I am able to find the y-intercept and x-intercepts of the parabola, just looking to find the slope now, and I know differentiation gives the slope of the quadratic equation
Would it be correct to say all formula are either atomic formula or can be construct from atomic formula through the use of the logical negation and conjunction connectives
When I was building TECS (The elements of computing systems) - The only primitive give was a NAND gate, with it I built the NOT gate, then the AND gate, then the OR gate. With NOT, AND and OR, it is possible to derive all other formulas though canonical representation.
I'm not sure if this is connected to your problem though.
Tell me @Lord_Farin, what is that Err... in your first comment at meta.math.stackexchange.com/q/9883/8581. Is that a code in Latex? I have thought many but couldn't find that?
@Babak Do you also correct the spelling and grammar errors? I always have fun with those, although it is usually not allowed to deduct points for them.
@Lord_Farin According to this meta.SO post the edits, where only tag were changed (and not the body) do not count to the archaeologist badge: Do tag edits count for Archaeologist?
This could explain the discrepancy between the data.SE queries and actually awarded badges.
@robjohn Why is $\log(C)$ acceptable? Typically in beginning calculus courses, the domain is $\mathbb{R}$, so we don't have negative results from $\log(C)$.
why and how are two different questions. the "how" is "difference of squares," the "why" is because then neither numerator nor denominator tends to zero
@JohnMerlino there are already many different named objects: (a) R[x], (b) R[x,x^-1], (c) R[[x]], (d) R(x), (e) R((x)). in order: (a) polynomials, (b) polynomials in x and x^-1, (c) formal power series, (d) rational functions, (e) formal laurent series.
asking why the word "stone" applies to various types of rock but does not apply to trees is very strange. we already have different words for them both - rock and tree - and that's exactly as it should be.
polynomials are the most basic and ubiquitous: algebraic relations in commutative rings (structures with addition and multiplication) can be encoded as polynomials. all of the other things are built from polynomials. it makes sense, both pedagogically and conceptually, to distinguish the foundational type of object from the more general types of objects that are made out of them.
If you already have the broader class of "rational functions", why indeed would you still need to expand the definition of "polynomial" to include negative powers of the independent variable?
When using truth tables in propositional logic, would it be an abuse of notation to substitute the bits $\{ 0, 1 \}$ in for sub formula appearing in formula, when attempting to evaluate there truth value.
evaluating the truth value for yourself or for others to see? (barely such a thing as abuse of notation in the former case). of course the subformula has to evaluate definitely to either true or false to put a constant value in for it.
@anon various types of stone look different and that's why they are grouped differently. In a polynomial you can divide a constant like so: 1/2. But in polynomial you cannot divide by variable like so: 2/x. Look, what causes the fabric of the two expressions to be different? Is it the graph looks different? What makes it look different: a green stone from a red stone, to use your analogy.
in addition to polynomials not having singularities (a graphical way of seeing a striking difference with reals or complex numbers), as I said earlier polynomials encode algebraic relations, and the 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...
@Lord_Farin to complex for me sorry lol, can you help me with one more thing though
@Lord_Farin The book I am reading defines the sub-formula of a formula as follows,
Definition $1.4$, The following rules define the subformulas of a formula. Any formula is a subformula of itself.
Any subformula of F is also a subformula of ¬F.
Any subformula of F or G is also a subformula of (F ∧ G).
-----
But then later goes on to say,
Example $1.5$,
Let A and B be atomic and let F be the formula ¬(¬A∧¬B). The formula A ∧ ¬B occurs as a substring of F, but it is not a subformula of F. There is no way to build the formula F from the formula A∧¬B.
one way to bridge the algebraic and geometric understanding is through the "evaluation maps": you can evaluate a polynomial anywhere, not so if you have negative powers of variables. so polynomials are more universal.
...as you will see, it's a house of cards: tweak the definitions of the most basic objects you're dealing with, and you will need to adjust all the theorems built on top of those definitions.
