The "Simple English" Wikipedia says (https://simple.wikipedia.org/wiki/Odd_number) that To find the set of odd natural numbers, we use $2N+1$ where $N$ is any integer. So, take $N=-3$. Then $2(-3)+1=-5$. So it is not in set of odd natural numbers. Actually, in the definition, $N$ should be any naturel number, shouldn't it?
To find the set of odd natural numbers, we use $2N+1$ where $N$ is any integer
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0According to that link's definition of *odd number*, you're incorrect: "An **odd number** is an integer which is not a multiple of two". – 2017-01-21
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$N $ is any integer and plugging in $N = -3$ produces the odd number $-5$. Negative numbers are also even or odd, depending on their remainder when divided by 2. As it turns out, a negative number $x $ is even if $-x$ is also even and the same for being odd (for example $-5$ is odd given that $5$ is also odd).
Because of how integer division works, we can see that any even number can be divided by 2, hence any even number is of the form $2N $, where $N $ is any integer. If a number is odd, then subtracting 1 from it gives an even number. But even numbers are of the form $2N $ so if I subtracted 1 to get to the number $2N $, then $2N+1$ is my original number, which was odd. Since this can be done with any odd number, all odd numbers are of the form $2N + 1$ where $N $ is any integer.
(Of course $N $ has to be an integer, otherwise setting $N=0.5$ gives $2\cdot0.5 + 1 = 2$ which is obviously not odd)
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0I agree with you. Probably, I couldn't explain clearly. Wikipedia says that ''To find the set of odd natural numbers, we use $2N+1$ where $N$ is any integer.'' Then, we can take $N=-3$. Then, we didin't find an odd number in set of odd natural numbers, right? – 2017-01-21
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0The explanation in the "Simple English" Wikipedia didn't make sense; I have changed it. – 2017-01-21
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1@Kahler you are right. Either $N $ is natural to give all natural odd numbers, or $N $ is an integer to give all integer odd numbers – 2017-01-21