Eric,....
I'll take that shirt with ketchup, if you don't mind (or is that catsup?).
(DUTW)??? Now you hurt my feelings... (FUEN)
Bernie said...
"Terry, help me, my mind is going!"
What... like I have a mind that either of us could hang onto? If we were floating in a river it would pull us both down like a rock!
When I said "Yes, even zero is EVEN", PLucas said...
"Zero cannot be divided by two. Does that make zero an odd number? "
"... the definition of an even number is 'an integer that is divisible by two with no remainder"
Zero most certainly can be divided by any number except zero!
The result is always zero... that is, an Integer with no remainder.
We must treat "Zero" with all the respect it deserves... and it deserves a HUGE amount of respect! We wouldn't be on-line (on the Internet or PLC's) without the VERY REAL concept of "My Hero Zero!"
Dividing zero is not a crime... dividing BY zero is the crime.
"If there are doubly-even numbers then there must be a doubly-odd number..."
How can you divide an ODD Number by 2, and then expect to have an ODD Integer with no Remainder?
There simply are no "doubly-odd" numbers.
"If I had 98 bottles of beer (I drank one while typeing this reply) and gave them to you and asked you to divide them between us, that would not be a problem, you could quite easily do that. But if I gave you NO (zero) bottles of beer and again asked you to divide them between us, you couldn't do it because there is nothing to divide.
In that case, zero is NOT divisible by two."
I'm going to give you a random amount of something. You take what I give you and share it equally with your buddy.
I give you 8 cookies. That's 4 for you, 4 for your buddy. No extras.
I give you 5 cookies. That's 2 for you, 2 for your buddy, and then you split the "odd one".
I give you no cookies, zip, nada, not one stinkin' cookie. Your buddy says, com'on, where's my cooookieees???
He pulls your hands open and grabs at nothing. Now, you both have the same fair share; none. And there ain't no extra cookie to split.
So, you both got the same share and you didn't have to break one. That's called, "Even-Steven".
"So, if Zero is not an odd or an even number is it really a number?"
IT IS MOST DEFINITELY A NUMBER and it is EVEN.
Then, BitMore said... (I can't hardly believe this...)
"Zero is nothing more than a place holder to indicate "none" as in none ones. 10 is one tens and no ones....and now on the right side of the decimal is none tenths, none hundreths, none thousanths, add ifinitum....
0 is not a number in the true sense of the word, keep in mind it is not of type NAN! Although it does become cumbersom when dealing with IEEE floaters!"
Repeating what I said earlier...
We must treat "Zero" with all the respect it deserves... and it deserves a HUGE amount of respect! We wouldn't be on-line (on the Internet or PLC's) without the VERY REAL concept of "My Hero Zero!"
Until the Arabs got the concept of "0" from the Indians (India), Mid-Eastern math (the leading math of the day) was at a stand-still. The Western World was at an even was impasse. They hadn't even developed a workable number system. Roman Numerals were fine for counting bales of wheat; and that was about all it was good for. Oh yeah, it also made the Date look interesting.
When the arabs got "zero" and the Western World got Arab math... then things started to bust wide open. It would not have happened without the recognition of the important place "none" (zero) has in the universe as we understand it to be. It's a REAL NUMBER and as valid as any other number you can pull out of your a$$.
BINARY, OCTAL, DECIMAL, HEXIDECIMAL,... pick a number system... they all start with "0" and carry it through the system as it progresses.
Zero is not just a beer-keg taking up space uselessly. There is real purpose in "My Hero Zero"! (Same applies to the beer-keg!)
Now... the rule defining an Even Number is as follows:
"n = 2k"
"k" is any member of the set of Integers.
"n" is a member of the set of EVEN Integers.
Two times any Integer ("k") = an Even Integer "n".
The Odd/Even Question is answered by means of a test. The test is to divide any number by 2....
"n / 2 = ?"
The result of REAL Division produces a Quotient where the Quotient is composed of an Integer value, followed by a dot, followed by some more numbers. The numbers after the dot represent a decimal fraction.
The result of MOD Division produces a Quotient AND a Remainder; both are Integers.
In the test, if "n" is an Even Integer, the division will produce neither a decimal fraction, nor a remainder.
So, in terms of MOD Div only...
If the test produces a remainder of "0" then the tested number is EVEN.
If the test produces a remainder of "1" then the tested number is ODD.
Akreel said...
"Look at it in the binary world. I can continue to divide by two (bit shift, if you like) until I have a 1 in the lowest bit. So, it all comes down to POWERS of two.
Take the binary number 10000000000000000, how "even" is that? 110110000000000000000 is "just as even," to use your terminology. But, in the end one of the divisions (the sixteenth) gets an odd number."
Interesting...
"Extended" testing, that is, repeating the test until there is a remainder of "1", shows something interesting.
0/2 = 0 r 0..................................."n"/EVEN/EVEN/EVEN... no end...
1/2 = 0 r 1..................................."n"/ODD
2/2 = 1 r 0 => 1/2 = 0 r 1..................."n"/EVEN/ODD
3/2 = 1 r 1..................................."n"/ODD
4/2 = 2 r 0 => 2/2 = 1 r 0 => 1/2 = 0 r 1....."n"/EVEN/EVEN/ODD
5/2 = 2 r 1..................................."n"/ODD
6/2 = 3 r 0 => 3/2 = 1 r 1...................."n"/EVEN/ODD
12/2 = 6 r 0 => 6/2 = 3 r 0 => 3/2 = 1 r 1...."n"/EVEN/EVEN/ODD
1024/2 = 256 r 0 => 256/2 = 128 r 0 =>...=> 4/2 = 2 r 0 => 2/2 = 1 r 0
"n"/EVEN/EVEN/.../EVEN/EVEN/ODD
hmmm.... now that's odd...
or as Rick says...
that's Terry Woods (I can really relate! I mean, I'm related... I mean... Hey, wait a minute!).
Except for "0", extended testing brings all numbers to ODD.
Could it be that, except for "0", there is a certain ODD-ness any number?
And why is "0" so special? Is "0" some kind of "perfect" number?
Try this hat on for size...
"n= 2k" => "n/2 = k"
This says... take any Even Number and divide it by 2 and the result is an Integer.
I can buy that.
But wait! There's more!
"n= 2k" => "n/k = 2"
hmmm... that says... take any Even Number "n" and divide it by any Integer "k" and the result will be 2.
How 'bout...
6 / 9 = ? It sure ain't 2!
14 / 15 = ? Nope! Not that one either!
Me thinks I might have stumbled onto the illusionary aspect of the ODD/EVEN game.
The algebraic equation that defines EVEN Numbers can not survive basic algebraic manipulation.
Interesting... both of those divisions produced Remainders. Does that mean that they are ODD?
Both of those values are between "0" and "1". Does that mean that all numbers between "0" and "1" are ODD?
And since there are an infinite number of numbers between "0" and "1", does that mean there are infinitely more ODD Numbers than EVEN Numbers???
Now I think is the time to act on David's advice.
(834) (LUEN)
