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$(a+b)^2=a^2+2ab+b^2$

$(a-b)^2=a^2-2ab+b^2$

What are these two called in English? Are they called anything at all? I looked around on some math websites and Wikipedia, but I didn't find these rules. I did find the conjugate rule for the difference of two squares. That was useful, and I have now memorized that name, but that was not exactly what I was looking for.

Maybe they are not considered to be noteworthy rules so they have not been given a name in the English speaking world?

Update:

These are called "kvadreringsregler" in Swedish. I have found some translations to other languages, but not to English.

  • Danish: Kvadratsætningen
  • Swedish: Kvadreringsreglerna
  • Norwegian: Kvadratsetningene
  • Polish: Wzory skróconego mnożenia
  • French: Identité remarquable
  • English: ???

The proposed "binomial theorem" is called "binomialsatsen" in Swedish. This is considered to be a more generalized form of "kvadreringsregelerna". The French name for this literally translates into "remarkable identities". That gave me some interesting web search results.

We call remarkable identities to [sic] some binomial products that appear very often in calculations with algebraic expressions.

Source: Mathspadilla

Could this be it?

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  • $\begingroup$ It is an example of the binomial theorem. $\endgroup$ Dec 25, 2015 at 22:04
  • $\begingroup$ A more general rule for multiplying binomials is sometimes called the FOIL method in schools. One will not see the name elsewhere. $\endgroup$ Dec 25, 2015 at 22:20
  • $\begingroup$ Ah, yes, there is an example on the Wikipedia page that uses the same form. So in a wider sense, I guess it is correct, this is the binomial theorem at work. By wider sense I mean that the binomial formula can be applied to higher powers than just 2. But in a more narrow sense, for powers of exactly 2, we have a special name for these two simple formulas in Swedish. We call them "kvadreringsregler". It's a plural for "squaring rules". $\endgroup$
    – Samir
    Dec 25, 2015 at 22:22
  • $\begingroup$ Also note the word "binomial" in "binomial formula" or "binomial theorem". As in polynomial! Binomial formula uses x and y in place of the a and b. They represent variables, algebraic expressions! The a and b are not commonly used to represent variables. $\endgroup$
    – Samir
    Dec 25, 2015 at 22:25
  • $\begingroup$ @André: One should not see that name elsewhere, but unfortunately it does happen. $\endgroup$ Dec 25, 2015 at 22:29

5 Answers 5

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I call both those identities binomial squares in my classes.

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    $\begingroup$ I like the sound of that! But "binomial square" would be the $(a+b)^2$ right? If you want to tell a student to use this identity or formula to expand an expression like $(3x+4)^2$, do you tell them to expand using the binomial identity?... $\endgroup$
    – Samir
    Dec 25, 2015 at 23:37
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    $\begingroup$ @sammyg: I would (and do) tell them to find (or expand) the binomial square. $\endgroup$ Dec 26, 2015 at 12:43
  • $\begingroup$ So you say "find the binomial square" when factorizing $9x^2+24x+16$, and "expand the binomial square" when expanding $(3x+4)^2$, right? That makes sense. But how do you refer to the rule or the identity itself? Binomial theorem? $\endgroup$
    – Samir
    Dec 26, 2015 at 14:23
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    $\begingroup$ @sammyg: I call it the binomial square identity/identities. My problem is not the name, but rather getting my students to remember and use the identity. They prefer using the more general FOIL identity, even though it is slower. $\endgroup$ Dec 26, 2015 at 14:53
  • $\begingroup$ Yes, I totally agree. But I think terminology is important, because I believe it helps to encode the rule or identity in the brain. It's like with computers, they need named identifiers in order to commit something to memory. This is also how FOIL works. It's a mnemonic that helps to recall the rule from the brain, piece by piece if necessary. I have seen "FOIL" in English texts before, but I didn't know it stood for First, Outer, Inner, Last. That's clever! We don't have anything like that in Swedish. In fact, we borrow the English "PEMDAS" acronym sometimes! ;-) $\endgroup$
    – Samir
    Dec 26, 2015 at 16:27
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Here in Spanish they are examples of "productos notables" ("remarkable products" might be the closest translation). The list includes:

$\begin{align} (a + b)^2 &= a^2 + 2 a b + b^2 \\ (a - b)^2 &= a^2 - 2 a b + b^2 \\ (a + b) (a - b) &= a^2 - b^2 \end{align}$

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  • $\begingroup$ Thank you! Yes, those are the ones! Yes, or perhaps "notable products". But when used in a sentence you have to add the word "formula" or "identity" if you are giving someone an instruction, such as "use the notable identity to expand..." $\endgroup$
    – Samir
    Dec 26, 2015 at 0:18
  • $\begingroup$ Notable product appears in sangakoo.com/en/unit/notable-products, apparently a Spanish site. "Produtos notáveis" is also used in Portuguese, but only in school mathematics. $\endgroup$
    – lhf
    Dec 26, 2015 at 11:21
  • $\begingroup$ Yes, so I noticed. It says Barcelona in the page footer. But how do you use this name in a sentence? How do you tell someone to use this "produtos notáveis" to expand or to factor? $\endgroup$
    – Samir
    Dec 26, 2015 at 11:37
  • $\begingroup$ @sammyg, "using sum-by-difference..." (the third identity), "this is the square of the binomial..." or something similar $\endgroup$
    – vonbrand
    Dec 26, 2015 at 12:13
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Identité remarquable!

In other words, remarkable identities!

We call remarkable identities to [sic] some binomial products that appear very often in calculations with algebraic expressions.

Source: Mathspadilla

Square of an addition:

$(a + b)^2 = a^2 + 2ab + b^2$

Square of a subtraction:

$(a - b)^2 = a^2 - 2ab + b^2$

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  • $\begingroup$ After reviewing the web search results, it seems to me that the term "remarkable identities" is mostly used by English speakers from India. It appears to be uncommon in the US. But there is at least one American site that collectively refers to these as "algebraic identities" and to the rules as "product formulas" or "binomial formulas". The difference of two squares is referred to as factoring formula. $\endgroup$
    – Samir
    Dec 26, 2015 at 0:01
  • $\begingroup$ Perhaps the French name for it should be translated as "notable identities" in English. $\endgroup$
    – Samir
    Dec 26, 2015 at 10:39
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I will use the information I have gathered so far to try to provide proper English translations. I will be using Swedish as the base language.

Kvadreringsreglerna


Första kvadreringsregeln:

$(a+b)^2=a^2+2ab+b^2$

This is often referred to as the first rule.

Andra kvadreringsregeln:

$(a-b)^2=a^2-2ab+b^2$

This is referred to as the second rule.

The first thing I realize is that there are no specific names for these in English. So I would propose the translation identities of binomial squares for the two of them. I would then suggest the name identity of binomial squares of a sum for the first rule, and identity of binomial squares of a difference for the second rule.

When used in a sentence:

  • Använd kvadreringsreglerna för att utveckla uttrycket.
  • Use the identities of binomial squares to expand the expression.
  • Använd kvadreringsreglerna för att faktorisera.
  • Use the identities of binomial squares to factorize.

What we refer to as rules ("regler") in Swedish, in this context, is referred to as identities in English. We do have the term "identitet" but we usually don't use it in this context. These identities are so important that we have decided to give them a special name. Apparently we are not alone in this, there are several other languages that have a special name for these identities.

  • French: Identité remarquable
  • Spanish: Productos notables
  • Portuguese: Produtos notáveis
  • Dutch: Merkwoardig product
  • Italian: Prodotto notevole

Dare we translate that into English as "notable products"? Or as "remarkable identities" I suggested earlier? Most of the sources I found that use those terms in English are web sites that are authored by people from India, Spain or Portugal. They may not be native English speakers, so they translate it literally from their own language.

For what I have seen so far, it's only in Nordic countries that these are referred to as rules or theorems.

  • Swedish: Kvadreringsreglerna (lit. squering rules)
  • Danish: Kvadratsætningen (square theorem)
  • Norwegian: Kvadratsetningene (square theorem)

This is really an example of the binomial theorem. But the binomial theorem is considered to be the more generalized, all-encompassing theorem for higher powers. When dealing with binomials with powers of 2, we call for the "kvadreringsregler" (Swedish), or "Kvadratsetningene" (Norwegian). For powers higher than 2 (or 3 in case of Swedish) we call for the "binomialsatsen" (binomial theorem).

Kuberingsregeln


Even though the question didn't call for cubes, but I might as well add these in.

Första kuberingsregeln:

$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

Andra kuberingsregeln:

$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

Similarly, these rules or identities apply to binomials of third power. I would propose the name identities of binomial cubes for the two of them, identity of binomial cube of a sum for the first rule, and identity of binomial cube of a difference for the second rule.

They might also be referred to as binomial theorem for cubes. Similarly, one might use a similar name for the powers of 2, by only replacing the word "cubes" with "squares". That might provide a more elegant description, without going into detail of "sum" and "difference".

Konjugatregeln


Again, the question didn't call for this one. But I will add this in, since it is one of those "notable product" identities used in other languages.

$a^2 - b^2 = (a+b)(a-b)$

This one actually does have an established English name. It's called the difference of two squares. The Swedish name for it might be translated as conjugate identity of two squares. One might use the "conjugate identity of two squares" to expand or to factorize an expression.

Algebraic identities


These are all collectively referred to as algebraic identities in English.

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I note in your longer answer to yourself that you skip over a subtle but important distinction: In Swedish, and the other Nordic tongues, the names you give refer to the formulae as a whole: the "squaring rules", the "cubing rules", the "conjugate rule". In English, the only one "named" is the conjugate rule, but that name does not actually refer to the formula.

Instead, it refers to one of the expressions, specifically the produced (I guess? neither "unfactored" nor "multiplied" feel right to me here) one, and this is why it has a name: the difference of two squares is easy to name as an English phrase (or a phrase in any natural language, really) while the two squaring rules produce trinomials which are harder to name without invoking symbols. If I were to try, it would be "the sum of two squares with twice the product of their roots" for the first squaring rule and "the difference between the sum of two squares and twice the product of their roots" for the second. The cubing rules, meanwhile, produce quadrinomials, which are even more of a nightmare to name.

Now, English could just as easily use the name for the factored expression for the formulae: "square of a sum", "square of a difference", "cube of a sum", "cube of a difference" and "conjugate product" (or, for the uninitiated, "the product of the sum of two values with their difference"), but it doesn't. But then, neither does it use "difference of two squares" for the conjugate rule, as you claim. As a native English speaker (General American dialect; results may vary), I never think "$a^2-b^2=$ $(a+b)(a-b)$" when I hear or say "difference of two squares". I think "$a^2-b^2$ which equals $(a+b)(a-b)$". The two expressions are equivalent mathematically but not syntactically and are separate in my mind.

Personally, I like the idea of a literal translation from the Swedish, with perhaps the modifier "binomial" for the squaring and cubing rules. They're simple and descriptive, perfect for teaching.

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