Proving a Division Algorithm for Real Numbers?

In summary, to prove the Division Algorithm for the Real numbers, one must show the existence of unique values for k and \delta such that x = k\alpha + \delta, where x, \alpha \in \mathbb{R} with 0 \leq \delta < \alpha and k \in \mathbb{Z}. This can be done by finding the lowest bound for the sequence na for n = 1,2,... and setting k = floor(x/a) and \delta = x - ka.
  • #1
Doom of Doom
86
0
How might I prove a Division Algorithm for the Real numbers?

That is to say, if [tex]x, \alpha \in \mathbb{R}[/tex], then [tex]x=k \alpha + \delta[/tex] for some [tex]k \in \mathbb{Z}, [/tex] [tex]\delta \in \mathbb{R}[/tex] with [tex]0 \leq \delta < \alpha[/tex] where [tex]k, \delta[/tex] are unique.
 
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  • #2
Assume that it is not unique, subtract one representation from the other. The resultant equation is obviously false.
 
  • #3
Yeah, but how can I show existence?
 
  • #4
Doom of Doom said:
Yeah, but how can I show existence?

The sequence na for n=1,2,... is unbounded. Therefore for some n, na>x. Find lowest bound, subtract 1 and you will have k. ka<=x, (k+1)a>x, so x-ka(remainder)<a
 
  • #5
or even more straightforward, let k = floor(x/a)
 

What is the division algorithm for R?

The division algorithm for R is a mathematical procedure used to divide two real numbers. It is based on the concept of long division and provides a systematic way to find the quotient and remainder when dividing two real numbers.

How is the division algorithm for R different from traditional long division?

The division algorithm for R is an extension of traditional long division to include real numbers. It uses the same principles and steps as long division, but allows for decimal numbers and infinite decimal expansions.

What are the steps involved in the division algorithm for R?

The division algorithm for R involves the following steps:
1. Write the dividend and divisor as decimal numbers.
2. Determine the first digit of the quotient by dividing the first digit of the dividend by the first digit of the divisor.
3. Multiply the first digit of the quotient by the divisor, and subtract this value from the first digit of the dividend.
4. Bring down the next digit of the dividend and repeat the process until all digits have been used.
5. The final answer will be the quotient and remainder, if any.

What is the significance of the remainder in the division algorithm for R?

The remainder in the division algorithm for R represents the part of the dividend that could not be evenly divided by the divisor. It is a crucial part of the algorithm and helps to determine the accuracy of the quotient.

Can the division algorithm for R be used for all types of real numbers?

Yes, the division algorithm for R can be used for all types of real numbers, including rational and irrational numbers. It is a universal method for dividing real numbers and is applicable in various mathematical and scientific fields.

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