## Numbers (Irrationals, etc.), Set, and Subsets

For college and university level advanced Mathematics
w_kneberg1
Posts: 5
Joined: Fri May 27, 2011 3:39 am

### Numbers (Irrationals, etc.), Set, and Subsets

I see that the real numbers have different subsets, and I wonder how to do Venn Diagrams about Sets and Subesets work with the Real Numbers. I also don't know how Sets and Subsets work, but I know that that may have something to do with Venn Diagrams as well, and possibly even mathematical logic. If someone can give me a really simple explaination that would be what I am looking for, but if anyone out there knows of some websites out there that sprecifically deal with these problems (and how they all fit together, if they do), I would be happy with that too.
From the title, you will see that I am interested in Irrationals. I am wondering how you do any of the basic operations on irrational and rational (addition, subtrations, multiplication, divisions, and different powers and their roots) work since with all of these (but divisions because it can keep on going on and on forever if it is not an irrational number dividing another irrational number), start with the last number to do the operation (multiplication starts from the last number, addition starts from the last number, as well as subtraction, and square roots can result in two irrational numbers that when mutliplied together result in the number that was square rooted, and the same thing can be with multiplying an irrational number by itself). While the square and square roots do not always involve irrational numbers, as division does, it happens sometime. There must be a rule that works all the time for addition, subtraction, multiplication, etc. in an opposite way of operating on the numbers (for example multiplication of two numbers that are irrational can't be done in the normal way because you have to start multiplying at the last number of both of them, and that doesn't work). Since all operations in math seem to have an opposite, I feel that there must be an opposite of the addition, subtraction, multiplication, division, powers, and roots, but not in the way that addition and subtraction are opposite, and multiplication and division are, and so on, but in a completely different way (and heck it may not be an opposite, it may be something that is not an opposite at all, like a a derivative is not an opposite, it is a rule for any tangent slope on a curve. Another intersting thought is that if you have a .9 + .1 = 0 .99 + .01 =1 .999 + .001, etc. What rule can you make for this? They all equal one, which sets them all equal, so what happens when they are equaled and you start using the principle of substitution? Another thing is (null solution) * 1 = -1? And this is for two lines on a graph whose slopes, when multiplied together, are -1. so the example given is vertical and horizontal line's slopes which results in -1. And I just wonder if you could take a number that is both positive and negative at the same time, so when it is square rooted, the answer will not be an imaginary number, but what would it be called and how would it be related to the imaginary number?
That's all for now,
Hope to get some feed back, I'd love to hear someone else's insight into this.
Sincerely,
w_kneberg

rakeen
Posts: 384
Joined: Thu Dec 09, 2010 5:21 pm
Location: Dhaka

### Re: Numbers (Irrationals, etc.), Set, and Subsets

I see that the real numbers have different subsets, and I wonder how to do Venn Diagrams about Sets and Subesets work with the Real Numbers. I also don't know how Sets and Subsets work, but I know that that may have something to do with Venn Diagrams as well, and possibly even mathematical logic. If someone can give me a really simple explaination that would be what I am looking for, but if anyone out there knows of some websites out there that sprecifically deal with these problems (and how they all fit together, if they do), I would be happy with that too.
I actually didn't understand that at all sorry. And why dont you post problem in School/college section instead of Social Longue
r@k€€/|/

nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### Re: Numbers (Irrationals, etc.), Set, and Subsets

I hardly understand what you are trying to say. It will be better if you divide it up into different sections. From what I have understood so far:

1.
w_kneberg wrote:From the title, you will see that I am interested in Irrationals. I am wondering how you do any of the basic operations on irrational and rational (addition, subtrations, multiplication, divisions, and different powers and their roots) work since with all of these (but divisions because it can keep on going on and on forever if it is not an irrational number dividing another irrational number), start with the last number to do the operation (multiplication starts from the last number, addition starts from the last number, as well as subtraction, and square roots can result in two irrational numbers that when mutliplied together result in the number that was square rooted, and the same thing can be with multiplying an irrational number by itself). While the square and square roots do not always involve irrational numbers, as division does, it happens sometime. There must be a rule that works all the time for addition, subtraction, multiplication, etc. in an opposite way of operating on the numbers (for example multiplication of two numbers that are irrational can't be done in the normal way because you have to start multiplying at the last number of both of them, and that doesn't work).
It is not true that these operations cannot be done with irrational numbers, you just cannot get a "visual" (maybe this is the right word) answer. Because for that we work in a finite base (e.g. base 10). And about addition and multiplication starting from the last number, we only found a "rule" (more specifically, algorithm), that is easy to perform, of obtaining the result when adding/mutiplying two numbers together, in which we start from the rightmost digit. Since irrationals have no rightmost digit our algorithm fails. But pure mathematicians aren't really interested in the "visual" bit, and applied mathematicians are satisfied with estimates (e.g. approximations). But if you really want to do it, why don't you just try and come up with a new rule yourself? Surely all the rules we use have been created by people before us.

2.
w_kneberg wrote:Since all operations in math seem to have an opposite, I feel that there must be an opposite of the addition, subtraction, multiplication, division, powers, and roots, but not in the way that addition and subtraction are opposite, and multiplication and division are, and so on, but in a completely different way (and heck it may not be an opposite, it may be something that is not an opposite at all, like a a derivative is not an opposite, it is a rule for any tangent slope on a curve.
I don't really understand what you mean but if you are interested in operations you may want to read some group theory.

3.
w_kneberg wrote:.9 + .1 = 0 .99 + .01 =1 .999 + .001, etc. What rule can you make for this? They all equal one, which sets them all equal, so what happens when they are equaled and you start using the principle of substitution?
So what's wrong if they all equal 1? I don't understand.

4.
w_kneberg wrote:Another thing is (null solution) * 1 = -1? And this is for two lines on a graph whose slopes, when multiplied together, are -1. so the example given is vertical and horizontal line's slopes which results in -1.
If I understand what you are saying: the slope of the vertical axis isn't defined so you cannot say that their product is -1.

5.
w_kneberg wrote:And I just wonder if you could take a number that is both positive and negative at the same time, so when it is square rooted, the answer will not be an imaginary number, but what would it be called and how would it be related to the imaginary number?
How can a real number be both positive and negative at the same time? Define a relation \$\sim\$ on the non-zero real numbers such that \$a\sim b\$ iff \$ab>0\$. This is an equivalence relation, and all non-zero real numbers are partitioned into the two equivalence classes: positive and negative reals; hence no such number exists.
"Everything should be made as simple as possible, but not simpler." - Albert Einstein