Does this small odd and even proof works?

In summary, we are proving that 101 is an odd number. Assuming 101 is even, we can find a number ##b## such that ##101=2b##. However, by adding 1 to both sides, we get ##102=2b+1##. This leads to a contradiction as the right side is not divisible by 2, but the left side can be divided by 2. We can also prove that ##2b+1## is odd by showing that there is no integer between ##2b## and ##2b+2##, leading to the conclusion that 101 is indeed an odd number.
  • #1
Seydlitz
263
4
This is taken from Peter J. Eccles, Introduction to Mathematical Reasoning, page 17. This is not a homework because it is an example in the text. Prove that 101 is an odd number. The text has given a way of proving it and I just want to do it with my own approach.

Prove that 101 is an odd number.

Assume 101 is even. There is a number ##b## such that ##101=2b##. Adds 1 to both side. ##102=2b+1## The right side shouldn't be divisible by 2 but the left side can be divided by 2. A contradiction? Hence 101 is an odd number.
 
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  • #2
Given the level of the question it seems like you should prove that 102 can be divided by 2 (by stating what the multiplication is). At that point it's just as easy to say 101 = 50*2+1 so is odd but them's the shakes
 
  • #3
Office_Shredder said:
Given the level of the question it seems like you should prove that 102 can be divided by 2 (by stating what the multiplication is). At that point it's just as easy to say 101 = 50*2+1 so is odd but them's the shakes

Well ok then, I just want to know if that contradiction works.
 
  • #4
Your proof also uses the fact that if a number isn't even, then it's odd. Again considering the level of the question, this fact may not have been proven yet.
 
  • #5
Tobias Funke said:
Your proof also uses the fact that if a number isn't even, then it's odd. Again considering the level of the question, this fact may not have been proven yet.

It is not proven but it is used in the definition that if a number is not even then it is odd.
 
  • #6
Seydlitz said:
It is not proven but it is used in the definition that if a number is not even then it is odd.

Then you should probably think about proving that if a number is of the form 2b+1 then it's odd (in particular you have to prove it's not even)
 
  • #7
Prove that ##2b+1## is odd.

Suppose ##2b+1## is even, then it exists an integer ##c##, where ##2b+1=2c##

##2c+1=2b+2## and ##2c-1 = 2b##, hence ##2b<2c<2b+2##. Further ##b<c<b+1##

But there is no integer which is larger and smaller than the next consecutive integer. So ##2b+1## must be odd.

I think this opens a new bag of cats, but at least I found this myself from scratch!

(The book standard proof implicitly assumes this I believe)
 
Last edited:
  • #8
That's a very nice proof
 
  • #9
Office_Shredder said:
That's a very nice proof

I think I can at least feel this small 'beauty' feeling after sketching it. Thanks!
 

1. Can you explain the proof of this small odd and even number?

The proof of this small odd and even number uses the fact that any even number can be written as the sum of two odd numbers. This means that if we have an even number, we can subtract an odd number from it to get another even number. We repeat this process until we get to 0, and then we have proven that the original even number is the sum of odd numbers.

2. Is this proof valid for all odd and even numbers?

Yes, this proof works for all odd and even numbers because the property of being able to write any even number as the sum of two odd numbers holds true for all even numbers. This means that the proof is valid for any odd and even numbers, regardless of their size.

3. How does this proof relate to mathematical induction?

This proof is related to mathematical induction because it uses the same principle of breaking down a larger problem into smaller, more manageable parts. In mathematical induction, we prove that a statement is true for a specific value, and then we use this to prove that it is also true for the next value. In this proof, we start with a large even number and break it down into smaller even numbers until we get to 0, which is the base case.

4. Can this proof be applied to other mathematical concepts?

Yes, the idea of breaking down a larger problem into smaller parts can be applied to other mathematical concepts, such as proving the sum of consecutive integers or the sum of a geometric series. This proof serves as a good example of how mathematical induction can be used to solve a variety of problems.

5. What are the real-world applications of this proof?

This proof has many real-world applications, particularly in computer science and programming. It can be used in algorithms that require breaking down a large problem into smaller parts, such as sorting and searching algorithms. It also has implications in cryptography, where odd and even numbers are used in encryption and decryption processes.

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