John

No. You've got the assumptions listed above, and you're assuming K is true. Starting with the assumption that K is true, you have to reason your way through the other assumptions you have to arrive at the conclusion that H is also true. This will prove that K implies H.

So far, we've seen that since K is true, ~K must be false. And since ~K is false, I must also be false. Now given that I is false, what can we conclude, given the various other statements we assume to be true?

Almost. The biconditional is true by assumption. Since [math]\neg K[/math] is false, the fact that [math]I \iff \neg K[/math] tells us [math]I[/math] is false.

Yes, and given that [math]I \iff \neg K[/math], what does this tell us?

Well, since [math]K \wedge \neg K[/math] is a contradiction, and [math]K[/math] is true, what do we know about [math]\neg K[/math]?

Hm. Well, the way a proof works in general is we start with some assumptions, and using new conclusions we can logically deduce from those assumptions (and further conclusions we can logically deduce from those conclusions, and so on), we work towards some end goal.

So for this proof, our assumption is that [math]K[/math] is true. We also have that [math]I \iff \neg K[/math]. Since [math]K[/math] is true, what do we know about [math]\neg K[/math]? And what does this tell us, given our biconditional?

I'm not sure what that program is, but I'm assuming

[math]\neg F \implies (G \vee H)[/math],

[math](F \vee G) \implies (I \wedge J)[/math], and

[math] I \iff \neg K[/math]

are propositions you can take to be true, and thus your goal is to use these to show that [math]K \implies H[/math]. Is that correct?

Is the problem that you don't see how to work from K to H, or is it that you're not sure how to express the proof using this program?

How can a negative radical sign applied to the principle square root denote a positive value?

You either have the positive and/or negative signs before the radical, which is then defined as always having a positive value, or you allow the value itself to be positive or negative, but not both. If the latter were to be the case the sign before the radical is worse than redundant since it can oppose the sing of the value.

In the case of 16, [math]-\sqrt{16} = -4[/math] and [math]\sqrt{16} = 4[/math]. The negative radical sign doesn't denote a positive value unless we take the radical sign to specifically refer to the negative root, which is something we probably agree would be unusual. As for the rest, I agree with you. If we take the radical to denote either root (thanks for the correction of my sloppy wording), then the [math]\pm[/math] is redundant. But usually we do prepend [math]\pm[/math] in contexts where either root is a solution, which lends further credence to my claim that the radical itself is usually taken to be the principal square root.

The radical sign by itself denotes not both roots, but either root. It is up to the employer of that radical to determine which particular one.

I absolutely agree. My point is that, without an explicit declaration by the mathematician in question, the radical sign is normally taken to denote the positive root.

And no, you have not stated many times. You have argued, many times that you are not discussing functions, that they are irrelevant etc.

I have stated that [math]\sqrt{x}[/math] denotes the principal square root of x in every post I've made in this thread (though a couple of times it was somewhat indirect). I've irritated even myself with the repetition, so surely you've noticed.

Let us examine the logic of this.

 

If the roots may be positive or negative then why does the radical also need a sign?

 

Surely if you negate the radical of a negative root you obtain a positive value?

 

 

 

What I don't understand is why so are so wedded to the idea that [math]\sqrt {16} [/math]may not equal -4 ?

I've explained and shown in several sources that the radical sign is used to denote specifically the principal square root. I've also explained that notation isn't sacred, and one can easily use the radical to denote both roots, but this isn't the common usage.

You have not presented a single reason beyond 'others also do it' for this view.

This is sort of the definition of "convention" in this context.

I can readily understand the mathematical benefits that flow from choosing only positive values for the square root function. It guarantees an inverse function and fits in with a heap of other mathematical structure about sets and functions etc.

 

 

But 16 is just a number, it is not a function.

and there are many ways to get to 16.

In the context of the square root function, 16 would be the input and the radical sign would be the function. In general, [math]\sqrt{16}[/math] represents the principal square root, as I've stated many times.

16 = 15+1

16=17-1

16 = 8*2

18= (-8)*(-2)

 

and so on

 

Do you also reject these?

I reject the last one.

Edit: Also, why are we even still discussing this? We agree on the concepts involved. The only disagreement is on the default meaning of a single symbol. It's silly to get this caught up in notation.

The passage you quoted doesn't seem to mention the radical sign at all, but simply explains that even roots may be positive or negative, which is not in dispute. I'm talking about the definition of the symbol itself, which is the principal square root.

As an example of ubiquitous use, check out the quadratic formula. Further examples are given by the Wikipedia articles on the square root and nth root, and the results of plugging something like x = sqrt(16) into Wolfram Alpha (which also has a page of information about the square root here). While these may not be century-old books, they are the results of the collaborative efforts of hundreds or thousands of individuals, which at least suggests the definition is common. These sources are also more up-to-date than a book written in 1880.

In any case, I think all points that can be made about this have been made. I appreciate you forcing me to reexamine my thoughts and statements on this, even if I have ended up reaching the same conclusions I had before.

I am following the (usual) definition.

[math]x^2 = 16[/math] has the solutions [math]x = \pm \sqrt{16} = \pm 4[/math].

[math]x = \sqrt{16}[/math] has the solution [math]x = 4[/math].

When Amaton says, in his post, that [math]\sqrt{16} = 4[/math], he is correct. There is no issue here.

I think we may be arguing about different things.

That's perfectly true, but [math]\sqrt{x}[/math] normally refers to the positive root specifically. There is no conflict here.

I'm stating the (usual) definition of notation. You might as well ask me to logically develop why we normally use "+" to denote addition. If you did, then I'd say the same thing: It's a matter of convention, and a Google search should provide sufficient evidence to show this.

Notation isn't sacred. If I like, I can define [math]\sqrt{x}[/math] to refer to [math]e^{x}[/math] or [math]x \choose {x-6}[/math] or anything else. It'd be odd, and there's no good reason--or at least, no reason that outweighs the confusion it'd likely cause--to do so, but I could. However, the usual definition of [math]\sqrt{x}[/math] is the principal square root of x. This doesn't require rigorous development. It's simply the way notation has developed over the years.

This definition also doesn't conflict with the quoted passage, unless you're referring to the fact that I translated [math]\sqrt{x}[/math] as "the square root of x." But if that bothers you, then ignore it. The rest of my post remains true.

We're just talking about the definition of the radical sign. By default, it indicates the principal (i.e. positive) square root of the radicand. In situations where both square roots of the radicand are desired, we prepend [math]\pm[/math] to specify this. So if someone asks for [math]\sqrt{16}[/math], i.e. "the square root of 16," the answer is 4. If someone asks for solutions to the equation [math]x^2 = 16[/math], i.e. "what numbers are square roots of 16," the answers are 4 and -4, and we can express these as [math]\pm\sqrt{16}[/math]. If you want to use [math]\sqrt{16}[/math] to mean both 4 and -4, then that's fine, but it may cause some confusion unless this usage is specified (though often the intended meaning will probably be clear from the context).

The difference between an equation and a function is irrelevant. The notation [math]\sqrt{x}[/math] refers to the principal square root of x. It's not strange. It's not controversial. I don't know why this is such a sticking point.

It's not my convention. It's the convention used by mathematicians in general. While the fact that it's a convention means there's probably no absolute authority to make the definition, a Google search for definitions of the square root or the radical sign should convince you.

And I'm honestly not sure what your point is. All I've said is that [math]\forall x \in \mathbb{R}, \sqrt{x^{2}} = |x|[/math]. It's simply a matter of notation. Questions about the roots of various polynomial equations are somewhat unrelated.

But for the particular example of [math]\sqrt {\left( {1 - 2x + 5{x^2} - 4{x^3} + 4{x^4}} \right)}[/math], so long as [math]1 - 2x + 5{x^2} - 4{x^3} + 4{x^4} > 0[/math], then yes, [math]\sqrt {\left( {1 - 2x + 5{x^2} - 4{x^3} + 4{x^4}} \right)} > 0[/math].

Using Wolfram Alpha, it seems the two square roots of [math]1 - 2x + 5{x^2} - 4{x^3} + 4{x^4}[/math] (which, as it turns out, is positive for all real x) are [math]\pm(2x^2 - x + 1)[/math], so [math]\sqrt {\left( {1 - 2x + 5{x^2} - 4{x^3} + 4{x^4}} \right)} = |2x^2 - x + 1|[/math], which is positive.

Edit: Just saw your edit. The notation indicates the principal square root, so [math]\sqrt{(-4)^{2}} = |{-4}| = 4[/math].

Er, no, why? Finding the roots of a general polynomial equation isn't the same as taking the square root of a real number. And as stated before, if you look at [math]x^{2} - 16 = 0[/math], then there are two solutions, [math]\sqrt{16} = 4[/math] and [math]-\sqrt{16} = -4[/math]. But the notation, [math]\sqrt{16}[/math], refers to the principal square root of 16, i.e. 4. Disagree if you like, but that is the convention.

There are two square roots of 16, but [math]\sqrt{16} = 4[/math] unless otherwise specified. In general, [math]\forall x \in \mathbb{R}, \sqrt{x^{2}} = |x|[/math]. If we have the equation [math]x^{2} - 16 = 0[/math], then [math]x = \pm \sqrt{16} = \pm 4[/math]. It's not a huge deal, regardless, but Amaton's post is correct.

The notation [math]\sqrt{x}[/math], for [math]x\in\mathbb{R}_{\ge 0}[/math], refers to the principal square root (which is positive) unless otherwise specified. Amaton's post is correct (except that notation is somewhat flexible--it just depends on how much you want to annoy your readers ).

Although it's not a book, you might like Wikipedia's Introduction to quantum mechanics. This is an article intended for readers who haven't necessarily had the physics and mathematics education necessarily for a rigorous introduction to the subject. Section 8 specifically discusses quantum entanglement. I admit I haven't read it yet, but if it's anything like the similar introductions to special and general relativity (which I have read), then it's a pretty good read. There are a few formulas, but these articles are primarily conceptual in nature.

I'm assuming these are questions you're asking for yourself, and not questions you have to answer for an assignment. Mods forgive me if I'm incorrect.

I'm assuming N is supposed to be the natural numbers. We have some function f that has the Cartesian product of N and X (i.e. all the ordered pairs (n, x) for n in N and x in X) as its domain and X as its codomain.

The corollary states that if we take any x in X, then there exists a unique sequence starting with x (this is what x(1) = x means) and defined by the recurrence relation x(n + 1) = f(n, x(n)), or alternatively x(n) = f(n - 1, x(n - 1)). Thus, letting x(1) = x, we have that x(2) = f(1, x(1)), x(3) = f(2, x(2)), etc. The precise form of this sequence will depend on the definition of f. All we know is that each term in the sequence must be an element of X (since X is the codomain of f).

As an example, let X be the rational numbers, and let f be defined such that f(n, x) = nx. Then, taking x = 3/5, we have that

x(1) = 3/5,

x(2) = f(1, x(1)) = (1)(3/5) = 3/5,

x(3) = f(2, x(2)) = (2)(3/5) = 6/5,

x(4) = f(3, x(3)) = (3)(6/5) = 18/5,

x(5) = f(4, x(4)) = (4)(18/5) = 72/5,

and so on.

"Mudslinging" and "preachiness" - not very helpful in trying to understand the meanings - replacing one verb or noun with a synonym. The Mod actions certainly prevented any 'flame wars' or 'soapboxing' but also prevented any productive discussion of what I think is an important topic. New method for cancer prophylaxis: kill every cell in your body before it has a chance to go awry

Replacement of an unfamiliar word with a perhaps more common synonym doesn't help? This is news to me.

To go with the cancer analogy, the situation is more akin to removing moles or polyps. The mods don't close every new thread in an effort to prevent problems, but they do close threads that seem likely to cause problems.

"If you find yourself on the end of a mod note, it's fair to say you made a mistake somewhere." In order for that statement to have any validity, you must have first given some independent thought as to the rationale of Mod actions following members' posts. In order to extrapolate to my particular situation, you would need to extend the same thought processes to this particular case, and make a judgment. You have already said that you "don't know" so the extrapolation is invalid. I think you actually agree with the Mod action but are too politically correct to say so. Say so! I really don't care

My statement is the result of hanging around these forums, talking to mods, and knowing roughly how their process works. I did see your thread, before it was locked, and I'm not sure whether I would have locked it, though I might have at least suggested a change in title. However, I'm not a mod, and even if I were, if several others had agreed with locking the thread, the outcome would have been the same.

As for extrapolation, if not knowing is grounds for rendering extrapolation invalid, then any extrapolation is invalid. Any extrapolation comes with a certain degree of uncertainty. It's the nature of inductive reasoning. In my experience, moderator actions are almost always justified, and those few that aren't are quickly reversed--it's that whole multiple moderators thing in action again. I can't be certain that all future moderator actions will be justified, but based on my experience, I think it's safe to say any given new action is.

Mod notes make the poster, who may have contributed a substantial number of quality posts, look like an idiot by immediately cutting them off and censoring their opinions not only on the topic at hand but also on their treatment at the hands of Mods who are, let's face it, probably younger and less qualified who nevertheless feel it is their noble duty to silence anything that might rock the boat. So the poster is disrespected and shamed, the Mod leaves on high horse, and there is no opportunity for the poster to justify their actions.

It's difficult to set aside one's ego in the face of criticism, especially when it's public. You seem to have a pretty good track record here, and no one thinks you're an idiot just because a couple of your posts have resulted in moderator response.

Whatever you think about the age, qualifications, or whatever else of a moderator, the fact remains that the moderators are given the authority to make decisions regarding possible violations of the forum rules. Also, you have the ability to justify yourself, through the proper channels.

"Flame war" usually connotes something closer to mudslinging, or at the very least, hostile back-and-forth with neither side really considering the other's position. "Soapboxing" refers to a certain level of preachiness that doesn't admit much in the way of debate.

Whether your thread constituted the latter or would have led to the former, I don't know. It's the moderators' responsibility to make these decisions, and they do a pretty good job at it. A mod decision (especially one that includes closing a thread entirely) isn't made on a whim, and if you find yourself on the receiving end of a mod note, it's fair to say you made a mistake somewhere. It's not the end of the world, but moderator actions do warrant consideration if you intend to continue posting here.

The public nature of mod notes serves the apparent purpose of reminding readers and thread participants (not just the targets of the notes) of the rules and providing examples of infractions, which helps to clarify the standards of the forums. Public responses to mod notes, on the other hand, lead either to thread derailment or (in the case of new mod response threads being created) ultimately to superfluous repetition of what the mod notes said in the first place.

If you're planning to go to college and do a STEM degree (or at least, take calculus), then I recommend holding off on physics until you've done calculus and can take calculus-based physics courses.

Otherwise, as mentioned previously, it's a matter of personal preference and deciding which instructor is more competent. Either class has the potential to be pretty interesting and useful.

You can also think of this as a parallelogram with sides of length 3 and 5. |a+ b| is the length of one diagonal, |a- b| is the length of the other.

 

In particular, since |a+ b|= 7, one half of that parallelogram, the one with the larger angle between the sides, is a triangle with sides of length 3, 5, and 7. Taking [tex]\theta[/tex] to be the angle between the two sides, by the cosine law, [tex]7^2= 3^2+ 5^2- 2(3)(5)cos(\theta)[/tex]. You can use that to solve for [itex]\theta[\itex]. then the other angle, [tex]\phi[/tex], is the supplement of [tex]\theta[/tex]. And, letting x= |a- b|, [tex]x^2= 3^2+ 5^2- 2(3)(5)cos(\phi)[/tex].

This is what I was alluding to earlier. There's actually an even easier formula to use.

For the first, you could let the first digit be x and the second be y, and use the equation 10x + y + 9 = 10y + x. Simplifying, you end up with x = y - 1, i.e. the second digit is one more than the first, which is the pattern you found. I'm not sure your second example (increasing by 6) is possible, since using the same reasoning (and assuming you're still talking about two-digit numbers), we end up with x = y - 2/3.

For the second problem, what you're looking for is the set of solutions to the equation 83 = nx + 2, or 81 = nx, i.e. x divides 81. The factors of 81 are 1, 3, 9, 27, and 81. Clearly 1 isn't a solution. For 275 and a remainder of 3, we have that x must divide 272. The factors of 272 are 1, 2, 4, 8, 16, 17, 34, 68, 136, and 272. A quick way to find these factors is to find the prime factorization of 272, which is 2^4 * 17, and realize that since each of the prime factors divides 272, any product consisting of the prime factors also divides 272.

For the last, since the number is divisible by 24, it must be divisible by both 8 and 3. The divisibility rule for 8 says a number is divisible by 8 if its last three digits are divisible by 8. So you have 65B, and you must find B. An immediate thing that pops into my mind is 640 is divisible by 8, so if we add two 8's to that, we arrive at 656 being divisible by 8. Therefore, B is 6. Now, the divisibility rule for 3 says a number is divisible by 3 if the sum of its digits is divisible by 3. At this point, we have 8A656, so we need to find A such that 8 + 6 + 5 + 6 + A = 25 + A is divisible by 3. We want the smallest number, so we can start with A = 0 and work our way up. Obviously 0 doesn't work, and letting A = 1 doesn't work either (the sum will be 26). But when A = 2, we have 8 + 2 + 6 + 5 + 6 = 27, which most certainly divides 3. Thus, 82656 is the smallest number of the form 8A65B that divides 24.