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[Graknorke]

So you can have:
x² - 14x + 49

[MagmaMcFry]

AltGr+2 is best symbol. ²²²²²

[3man75]

So I know that to get rid of a radical you square it.

But How do you square Radical over 2x-3 Radical over x+2 (end radical) +2?

The textbook I have shows the left part turning into 2x-3 because the radical dissolved but on the right the radical not only dissolves but it adds a 4*radical over x+2 (end radical) +4.

In total the right side of the equation looks like: x+2 4(radical)x+2(end rad)+4. I can add the 2 and the 4 since their like terms but WHERE does this new radical come from? I know it has to do something with the distribution property rule of math is the trigger for this new radical but can someone show me exactly how that works?

[frostshotgg]

You multiply (2x-3)*root(x+2)+2, the whole expression, by (root(x+2)), so you get (2x-3)*(root(x+2))^2 + 2*(root(x+2)) which simplifies to (2x-3)*(x+2) + 2*(root(x+2)). If I understood what the original problem was, it's hard to tell with the way you wrote it.

[MagmaMcFry]

3man, how about you use "root(ax+b)" to denote roots, instead of "Radical ax+b (end radical)"? Parentheses go a long way towards legibility. Right now I can't read whether you mean root(2x-3)*root(x+2)+2 or root(2x-3*root(x+2))+2 or root(2x-3*root(x+2)+2) or what.

[Graknorke]

-snip-

I'll have to jump on the "could you please clear that up with some brackets" bandwagon.

[bahihs]

Reading through "What is Mathematics" I come across this mind-blowing concept of different kinds of infinity. The author proceeds with a proof of the denumerability of the rational numbers using some sort of listing of the numbers which he then traverses through using a weird path.

pg 105 "What is Mathematics?"



I can't make heads or tails of this, can someone please explain?

I can see that the path is probably necessary to create a "way to count" the rational numbers and that counting, for example, right to left or up and down, will not "hit" every number (since everything is extending to infinity). What I don't get is how the number list(s) were generated.

In addition to this, one of the exercises is to prove the denumerability of the all positive and negative integers and then all positive and negative rational numbers (the above only covers positive rational numbers). I'm assuming a similar method to the above can be used, but again, how are the number lists to be generated?

[frostshotgg]

A rational number is defined as "any number than can be expressed as the ratio of two integers", which is to say, anything that can be written as a fraction with a whole number in the top and bottom. That system in that chart has increasing numerators on the x-axis, and increasing denominators on the y-axis. The x-axis has every number from 1 to infinity (a particular kind of infinity called a countable infinity, or "Aleph null/zero" (א₀)) and the y-axis has the same. Because of this, there is some point on that chart that has any given ratio of one integer to another, and therefore the number of rational numbers MUST be countable, because the area of the chart is (א₀)^2 which is STILL (א₀), the area of the chart has all of the same properties, which includes countability.

EDIT: There's literally no way to get this post to do it right, but the 0 is supposed to be after the "א", but because it's a hebrew character and hebrew reads right-left, the 0 always appears to the left. Either it's after the character and as such considered hebrew and it goes at the end of the line, the left, or it's before in english text and to the left. That's dumb.

[Reelya]

Infinite sets are of the same "size" if you can map one set to the other without any "leftovers". Any system which maps the elements of one set to the elements of another set defines the sets as having the same size. You can do this for finite sets or infinite sets.

What I don't get is how the number list(s) were generated.

The number list is just using the x and y location as the top and bottom of the fraction. So the third one across and second down is 3/2. This gives all possible fractions a unique x/y location.

The image provided by bahihs is a visual proof that the set of whole-number fractions is of the same size as the set of whole numbers. First, each fraction consists of two numbers (top and bottom) and hence, they can be mapped to a grid using top and bottom as x/y values. But then you can also overlay that squiggly line on top of the grid, which maps the grid to a single value (the start of the line is 0, and it counts up from there). This proves that the set of 2D grid points is of the same size as the set of whole numbers. Thus, the set of fractions is the same size as the set of whole numbers. Every fraction can code exactly one whole number, and every whole number can code exactly one fraction. Thus proving that they are the same size.

But there are some sets which can be proven to be too large to code to the set of whole numbers (which is itself infinite). Hence, there are sets bigger than "infinity", and it can in fact be shown that there are ascending infinities each one of which is too complex to be rendered down into any "lesser" infinity. The most basic example of a higher order infinity is the number of points in a line (e.g. the number of decimal points between zero and one, each of which may have infinite decimal points). This cannot be mapped to the whole numbers. But it makes sense when you look at it this way: whole numbers are strings of digits, of finite length. Although each whole number may have an arbitrary number of digits, it's always finite. But a real number between one and zero is a string of digits of infinite length. Hence, there will always be infinity of these for each of the whole numbers.

[Reelya]

Sorry for double post but I wanted to split this up...

all positive and negative integers and then all positive and negative rational numbers. I'm assuming a similar method to the above can be used, but again, how are the number lists to be generated?

It's entirely arbitrary, there are many ways to map numbers to other numbers. You just come up with a pattern and stick to it. Consider this for proving that even numbers are the same as odd numbers (I'm using <=> as "maps to"):

1 <=> 2
3 <=> 4
5 <=> 6
[...]
2n-1 <=> 2n (for all n)

To map positive to negative numbers it's really simple: "-n <=> n (for all n)"

To map all positive numbers to the whole range of +- numbers you can do it this way:

0 <=> 0
1 <=> -1
2 <=> 1
3 <=> -2
4 <=> 2
5 <=> -3
6 <=> 3
...
n <=> n/2 for all even n
n <=> -((n+1)/2) for all odd n

You can see this maps the positive numbers unambiguously to the set of all whole numbers, i.e. every number is accounted for.

For the set of positive and negative fractions, it's done the same as in your image. But the grid extends to the left and up, to account for negative x/y locations. Then, instead of the single line that zig-zags, you draw a diagonal (diamond shaped) spiral starting at the origin and working outwards to inifinity.

[ZetaX]

Infinite sets are of the same "size" if you can map one set to the other without any "leftovers".

To be precise: without any "leftovers" (the map is "surjective") and without sending two things to the same thing (it is  "injective").

Which is also a problem in this approach: you hit each rational number not only multiple times but infinitely often. There are several easy fixes, which amount to either showing that "a non-finite subsets of the positive integers is of the same size as the integers" or "a non-finite image of the positive integers is of the same size as the integers". Both these arguments often use the well-ordering (every non-empty subset has a smallest elements) of the positive integers somwhere.

Alternatively, one can use the (Cantor-)Schröder-Bernstein property of sets:

Let A, B be sets.
1. Write A ≤ B if there is an injective map from A to B. Then A ≤ B ≤ A gives A = B.
2. Write A ≤ B if there is an surjective map from B to A. Then A ≤ B ≤ A gives A = B.

[MagmaMcFry]

Alternatively, one can use the (Cantor-)Schröder-Bernstein property of sets:

Let A, B be sets.
1. Write A ≤ B if there is an injective map from A to B. Then A ≤ B ≤ A gives A = B.
2. Write A ≤ B if there is an surjective map from B to A. Then A ≤ B ≤ A gives A = B.

Actually, the CSB property is only part 1. Part 2 is dependent on the axiom of choice, I believe.

[ZetaX]

Alternatively, one can use the (Cantor-)Schröder-Bernstein property of sets:

Let A, B be sets.
1. Write A ≤ B if there is an injective map from A to B. Then A ≤ B ≤ A gives A = B.
2. Write A ≤ B if there is an surjective map from B to A. Then A ≤ B ≤ A gives A = B.

Actually, the CSB property is only part 1. Part 2 is dependent on the axiom of choice, I believe.

Both are CSB properties. The first one is the well-known CSB theorem, the latter is its dual (and indeed requires the axiom of choice). In general, a CSB property in its most general form (that is known to me) is a statement of the following type:
Let C be some category. For objects A, B write A ≤ B is there is a monomorphism from A to B. We say that C satisfies the CSB property if A ≤ B ≤ A implies that A, B are isomorphic in C.

One could replace "monomorphism" by any property of morphisms and "isomorphic" by "equivalent in regard to some equivalence relation on objects". But I never saw those generalisations in actual use.

If C are all sets, then "monomorphism" simply means "injective" and we get the first one from above.
Now let C be the dual category of sets. Then a monomorphism in C is an epimorphism of sets, i.e. a surjective map in the opposite direction. This gives the second one.

Wikipedia only really talks about injective versions in http://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_property, but their formulation is actually just what I wrote above.

[3man75]

Sorry for double post but I wanted to split this up...

all positive and negative integers and then all positive and negative rational numbers. I'm assuming a similar method to the above can be used, but again, how are the number lists to be generated?

It's entirely arbitrary, there are many ways to map numbers to other numbers. You just come up with a pattern and stick to it. Consider this for proving that even numbers are the same as odd numbers (I'm using <=> as "maps to"):

1 <=> 2
3 <=> 4
5 <=> 6
[...]
2n-1 <=> 2n (for all n)

To map positive to negative numbers it's really simple: "-n <=> n (for all n)"

To map all positive numbers to the whole range of +- numbers you can do it this way:

0 <=> 0
1 <=> -1
2 <=> 1
3 <=> -2
4 <=> 2
5 <=> -3
6 <=> 3
...
n <=> n/2 for all even n
n <=> -((n+1)/2) for all odd n

You can see this maps the positive numbers unambiguously to the set of all whole numbers, i.e. every number is accounted for.

For the set of positive and negative fractions, it's done the same as in your image. But the grid extends to the left and up, to account for negative x/y locations. Then, instead of the single line that zig-zags, you draw a diagonal (diamond shaped) spiral starting at the origin and working outwards to inifinity.

This looks like a function..Is this Calculas? Not that I would know its simply that I am actually very nosy 

Also I've recently found an issue with a certain problem. (2 - 3i) + (9 +8i).

The Math book I am working with is asking me to write it in a + bi form which involves rewriting the equation. I recently tried it and found that I got 6 + 1i but the answer book tells me its -6-11i.

I erased it and tried it again doing so:

(2 - 5i) - (8 + 6i)
(2 - 8 ) - (5 + 6) i ((Note: I planned to use the distributive prop.))
6 - 11i ((This is close but I did something wrong here and I can't find it.))

EDIT: Just saw that my 8 turned into an emote hehe. Sorry guys.

[frostshotgg]

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