What is the Relationship Between Even Bases and Even Numbers in a Number System?

In summary, the conversation discusses the evenness of a number in base b, where b is an even integer. The statement states that a number H=(d(n-1)d(n-2)...d1d0)b is even if and only if d0 is even. The person is seeking advice on how to proceed with proving the backwards direction, starting with the fact that d0 is even. The suggestion is to consider the properties of even and odd numbers when multiplied or added together. The person later rewrites the problem to better understand it and concludes that it should not be too difficult to prove the backwards direction from there.
  • #1
Noxide
121
0

Homework Statement



Whenever base b is even (b=2k for some integer k) a number H=(d(n-1)d(n-2)...d1d0)b is even <=> d0 is even.

Homework Equations



The Attempt at a Solution


I have formulated a proof for the forward direction (=>) but I am having trouble getting started on a proof for the backwards direction (<=) starting with the fact that d0 is even.
Any advice would be much appreciated. I know all the definitions I'm just unsure how to build from d0 is even.
 
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  • #2
Noxide said:

Homework Statement



Whenever base b is even (b=2k for some integer k) a number H=(d(n-1)d(n-2)...d1d0)b is even <=> d0 is even.

Homework Equations



The Attempt at a Solution


I have formulated a proof for the forward direction (=>) but I am having trouble getting started on a proof for the backwards direction (<=) starting with the fact that d0 is even.
Any advice would be much appreciated. I know all the definitions I'm just unsure how to build from d0 is even.
You don't show what you've done, so this might or might not be useful.

H=(d(n-1)d(n-2)...d1d0)b
= d(n-1) * bn - 1 + d(n-2) * bn - 2 + ... + d1*b + d0

You have d0 being even. What can you say about b raised to any positive power? Is it even or odd? What about multiples of even or odd numbers?
 
  • #3
even * even = even always
odd * even = even always
even + odd = odd

I think my problem was identifying the problem correctly

I rewrote it this way

b = 2k, k is a natural number, H=(d(n-1)d(n-2)...d1d0)b is even <=> do is even

shouldnt be too hard from there
 

FAQ: What is the Relationship Between Even Bases and Even Numbers in a Number System?

What is base b proof?

Base b proof is a type of mathematical proof that uses a number system other than base 10 to illustrate a given statement or theorem. This can be useful in situations where a particular base allows for a more elegant or concise proof.

Why use a base other than 10 for proof?

Using a different base for proof can sometimes reveal patterns or relationships that may not be as clear in base 10. It can also simplify calculations or reduce the number of steps in a proof.

How do you convert a base b proof to base 10?

To convert a base b proof to base 10, you can use the following formula: base b number = (base 10 number) / (bn), where n is the number of digits in the base b number.

Can base b proof be used in all types of mathematical proofs?

Base b proof can be used in many types of mathematical proofs, but it may not always be necessary or applicable. It is most commonly used in number theory and abstract algebra, but can also be used in other areas of mathematics.

Are there any disadvantages to using base b proof?

One potential disadvantage of using base b proof is that it may be more difficult for readers who are not familiar with the particular base being used to understand the proof. It may also be more complex and require additional calculations compared to using base 10.

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