So are there more even numbers than odd ones?

[NikPreviousAcct]

Originally posted by: RaistlinZ
Yeah, what about 0? Given that there is no -0 and that 0 is an even number, there will always be one more even number than odd number.

Bingo.

 

[Orsorum]

Hehe, this thread made my head hurt.

 

[oboeguy]

Lots of good math people answering here. How about a trivial proof that there are the same number of evens as odds, degrees of infinity aside? For each even number you get a unique odd number by adding one to it, and vice versa. QED.

 

[Legendary]

Originally posted by: Nik

Originally posted by: RaistlinZ
Yeah, what about 0? Given that there is no -0 and that 0 is an even number, there will always be one more even number than odd number.

Bingo.

Edit: looked it up, apparently zero is even. My mistake.

 

[Howard]

Originally posted by: RaistlinZ
Yeah, what about 0? Given that there is no -0 and that 0 is an even number, there will always be one more even number than odd number.

Starting counting even/odd from zero, then.

{0,1} is the first set, {2,3} is the second... then there are an infinite number of sets if we're talking about non-negative integers.

 

[Kev]

Originally posted by: trilks
Infinite evens !> infinite odds.

:thumbsup:

 

[SpecialEd]

here we go...


write out all integers ...-3, -2, -1, 0, 1, 2, 3....

Lets assume 0 is even.

just group every two numbers based off the group {0,1}

we get ..., {-4,-3}, {-2,-1}, {0,1}, {2,3}, {4,5}, ...

Clearly there is 1 even and 1 odd in every pair.

Thus there must be the same number of even and odd integers.



edit: I didn't see you post howard. Good job.

 

[SinNisTeR]

Originally posted by: mchammer187

Originally posted by: SinNisTeR

Originally posted by: mchammer187

Originally posted by: SinNisTeR
there are the same amount..

what about rationals vs irrationals?

same

sqrt of any number that is not a perfect square

rationals have a countable infinity, while irrationals do not. there are different types of infinity..

i dont understand how you can count the rationals

.01 .001
.0001 .00001

isn't there an uncountable infinite number of rationals between .01 and .0100000000000000000001

now if you are talking integers that is one thing but rationals is another

i like to draw a little table to visualize it better..
-----1---2---3---4---5---...........
1- 1/1 2/1 3/1 4/1 5/1
2- 1/2 2/2 3/2 4/2 5/2
3- 1/3 2/3 3/3 4/3 5/3
4- 1/4 2/4 3/4 4/4 5/4
5- 1/5 2/5 3/5 4/5 5/5
.
.
.
.
.

for irrationals you have things like square root 2 and pi. etc. those all fit in between all rationals, ifinitely packed.

think of this graph
f(x)
{
0 while x = irrational
1 while x = rational or p/q while q != 0
}

what does it look like?

picture an xy coordinate system with
y=0 completely covered, a solid line pretty much and
y=1 very dense line ............ with little holes pretty much.

make sure you zoom in on it

if you put y=1 on top of y=0 you get the Real number system

 

[eigen]

The number of evens equals the number of odds because you can form a bijection between the two....
The kicker is this... there are as many even number as there integers .....