`площадь равна единице`
`функция равна единице`

ребро, число, ускорение, расстояние, произведение == `средний род`

`ребро равно единице`
`число равно единице`
`ускорение равно единице`
`расстояние равно единице`
`произведение равно единице`

Желтое, красное, одно, первое, третье == `средний род`

`Желтое равно красному`
`красное равно единице`
`одно равно другому`
`первое равно второму`
`третье равно пятому`

Неизвестные, коэффициенты, ребра, стороны, первые == `множественное число`

`Неизвестные равны единице`
`Коэффициенты равны единице`
`ребра равны единице`
`стороны равны единице`
`первые равны единице`

Next,

How to translate `ObjectA = 1 ObjectB` into Russian, for ex.: `a value equal to one meter`?

``````If ObjectA is `мужской род` Then

If ObjectB is `мужской род` Then
ObjectA равен одному ObjectB       // мегабит равен одному мегабиту

else If ObjectB is `средний род` Then
ObjectA равен одному ObjectB       // мегабит равен одному числу

else If ObjectB is `женский род` Then
ObjectA равен одной ObjectB        // мегабит равен одной части

else If ObjectB are `множественное число` Then
ObjectA равен одним ObjectB        // показатель равен одним суткам

EndIf
EndIf

If ObjectA is `средний род` Then

If ObjectB is `мужской род` Then
ObjectA равно одному ObjectB       // ребро равно одному метру

else If ObjectB is `средний род` Then
ObjectA равно одному ObjectB      // ребро равно одному числу

else If ObjectB is `женский род` Then
ObjectA равно одной ObjectB       // ребро равно одной сажени

else If ObjectB are `множественное число` Then
None

EndIf
EndIf

If ObjectA is `женский род` Then

If ObjectB is `мужской род` Then
ObjectA равна одному ObjectB       // длина равна одному метру

else If ObjectB is `средний род` Then
ObjectA равна одному ObjectB       // длина равна одному числу

else If ObjectB is `женский род` Then
ObjectA равна одной ObjectB       // длина равна одной сажени

else If ObjectB are `множественное число` Then
ObjectA равна одним ObjectB       // Норма равна одним суткам

EndIf
EndIf

If ObjectA are `множественное число` Then

If ObjectB is `мужской род` Then
ObjectA равны одному ObjectB       // неизвестные равны одному метру

else If ObjectB is `средний род` Then
ObjectA равны одному ObjectB       // неизвестные равны одному числу

else If ObjectB is `женский род` Then
ObjectA равна одной ObjectB       // неизвестные равны одной сажени

else If ObjectB are `множественное число` Then
ObjectA равны одним ObjectB       // периоды равны одним суткам

EndIf
EndIf
``````

Икс квадрат равен одному метру. Чему равен икс?
Расстояние равно одному дюйму.
Неизвестное равно одной сажени.
Сумма двух чисел равна одному из них. Чему равно это число?

The `равен/равна/равно/равны` are setted by genders or plural forms of operations or functions.

`Результат возведения в квадрат -- квадрат [числа]; мужской род`
`Результат вычисления синуса -- синус [числа]; мужской род`
`Результат произведения -- произведение [чисел]; средний род`

Мы полностью последовательны и логичны (there is a full consistency):

`i^2 = 4 и квадрат равен четырем`
`i^2 = 5 квадрат и равен четырем`
`sin(i) = 1/2 синус и равен одной второй`
`Div E = 0 дивергенция йэ равна нулю`
`Div X = 0 дивергенция икс равна нулю`

`Результат сложения -- сумма [чисел]; женский род`

`a+b = 2` `а плюс бэ равно двум` ( Plus was read aloud as a letter or glyph )
`a+b = 2` `сумма а и бэ равна двум` ( setted by genders )

• You wrote an enormous number of examples that could all be summarized more succinctly as "равен/равна/равно/равны + dative." I'm not a fan of presenting answers as a computer program, but I understand it all the same. You say at the very start that pronouncing x^2 = 1 as икс квадрат равно единице is illiterate. None of the other answers say it is bad and that is how I have heard math pronounced, the point being that x^2 is regarded as выражение, hence средний род. Have you ever heard a power of x on the left side of an equation followed by равно?
– KCd
Apr 6 '15 at 5:08
• It doesn't matter, has anybody heard this or not, though making mistakes is in human nature. We should be able give the rule and logical substantiation. How to translate `f(x) = 1` into Russian? (pronunciation): `эф от икс`. Each character — or glyph — is placed inside a box, either an expression or word. The attribute genders is not connected to a character. So this attribute should be determined in relation to the word 'expression', hence `средний род`.Thus `f(x) = 1` pronounced as `эф от икс равно единице`. But the word 'square' is a word, not a character. Apr 6 '15 at 5:58
• How would you say "z^2 =": зэд квадрат равен or зед квадрат равно? I'm trying to check if your focus with "x^2 =" is on x being masculine or квадрат being masculine. I'm pretty sure that Latin letters other than x and y are considered neuter.
– KCd
Apr 6 '15 at 16:41
• I also think in this case that it really does matter if anybody has heard this or not because mathematical expressions are rarely written out in words. The oral tradition carries a great deal of weight here.
– KCd
Apr 6 '15 at 20:06
• @KCd I was update my answer. If there are objections, I hope someone represent by writing, not just flagged. I focus with "x^2 =" on `квадрат`. I'm not sure about letters, whether they could be considered as attributed by genders. Apr 6 '15 at 21:06