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

[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.

 

[mathwonk]

to me, after these basic facts, the more interesting formula is (r+s)^2 -4rs = (r-s)^2, since this solves quadratic equations x^2 - bx + c, in which b = r+s, and c = rs, where r,s are the roots. I.e. then r,s = (1/2)(r+s ± [r-s]) =
[b ± sqrt(b^2-4c)]/2.

 

[MidgetDwarf]

Maybe they are nameless in america, since students traditionally encounter the binomial theorem near the end of algebra 2?

Is the binomial theorem shown earlier in germany?

 

[fresh_42]

Maybe they are nameless in america, since students traditionally encounter the binomial theorem near the end of algebra 2?

Is the binomial theorem shown earlier in germany?

Wikipedia says in 7th-8th grade (including the numbering we discuss here):
https://de.wikipedia.org/wiki/Binomische_Formeln
(Chrome plus right-click provides a translation to English)

 

[mathwonk]

This discussion sure lasted a long time apparently without anyone, including me, noticing the second formula in post #1, namely (a-b)^2 = a^2 -2ab -b^2, is properly called "nonsense" (or Unsinn?), since it is false when b≠0. Presumably that is not in fact what is usually taught children in Germany, although a few of my American students did think this was true.

 

[Mark44]

This discussion sure lasted a long time apparently without anyone, including me, noticing the second formula in post #1, namely (a-b)^2 = a^2 -2ab -b^2, is properly called "nonsense" (or Unsinn?), since it is false when b≠0. Presumably that is not in fact what is usually taught children in Germany, although a few of my American students did think this was true.

More likely it was a typo by the OP in this thread. Clearly it should be $(a - b)^2 = a^2 - 2ab + b^2$.

 

[WWGD]

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.

Which ignore the assumption of commutativity of product needed for the result, specifically, so that the cancelation$AB-BA=0$ is justified. Just try any two non-diagonal $n\times n$ matrices. Edit: Similar assumption is needed, used, to conclude $(A \pm B)^2=A^2 \pm 2AB +B^2)$

 

[fresh_42]

Which ignore the assumption of commutativity of product needed for the result, specifically, so that the cancelation$AB-BA=0$ is justified. Just try any two non-diagonal $n\times n$ matrices. Edit: Similar assumption is needed, used, to conclude $(A \pm B)^2=A^2 \pm 2AB +B^2)$

Don't tell it physicists, but BCH is the only reason to stop a series after the second term.

 

[mathwonk]

Of course, the rearranged "second formula" quoted in post #41, namely "a^2 - b^2 = (a-b)^2 + 2ab", is just as false as the original second formula; in particular it ignores more than commutativity. Presumably, that's why it has no special English name.