Probably you have an "inductive" definition of formula. E.g. "A variable is a formula; if $F$ is a formula, so is $\neg F$; if $F,G$ are formulae, so is $(F\land G)$."
@Lord_Farin Yes, All formula are either atomic formula or can be constructed from atomic formula through the use of the logical negation and conjunction connectives
@Ethan Ok; due to the proper insertion of brackets, we have an (essentially) unique way of reading any formula. That is to say, there is a unique way to construct a formula from atomic ones using the rules of formation just given. (This is a theorem.)
Those formulae that we encounter in the course of this unique construction are the subformulae.
@Ethan Does this definition work better for you? Can you see they are equivalent?
you can even allow 1/x that is no problem. Taking the polynomial ring over any field there is a natural way of choosing the field of quotients of the polynomial ring
@0x4A4D and what about the negative exponents? Why aren't polynomials allowed to have negative exponents? Is it because X^-2 = 1/x^2, and if x is 0, then there is a singularity, as you suggested before?
@JohnMerlino Yes, you can say that. It was agreed upon in advance that polynomials cannot have singularities. $1/x^2=x^{-2}$ fails to uphold that agreement.
@anon and I didnt understand what you meant by this "he things they encode the relations for need not be invertible. note that the coefficients of polynomials don't even need to be invertible depending on the underlying ring" What is ring?
@skullpatrol I just did evaluate (again) the Dirichlet integral in one line $\int_0^{\infty} \frac{\sin(x)}{x} \ dx=\int_0^{\infty}\sin(x)\int_0^{\infty}e^{-xt}\ dt \ dx=\int_0^{\infty}\int_0^{\infty}\sin(x)e^{-xt} \ dx \ dt=\int_{0}^{\infty} \frac{1}{1+t^2} \ dt = \frac{\pi}{2}$
@Ethan I know that it works, because that's what the book says how to do it. I mean (x)*(x)=x^2. that makes sense. And if M is -7 and N is -1, then -7 + -1 = -8 makes sense. And -7 * -1 = 7 makes sense. All that makes sense. What is a little confusing is (x + M)(x + N) only has two x's. Yet the expanded form has three x's: x^2 -8x + 7. Now I know the FOIL method works in such a way to create that third x. But logically how is a third x become part of the expanded form.
to take another example, 2x=x+x. left side has one x, right side has two x's. there is absolutely no reason to believe that if two expressions are equal then the number of times something is written on the left is equal to the number of times it is written on the right, e.g. 2=1+1 doesn't have any 2 on the right, and similarly has two 1s on the right but no 1 on the left
In general the kth coefficient of $\prod_{n\in s}(x+a_n)$ is the sum of the distinct $|s|-k$ sub products of the variables $a_i: i\in s$
@JohnMerlino if $a_i=a_j$, $\forall i,j \in s$ sense multiplication is communative we can count each product the same, and sense there are $\binom{|s|}{|s|-k}$ coeiffients, it follows the kth coeiffient of $(a_i+x)^{|s|}$ is $\binom{|s|}{|s|-k}=\binom{|s|}{k}$
@JohnMerlino if $a_i=a_j$, $\forall i,j \in s$ sense multiplication is communative we can count each product the same, and sense there are $\binom{|s|}{|s|-k}$ coeiffients, it follows the kth coeiffient of $(a_i+x)^{|s|}$ is $\binom{|s|}{|s|-k}=\binom{|s|}{k}$
is not going to help someone who is struggling with FOIL.
I figured it out: x^2 – 18x = 0. There is no constant term, meaning that the constant term is 0, meaning that the y-intercept is 0. If that is the case, then when we factor out the equation (using the greatest common factor), we will then realize x itself is the value of b in ax^2 + bx + c, which in the example is 18, because b(b – b) = 0. x can also be 0, because 0(0 – 18)=0. Hence, our parabola crosses the x-intercept at 0 and 18.
If $d_k(n)$ counts how many ways $n$ can be written as a product of $k$ distinct integers $$\sum_{n=1}^\infty\frac{d_k(n)}{n^2}=\frac{\pi^{2k}}{(2k+1)!}$$
Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
