Proving Density of Dyadic Rationals in Q | Rational Numbers

  • Thread starter cragar
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In summary, we can prove that the dyadic rationals are dense in Q by finding the midpoint between two arbitrary rationals x and y, where x<y, and repeating this process until we find a rational in between x and y. This is achieved by picking a rational smaller than x and a rational larger than y, both of the form \frac{m}{2^n}, and finding the midpoint by adding them together and dividing by 2. This process can be repeated with increasingly smaller intervals until we reach a rational in between x and y. This method can be considered creative and can be proved in a less elaborate way.
  • #1
cragar
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Homework Statement


Prove that the dyadic rationals are dense in Q.
That is the rationals of the form [itex] \frac{m}{2^n} [/itex]
m is an integer and n is a natural

The Attempt at a Solution


Let's say we have two arbitrary rationals x and y. where x<y
Now I will pick a rational smaller than x such that it is of the form
[itex] \frac{s}{2^k} [/itex] and i will call this P ,
now I will pick a rational larger than y that is of the same form
and i will call it O .
Now I will add P and O together and then divide by 2, find the midpoint
Now this new rational has a denominator that is a power of 2 because
everything we did had a denominator of 2. Now I will keep doing this,
I will keep finding mid points between these sets of rationals
that I created and I might have to pick the left or right one and then
keep finding the midpoints. Eventually i will get in between x and y.
I realize this is informal but Is my general idea in the right direction.
 
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  • #2
cragar said:

Homework Statement


Prove that the dyadic rationals are dense in Q.
That is the rationals of the form [itex] \frac{m}{2^n} [/itex]
m is an integer and n is a natural

The Attempt at a Solution


Let's say we have two arbitrary rationals x and y. where x<y
Now I will pick a rational smaller than x such that it is of the form
[itex] \frac{s}{2^k} [/itex] and i will call this P ,
now I will pick a rational larger than y that is of the same form
and i will call it O .
Now I will add P and O together and then divide by 2, find the midpoint
Now this new rational has a denominator that is a power of 2 because
everything we did had a denominator of 2. Now I will keep doing this,
I will keep finding mid points between these sets of rationals
that I created and I might have to pick the left or right one and then
keep finding the midpoints. Eventually i will get in between x and y.
I realize this is informal but Is my general idea in the right direction.

You can probably prove it in a less elaborate way. If x<y then y-x is positive and there must be an n such that 1/2^n is less than y-x, yes?
 
  • #3
yes I could do it that way. Thats the cool thing about pure math is that it is very creative.
 

FAQ: Proving Density of Dyadic Rationals in Q | Rational Numbers

What is the definition of dyadic rationals in Q?

Dyadic rationals are rational numbers that can be expressed in the form of p/2^q, where p and q are integers and q is non-negative.

Why is it important to prove the density of dyadic rationals in Q?

It is important to prove the density of dyadic rationals in Q because it helps us understand the distribution of rational numbers on the number line and their relationship with other types of numbers, such as irrational numbers. It also allows us to approximate any real number with a dyadic rational number, which has practical applications in fields such as engineering and computer science.

How is the density of dyadic rationals in Q proven?

The density of dyadic rationals in Q is proven using the Archimedean property, which states that for any two real numbers x and y, there exists a natural number n such that nx > y. By applying this property to the rational numbers, we can show that between any two rational numbers, there exists a dyadic rational number.

What are some examples of dyadic rationals in Q?

Some examples of dyadic rationals in Q include 1/2, 3/4, -2/8, and 5/16. These numbers can be expressed in the form of p/2^q, where p and q are integers and q is non-negative.

How does proving the density of dyadic rationals in Q relate to other mathematical concepts?

Proving the density of dyadic rationals in Q is closely related to other mathematical concepts such as decimal representation of numbers and continued fractions. It also has implications in fields such as topology and measure theory, where the concept of density is important in understanding the properties of mathematical structures.

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