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[3man75]

*Snip*
2 - 8 is -6, not 6.

I feel througoughly ashamed...

On a more realistic note thank you Ispil.

[bahihs]

That's not actually calculus. That's more number theory... sorta?

"What is Mathematics?" puts this under: "The Number System of Mathematics" so yeah, number theory is more or less right.

[3man75]

If mathematics is the grand tree trunk of math does that make number theory and calculus branches of mathematics? Also who determines what Axioms and theorems are 'correct'.

EDIT: Okay I thought I had gotten the concept behind a+bi but apparently not. (-6 + i) (6 - I) = 37.

The way I tried it left me with -12.

(-6-6) (i-i)= -12. ((The two i's cancel out since one is positive and the other negative.))

[TheDarkStar]

If mathematics is the grand tree trunk of math does that make number theory and calculus branches of mathematics? Also who determines what Axioms and theorems are 'correct'

First question: Yep. They interact in some ways, but number theory and calculus tend to answer different questions in different ways.

Second question: Axioms are statements assumed to be true for the purposes of a theorem. If they're later proven wrong, then you have a theorem that's correct but doesn't reflect reality. Some of them end up being necessary - for example, there's no mathematical way to prove that two parallel lines don't intersect or that triangles have 180 degrees, despite the world working that way.

Theorems are ideas that are proven true based on certain axioms. A simple theorem is (a+b)² = a² + 2ab +b. This is true assuming that multiplication is associative (meaning the order of multiplying doesn't matter; ab = ba) and assuming that multiplication functions as expected.

[Karlito]

EDIT: Okay I thought I had gotten the concept behind a+bi but apparently not. (-6 + i) (6 - I) = 37.

The way I tried it left me with -12.

(-6-6) (i-i)= -12. ((The two i's cancel out since one is positive and the other negative.))

You can't swap the terms of factors around like that. (a+b)(c+d) ≠ (a+c)(b+d) Also (-6-6) (i-i) = 0, but that is perhaps beside the point.

(-6 + i) (6 - i)
=-6*6 + (-6)(-i) + 6i - i²
=-36 + 12i -(-1)
=-35 + 12i

If it's supposed to be 37, then you misplaced a minus sign when you copied the problem here.

[3man75]

EDIT: Okay I thought I had gotten the concept behind a+bi but apparently not. (-6 + i) (6 - I) = 37.

The way I tried it left me with -12.

(-6-6) (i-i)= -12. ((The two i's cancel out since one is positive and the other negative.))

You can't swap the terms of factors around like that. (a+b)(c+d) ≠ (a+c)(b+d) Also (-6-6) (i-i) = 0, but that is perhaps beside the point.

(-6 + i) (6 - i)
=-6*6 + (-6)(-i) + 6i - i²
=-36 + 12i -(-1)
=-35 + 12i

If it's supposed to be 37, then you misplaced a minus sign when you copied the problem here.

The book is asking for it to be put in that form.

If it was me I would just say equate to 0 also.

It looks like you distributed the numbers here. What allows (6*6) + (-6)(-i) + 6i - i^= (-6 + i) (6 - i)? It looks from here that the numbers can be set up with any cumulative sign you want.

EDIT: NVM. I forgot about distribution.

[frostshotgg]

Axioms have to be accepted by their very nature. You can't prove them without using themselves. The thing about axioms is that there's not really anybody to say certain ones are valid or not, there's a bunch that are almost universally accepted because all of mathematics relies on them, and some that are significantly more questionable. One example of the latter is called the Axiom of Choice. Basically, it says that in any given set you're allowed to specifically choose which piece of the set and make a new set with it by some criteria. Without this axiom in effect, you cannot preform a lot of operations on infinite sets. However, with it in effect, you can make some really stupid paradoxes, like being able to take one sphere, and make 2 out of it that are perfect copies. So there's no universal way to determine "this axiom is correct".

Theorems on the other hand, all have proofs for them, sometimes a couple, sometimes hundreds. Which are reliant on various other theorems or rules, which in turn are reliant on various axioms.

[3man75]

Axioms have to be accepted by their very nature. You can't prove them without using themselves. The thing about axioms is that there's not really anybody to say certain ones are valid or not, there's a bunch that are almost universally accepted because all of mathematics relies on them, and some that are significantly more questionable. One example of the latter is called the Axiom of Choice. Basically, it says that in any given set you're allowed to specifically choose which piece of the set and make a new set with it by some criteria. Without this axiom in effect, you cannot preform a lot of operations on infinite sets. However, with it in effect, you can make some really stupid paradoxes, like being able to take one sphere, and make 2 out of it that are perfect copies. So there's no universal way to determine "this axiom is correct".

Theorems on the other hand, all have proofs for them, sometimes a couple, sometimes hundreds. Which are reliant on various other theorems or rules, which in turn are reliant on various axioms.

I think that makes sense. Basically if I have two apples (a unit of something) and I pair it with another two apples (adding the same unit) then I come up with a higher unit of that same unit. It's kinda interesting although math looks like it can get very dirty with lines and numbers ridding your clean white paper into a black lead infested one.

So would the Axiom of choice be called a "theoretical" math?

[Culise]

Second question: Axioms are statements assumed to be true for the purposes of a theorem. If they're later proven wrong, then you have a theorem that's correct but doesn't reflect reality. Some of them end up being necessary - for example, there's no mathematical way to prove that two parallel lines don't intersect or that triangles have 180 degrees, despite the world working that way.

Nope.  Since frostshotgg pointed out the problem with the definition, I'll simply add that changing your axioms can create all sorts of interesting world-views that do pertain to reality.  For example, what if I told you that two parallel lines can in fact intersect?  Euclid's fifth is an awkward, cumbersome, and actually unnecessary axiom.  Its use defines realm of Euclidian space, but discarding it and making either one of two assumptions - either that parallel lines always intersect or that they always diverge (get further away from each other) creates all sorts of interesting and new realms of mathematical theory, some with practical applications.  While mostly famous for its use as a term in Lovecraft's works, the most famous non-fictional example of non-Euclidean geometry is actually one everyone is familiar with: the Earth's surface.  Lines of longitude are parallel lines that intersect at precisely two points (the north and south poles), and triangles do not sum to 180 degrees (for example, draw a triangle using two longitudinal lines and the equator).  Any curved space is non-Euclidian in nature - a two-dimensional sheet with positive curvature (elliptic) describes, at its simplest, a sphere, and with negative curvature (hyperbolic) a saddle - and this has seen great use in understanding space and gravity. 

EDIT:
Oh, I must also hasten to add that this does not mean that Euclidean space is worthless, any more than general relativity made Newton worthless.  It's still all sorts of useful for what it does describe - flat space, or more generally space where the local curvature approaches zero.  When dealing with problems on the scale of meters, it's not as important to worry about a radius of curvature measured in thousands of kilometers. 

[frostshotgg]

Using your example, it's more, I have one apple. Axiom of Choice lets me pick out infinitesimally tiny bits of the apple and then put them together into another apple, while there's still the first apple in one piece. By this method, I now have 2 apples the same size/content/density/whatever of the first apple. At its very base it violates the physics law of conservation of matter. But then that's because reality is macroscopic minecraft, not anything like conventional logic would lead you to believe.

[Culise]

Using your example, it's more, I have one apple. Axiom of Choice lets me pick out infinitesimally tiny bits of the apple and then put them together into another apple, while there's still the first apple in one piece. By this method, I now have 2 apples the same size/content/density/whatever of the first apple. At its very base it violates the physics law of conservation of matter. But then that's because reality is macroscopic minecraft, not anything like conventional logic would lead you to believe.

Well, yeah, that one isn't so realistic, much like Tarski's method of squaring the circle (a favorite historical problem, nowadays considered impossible in reality outside of abuse of the axiom of choice).  I'm just pointing out that there are axioms (like Euclid's Fifth/the parallel postulate) that can be altered and still have meaningful applications to reality, since TheDarkStar's comment was that an axiom, if altered, will automatically produce non-realistic results, using a patently incorrect example.  An axiom is only a premise that is assumed to be true for the purposes of reasoning, from which logical theorems can be proven.  For example, natural numbers are defined as a set of axioms; change the axioms, and you change arithmetic. 

^{What's a mathematician's favorite anagram of Banach-Tarski?  Banach-Tarski Banach-Tarski.}

EDIT: Oh, whoops; you were responding to 3man75.  Sorry...well, I'll leave this post mostly for the joke at the end.

[ZetaX]

If you dislike Banach-Tarski for it being a "paradox", you should consider that the paradox divides the sphere/ball into 5 parts that are not only "unrealistic" (for being infinitely finer than atoms or any other real world thing), but also "unmeasurable" (which is a mathematical way of saying: those parts are so weird, one cannot even say what their volume is; generally, if talking volume, one only consideres measurable things).

That's not actually calculus. That's more number theory... sorta?

"What is Mathematics?" puts this under: "The Number System of Mathematics" so yeah, number theory is more or less right.

No, this is set theory. Number theory is concerned about the inherent structure of the ring of integers, diophantine equations and such. Those two are very different things.

A simple theorem is (a+b)² = a² + 2ab +b. This is true assuming that multiplication is associative (meaning the order of multiplying doesn't matter; ab = ba)

ab=ba is being commutative, not associative. Associativity is a(bc) = (ab)c, which is not needed here. But distributivity, i.e. a(b+c) = ab+ac, is used twice.

[bahihs]

If you dislike Banach-Tarski for it being a "paradox", you should consider that the paradox divides the sphere/ball into 5 parts that are not only "unrealistic" (for being infinitely finer than atoms or any other real world thing), but also "unmeasurable" (which is a mathematical way of saying: those parts are so weird, one cannot even say what their volume is; generally, if talking volume, one only consideres measurable things).

That's not actually calculus. That's more number theory... sorta?

"What is Mathematics?" puts this under: "The Number System of Mathematics" so yeah, number theory is more or less right.

No, this is set theory. Number theory is concerned about the inherent structure of the ring of integers, diophantine equations and such. Those two are very different things.

A simple theorem is (a+b)² = a² + 2ab +b. This is true assuming that multiplication is associative (meaning the order of multiplying doesn't matter; ab = ba)

ab=ba is being commutative, not associative. Associativity is a(bc) = (ab)c, which is not needed here. But distributivity, i.e. a(b+c) = ab+ac, is used twice.

This is talking about the denumerability of the rational number system, I don't see how that doesn't fall into number theory. Although its kinda a trivial argument...

[ZetaX]

This is talking about the denumerability of the rational number system, I don't see how that doesn't fall into number theory. Although its kinda a trivial argument...

The denumerability is a set theoretic property. Nothing number theory is really concerned about. Obviously, set theory is. Why would you associate it into a field that does not care about this instead of the field that is by definition concerned with the question¿
Edit: Additionally, this is not really about the rational numbers, but about denumberability of finite products of denumerable sets. And the denumberability of their subsets. See, no mentioning of the rationals at all.

[frostshotgg]

I would try to avoid using the sarcasm mark. It shows up as a broken character on a very significant proportion of computers/fonts.