The basis of the Real numbers over the Irrationals

In summary, the dimension of the basis of the Reals over the Irrationals is infinite, as any real number can be made from the combination of irrational vectors multiplied by the same irrational coefficient. However, this statement is not entirely accurate as it involves incorrect calculations and does not address the concept of a vector space.
  • #1
nautolian
34
0
1. What can be said of the dimension of the basis of the Reals over the Irrationals
2. Homework Equations
3. I believe the basis is infinite because any real number can be made out of the combination of irrational vectors multiplied by the same irrational coefficient to make any real number. ie. sqrt(2)*sqrt(2)=2 + sqrt(51)*sqrt(51)=53, etc. Could you please help me figure out if this is the correct solution or what a better way to phrase the proof would be? Thanks!
 
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  • #2
nautolian said:
1. What can be said of the dimension of the basis of the Reals over the Irrationals



Homework Equations





3. I believe the basis is infinite because any real number can be made out of the combination of irrational vectors multiplied by the same irrational coefficient to make any real number. ie. sqrt(2)*sqrt(2)=2 + sqrt(51)*sqrt(51)=53, etc. Could you please help me figure out if this is the correct solution or what a better way to phrase the proof would be? Thanks!

The irrationals aren't a vector subspace of R, so the question is pretty meaningless. Did you maybe mean the rationals?
 
  • #3
Actually, I think it's a trick question because I thought that at first but I wasn't sure. How would I prove that? Does it break the multiply by zero axium since zero is not an irrational? Thanks for your help!
 
  • #4
nautolian said:
Actually, I think it's a trick question because I thought that at first but I wasn't sure. How would I prove that? Does it break the multiply by zero axium since zero is not an irrational? Thanks for your help!

It sure does. 0 is not an irrational. Lot's of other things break. You can also add two irrationals and get a rational etc etc.
 
  • #5
Hey thanks, how would adding two irrationals to get a rational break the vector space rule?
 
  • #6
nautolian said:
Hey thanks, how would adding two irrationals to get a rational break the vector space rule?

To be a vector space you first have to be an additive group. The addition operation has to be closed. I.e. if x and y are irrational then x+y has to be irrational. Give a counterexample.
 
  • #7
nautolian said:
1. What can be said of the dimension of the basis of the Reals over the Irrationals



2. Homework Equations



3. I believe the basis is infinite because any real number can be made out of the combination of irrational vectors multiplied by the same irrational coefficient to make any real number. ie. sqrt(2)*sqrt(2)=2 + sqrt(51)*sqrt(51)=53, etc.
What are you doing here? √2 + √2 = 2, and 2 + √51 + √51 = 53, but in connecting expressions as you did above, you are saying that 2 = 53, which is clearly not true.
nautolian said:
Could you please help me figure out if this is the correct solution or what a better way to phrase the proof would be? Thanks!
 

1. What is the difference between real numbers and irrationals?

Real numbers are a set of numbers that include both rational and irrational numbers, while irrationals are a subset of real numbers that cannot be expressed as a ratio of two integers.

2. How are real numbers over the irrationals represented?

Real numbers over the irrationals are usually represented as an interval on the number line, with the irrational numbers being the points in between the rational numbers.

3. Can irrationals be written in decimal form?

Yes, irrationals can be written in decimal form, but they will either have an infinite number of non-repeating digits (e.g. pi) or they will have a repeating pattern of digits (e.g. square root of 2).

4. How do irrational numbers behave in mathematical operations?

Irrational numbers behave in the same way as real numbers in mathematical operations such as addition, subtraction, multiplication, and division. However, the result may be a non-terminating decimal, unlike rational numbers where the result is always a terminating or repeating decimal.

5. Why are irrational numbers important in mathematics?

Irrational numbers play a crucial role in mathematics, as many important constants and mathematical concepts (such as pi and the Pythagorean theorem) involve irrational numbers. They also help to bridge the gap between rational numbers and real numbers, providing a more complete understanding of the number system.

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