Are there English names for those formulas? (binomial formulas)

[Trysse]

In Germany the formulas
$$(a+b)^2=a^2+2ab+b^2$$
$$(a-b)^2=a^2-2ab-b^2$$
$$(a+b)(a-b)=a^2-b^2$$
are referred to as the first, second, and third "binomische Formel" (i.e. binomial formula). In German schools, these formulas are usually part of 8th grade syllabus.

See here for example https://de.wikipedia.org/wiki/Binomische_Formeln

However, when I search for "Binomial formula" in English, I find only links to the "binomial theorem". https://en.wikipedia.org/wiki/Binomial_theorem.

The German Wikipedia entry has no English counterpart.

Is there any specific name in English mathematics for these three formulas?

 

[PeroK]

Is there any specific name in English mathematics for these three formulas?

Not that I know.

 

[jedishrfu]

Sometimes folks refer to them as quadratic binomials:

https://en.wikipedia.org/wiki/Degree_of_a_polynomial

There is no wiki page specific to that though only the one mentioned above.

 

[fresh_42]

The lack of it is indeed a bit annoying. Speaking of a "theorem" or referencing the entire Pascal triangle if you want to refer to these more or less elementary formulas that are more than only often used at school appears to me to be a bit like "mit Kanonen auf Spatzen schießen" (="shooting at sparrows with canons"). It bothered me more than once that there is no adequate term for it. I always thought that I made things more complicated than would have been necessary.

 

[jedishrfu]

what about quadratic binomial?

 

[fresh_42]

what about quadratic binomial?

That still lacks the numbering and we cannot assess whether this is well understood if there isn't even a Wikipedia page about it. Wikipedia lists "quadratic formula" if you search for "quadratic binomial". And that would be what I learnt as p-q-formula:
$$
x^2+px+q=0 \Longrightarrow x_{1,2}=-\dfrac{p}{2}\pm \sqrt{\left(\dfrac{p}{2}\right)^2-q}\;,
$$
but the quadratic formula works as well.

Particularly $a^2-b^2=(a-b)\cdot (a+b)$ is so important that it is not only useful at school but also for many mathematical "tricks" in proofs. We are so used to referring to it as "third binomial formula" that everybody understands it immediately. The quadratic binomial cannot even come close to filling this gap. It's just too unspecific.

 

[jedishrfu]

I mention it because the wiki page has a naming convention for these we learn in school.

quadratic binomial --> quadratic function --> quadratic formula

Perhaps this disconnect explains why German education is more thorough than US math education.

 

[Trysse]

OK. So there is a gap between German and English nomenclature: There is no English equivalent for the German names. Good to know.

Are these formulas part of the UK or US syllabus or are they not specifically taught?

 

[jedishrfu]

Of course, they are taught.

Usually, in the context of factoring quadratics. We use the formula only when all else fails. Of course, kids being kids will jump to the formula and then struggle to reduce it to get a solution and mix up a sign or two before arriving at the correct answers.

Here's one curriculum for Algebra 1 as taught in a community college. High school Algebra 1 is similar but not as extensive.

https://mathispower4u.com/algebra.php

 

[fresh_42]

Perhaps this disconnect explains why German education is more thorough than US math education.

I'm not sure I would sign this as far as mathematics is concerned. According to my nephews, this is true for geography or history classes. I just thought today (watching Star Trek the original series) that American sci-fi stories, especially the older ones barely speak about the language problem. Everybody assumes that English is the norm - no questioning. Douglas Adams on the other hand invented the babblefish. This comparison explains a lot if you think about it. It also reflects my own experiences in the UK versus the US.

I think that many students automatically associate Pythagoras if someone says $a^2+b^2=c^2$ and vice versa no matter where. We associate the quadratic formula by p-q-formula, and the three formulas in the OP of this thread as 1st, 2nd and 3rd binomial formula. It is more a matter of convenience than it is a matter of diligence.

I could add quite a few examples of school mathematics where I disagree with either system. I just recently had an example where an English schoolbook (not sure of which country) used differentiability to explain monotone behavior. I didn't know that I was talking to a kid, so I expressed my dislike of comparing a global statement (monotony on intervals) by means of a local statement (differentiability in a neighborhood). And I remember that I got mad at German schoolbooks more than once when tutoring kids. The phrase n-th binomial formula is in itself hyperbolic since in the end it is simply the distributive law. However, it is pretty convenient, especially the third one $(a^2-b^2)$ as it occurs so often. One of my standard advice is "always automatically think of the third binomial formula whenever you see a square" sounds so much better than it would sound by describing it with the distributive law, or explicitly.

 

[renormalize]

OK. So there is a gap between German and English nomenclature: There is no English equivalent for the German names. Good to know.

I don't know how universal their nomenclature is, but the authors of this US introductory algebra text: https://www.opentextbookstore.com/sousa/CK12IntroAlg.pdf use the terms "Sum and Difference Formula" and "Square of a Binomial Formula" (pg. 464):

 

[PeroK]

OK. So there is a gap between German and English nomenclature: There is no English equivalent for the German names. Good to know.

Are these formulas part of the UK or US syllabus or are they not specifically taught?

I must admit I've never worried about not having names for these things. We don't need a name for everything. I would simply describe these as the expansion of $(a + b)^2$ etc. The mathematics is inherent in the expansion itself, and not in the name.

Roberts Burns (1759-96), the Scottish national poet, famously wrote a rose by any other name would smell as sweet.

 

[fresh_42]

We don't need a name for everything.

My experience on PF is, that especially formulas and theorems have a lot more names in English than they have in German, especially if they are named after people. Not always, as the lack of Weierstraß or Graßmann shows, and not always complete as the lack of Raphson in Newton's algorithm shows. This
even made me write Abelian and Cartesian with caps because everybody writes all the others like Hamiltonian or Lagrangian with caps, too. We don't need a name for those formulas, but it is convenient.

 

[Trysse]

One of my standard advice is "always automatically think of the third binomial formula whenever you see a square" sounds so much better than it would sound by describing it with the distributive law, or explicitly.

I like this advice. And actually, that is why I came up with my original question.

 

[pbuk]

OK. So there is a gap between German and English nomenclature: There is no English equivalent for the German names. Good to know.

Are these formulas part of the UK or US syllabus or are they not specifically taught?

Unfortunately I believe all of the replies have been either from German natives or (say it quietly) Americans (with a possible Canadian as well who should know better ).

The most useful of these formulae, die dritte binomische Formel is known by the much more descriptive and memorable name as the "difference of two squares" and is taught in the National Curriculum at ages 12-16: see e.g. https://www.bbc.co.uk/bitesize/guides/z94k7hv/revision/3

In the UK we usually write this the other way round: $a^2-b^2 = (a+b)(a-b)$, and so we also have a different arrangement of the first theorem as the "sum of two squares" $a^2 + b^2 = (a+b)^2 - 2ab$.

A similar rearrangement of the second formula leads to $a^2-b^2 = (a-b)^2+2ab$ which gives a different expression for the difference of two squares: I don't know of an English name for this.

 

[DrClaude]

Roberts Burns (1759-96), the Scottish national poet, famously wrote a rose by any other name would smell as sweet.

I don't want to derail the thread, but Shakespeare wrote that about 2 centuries earlier:

What's in a name? that which we call a rose
By any other name would smell as sweet;

 

[PeroK]

I don't want to derail the thread, but Shakespeare wrote that about 2 centuries earlier:

Yes, I got the quotations mixed up. It was Shakespeare. Burns wrote a poem called A Red, Red Rose, which must have confused me.

 

[pines-demon]

This reminds me of French relation de Chasles, which is a pretty useful property of sums and vectors but it is completely trivial and it is understandable why other languages do not care to name it.

In France, one also learns about the set of terminating decimal numbers $\mathbb{D}$ as if it was a very common set to introduce in the hierarchy of naturals, integers, rationals and reals:

 

[fresh_42]

The quotation from Shakespeare is inappropriate since it misses the point, as so often in such discussions that affect what people are used to or not. We are not discussing the name of the rose but rather whether it has a name at all! This is a polemic, not an argument. Sigh.

We are also discussing didactic not mathematics. The term 3rd binomial formula is introduced in schools to make life easier for teachers! I have seen a lot more examples of terms in schoolbooks that I found questionable to say the least. This one does not belong in that category. Yes, it is not necessary and it is closer to the distributive law than it is related to Pascal's triangle, but it is simply convenient to call some of the Rosaceae roses.

 

[martinbn]

The quotation from Shakespeare is inappropriate since it misses the point, as so often in such discussions that affect what people are used to or not. We are not discussing the name of the rose but rather whether it has a name at all! This is a polemic, not an argument.

Well, a rose without a name would smell the same.

 

[weirdoguy]

In polish we call them (per google translator), "abbreviated multiplication formulas", although I feel like "reduced" is a better word than "abbreviated". Wzory skróconego mnożenia. And as a teacher, names are sometimes important, because they facilitate communication (again I have some problem with word "facilitate", my english is not englishing lately). Teaching is mostly about proper communication and flexibility in that.

 

[martinbn]

Since we are quoting poets here is one from Feynman

The next Monday, when the fathers were all back at work, we kids were playing in a field. One kid says to me, “See that bird? What kind of bird is that?” I said, “I haven’t the slightest idea what kind of a bird it is.” He says, “It’s a brown-throated thrush. Your father doesn’t teach you anything!” But it was the opposite. He had already taught me: “See that bird?” he says. “It’s a Spencer’s warbler.” (I knew he didn’t know the real name.) “Well, in Italian, it’s a Chutto Lapittida. In Portuguese, it’s a Bom da Peida. In Chinese, it’s a Chung-long-tah, and in Japanese, it’s a Katano Tekeda. You can know the name of that bird in all the languages of the world, but when you’re finished, you’ll know absolutely nothing whatever about the bird. You’ll only know about humans in different places, and what they call the bird. So let’s look at the bird and see what it’s doing—that’s what counts.” (I learned very early the difference between knowing the name of something and knowing something.)

 

[martinbn]

In polish we call them (per google translator), "abbreviated multiplication formulas", although I feel like "reduced" is a better word than "abbreviated". Wzory skróconego mnożenia. And as a teacher, names are sometimes important, because they facilitate communication (again I have some problem with word "facilitate", my english is not englishing lately). Teaching is mostly about proper communication and flexibility in that.

Interesting, we call them "Formuli za sukrateno umnozhenie"

 

[fresh_42]

Well, a rose without a name would smell the same.

Yes, but I wouldn't have understood this comment if you were not able to call it a rose. That's exactly the point.

 

[PeroK]

Yes, but I wouldn't have understood this comment if you were not able to call it a rose. That's exactly the point.

Sigh! My point is that we can (and in my case I do) understand these factorizations without having a name for them:
$$(a+b)^2=a^2+2ab+b^2$$$$(a-b)^2=a^2-2ab-b^2$$$$(a+b)(a-b)=a^2-b^2$$

 

[fresh_42]

Sigh! My point is that we can (and in my case I do) understand these factorizations without having a name for them:
$$(a+b)^2=a^2+2ab+b^2$$$$(a-b)^2=a^2-2ab-b^2$$$$(a+b)(a-b)=a^2-b^2$$

Yes, we can describe it. But that is not as convenient as naming it. That's all. And it is still about school math.
https://en.wikipedia.org/wiki/Polemic

Polemics often concern questions in religion or politics.

 

[PeroK]

Yes, we can describe it. But that is not as convenient as naming it.

What about calling them the "first, second and third nameless identities"?

 

[fresh_42]

What about calling them the "first, second and third nameless identities"?

I long gave up discussing with the didactic department. But I can imagine that this particular case makes things easier for teachers.

 

[pines-demon]

I long gave up discussing with the didactic department. But I can imagine that this particular case makes things easier for teachers.

I think they just write (1) (2) (3) next to them and that's it.

 

[fresh_42]

What about calling them the "first, second and third nameless identities"?

Even the one who cannot be named finally received a name.

 

[fresh_42]

I think they just write (1) (2) (3) next to them and that's it.

Not here. 1st , 2nd, and 3rd are standing references.

 

[SammyS]

. . .
In the UK we usually write this the other way round: $a^2-b^2 = (a+b)(a-b)$, and so we also have a different arrangement of the first theorem as the "sum of two squares" $a^2 + b^2 = (a+b)^2 - 2ab$.

A similar rearrangement of the second formula leads to $a^2-b^2 = (a-b)^2+2ab$ which gives a different expression for the difference of two squares: I don't know of an English name for this.

The last expression is in error.
$\displaystyle (a-b)^2+2ab = a^2-2ab+b^2+2ab=a^2 + b^2 \,,\$ so it gives aa alternate expression for the sum of two squares.

 

[pbuk]

The last expression is in error.
$\displaystyle (a-b)^2+2ab = a^2-2ab+b^2+2ab=a^2 + b^2 \,,\$ so it gives aa alternate expression for the sum of two squares.

Thanks, good catch.

 

[gmax137]

When I was in school (USA), we called it:
"a squared minus b squared equals a plus b times a minus b"

A long winded name, but it avoids the kids wondering, "wait, is that the second or the third binomial." Plus, if you say the long winded name often enough, it sticks on your mind.

 

[PAllen]

Of course, they are taught.

Usually, in the context of factoring quadratics. We use the formula only when all else fails. Of course, kids being kids will jump to the formula and then struggle to reduce it to get a solution and mix up a sign or two before arriving at the correct answers.

Here's one curriculum for Algebra 1 as taught in a community college. High school Algebra 1 is similar but not as extensive.

https://mathispower4u.com/algebra.php

I learned them in 9th grade, no name needed to use them. Actually this thread is the very first time I heard that anyone ever named them, or that anyone thought they needed a name. There are a great many useful formulas in special relativity that I think most anyone using it much knows, that don’t have any names.