Can this expression be expressed in another way?

Question

I saw this meme, pretty daft IMO which "ironically" shows how $9+1\div10=1$ would be expressed (from what I understood) by inverting the order. I was asking asking myself, how many errors does it have? Also, why is $(x+y)^2$ produced as $x^2+y^2$ and not as $x^2+2xy+y^2$ using the binomial theorem (isn't the first wrong?).

Answer 1

The $(x+y)^2$ is produced as $x^2+y^2$ for humoristic reasons, and you are correct, it should be $x^2+2xy+y^2$. This meme tries to humoristically make fun of some mathematical mistakes.

Basically all of the calculations are wrong, with each another one in more and more ridiculous way.

Answer 2

All of the calculations are wrong, that's the point.

Answer 3

They're all wrong.

1. $ 9 + 1 \div 10 = 9 + (1 \div 10) = 9.1 $ The mistake is grouping $(9+1) \div 10 = 1$, which ignores the order of operations.

2. $(x+y)^2 = x^2 + 2xy + y^2$ as you noted. This is a reference to the common mistake, Freshman's dream.

3. $(x + y)$ is an entire term that you can't cancel with the numerator, i.e. $\frac{x}{x+y}$ is fully simplified.

4. $\sin$ is a function. The letters are not variables and can't be divided out.

Answer 4

They are all wrong, but they are common mistakes that students make.

$9+1\div10=1$ seems to be a stupid and popular meme around the interwebs. In my mind the $\div$ symbol makes for confusing notation and should be avoided whenever possible. However, in this notation, division should happen before addition.

The second one. $f(x) = x^2$ is not a linear function.... at least that would be the high-fallut'n way of saying it. There is a small (and important) subset of linear functions, but most are not. So $\sqrt{x+y} \ne \sqrt x + \sqrt y$ and $\sin (x+y) \ne \sin x + \sin y$

Multiplication distributes over addition. And, that is what the binomial theorem is all about.

The next one.

Division is not commutative, and it distributes one direction but not the other.

It distributes this way: $\frac {a+b}{c} = \frac ac + \frac bc$

but not this way: $\frac {a}{b+c} \ne \frac ab + \frac ac$

And if you choose to cancel, you must cancel in every term in the numerator and in the denominator.

and the last one is just silly.