Algebra Question: 2r - 2s - 1 = 2(r-s-1) + 1 (?)

[Animuo]

Homework Statement

Well, the problem is a discrete math problem to prove that the difference of an even integer and an odd integer is an odd integer.

Homework Equations

If you let m = an even integer, and n = an odd integer then m = 2r for some integer r, while n = 2s + 1 for some integer s.

The Attempt at a Solution

Then
m - n = 2r - (2s + 1) = 2r - 2s - 1

at the next step, the answer in the book shows
2(r-s-1)+1, from there I can follow the rest of the steps to show that an integer is odd but I'm confused as to where this extra one is coming from. I know it's just a basic factoring question, but I'm a little rusty, maybe somebody could help me.. thanks.

 

[Mark44]

Homework Statement

Well, the problem is a discrete math problem to prove that the difference of an even integer and an odd integer is an odd integer.

Homework Equations

If you let m = an even integer, and n = an odd integer then m = 2r for some integer r, while n = 2s + 1 for some integer s.

The Attempt at a Solution

Then
m - n = 2r - (2s + 1) = 2r - 2s - 1

at the next step, the answer in the book shows
2(r-s-1)+1, from there I can follow the rest of the steps to show that an integer is odd but I'm confused as to where this extra one is coming from. I know it's just a basic factoring question, but I'm a little rusty, maybe somebody could help me.. thanks.

2r - 2s - 1 = 2r - 2s - 1 - 1 + 1 = 2r - 2s - 2 + 1 = 2(r - s - 1) + 1

In the second step above, I added 0, in the form of -1 + 1. You can always add 0 to an expression to get an identically equal expression.

 

[Ray Vickson]

Homework Statement

well, the problem is a discrete math problem to prove that the difference of an even integer and an odd integer is an odd integer.

Homework Equations

if you let m = an even integer, and n = an odd integer then m = 2r for some integer r, while n = 2s + 1 for some integer s.

The Attempt at a Solution

then
m - n = 2r - (2s + 1) = 2r - 2s - 1

at the next step, the answer in the book shows
2(r-s-1)+1, from there i can follow the rest of the steps to show that an integer is odd but I'm confused as to where this extra one is coming from. I know it's just a basic factoring question, but I'm a little rusty, maybe somebody could help me.. Thanks.

-1 = -2 + 1

rgv

 

[Animuo]

Aha, can't believe I missed such an easy solution, guess it's good to humble one down, thanks for the help mates... I'm new to this forum is there anything like +rep or likes that I can give?

 

[skiller]

What am I missing? Why do you have to put it as 2(r-s-1)+1 to show it is odd? Why isn't it acceptable (and slightly simpler) just to put it as 2(r-s)-1 ?

 

[Bohrok]

Aha, can't believe I missed such an easy solution, guess it's good to humble one down, thanks for the help mates... I'm new to this forum is there anything like +rep or likes that I can give?

Unfortunately no, you can't give rep; but I know some would have a whole lot more than the rest of us!

What am I missing? Why do you have to put it as 2(r-s-1)+1 to show it is odd? Why isn't it acceptable (and slightly simpler) just to put it as 2(r-s)-1 ?

It's probably because odd numbers are usually written in the form 2(some letter or quantity) + 1 instead of 2(some letter or quantity) - 1.

 

[skiller]

It's probably because odd numbers are usually written in the form 2(some letter or quantity) + 1 instead of 2(some letter or quantity) - 1.

Yep, that's what I assumed. Seems a bit daft to me.