Game Score - Game Talk Discussion Forum

Daniel · Posted: Tue May 06, 2003 11:40 am Post subject: Math Problem #1

Here is a math problem for you to solve.

For all x : (x + 1)^2 = x^2 + 1

What assumption makes this true?
And a proof of why this assumption makes this true.

There is likely to be many interpretations and possible answers, so alternative solutions will probably be encountered. Only the solution I am thinking of will be accepted as the correct solution although other correct solutions may be possible.

Daniel · Posted: Tue May 06, 2003 1:41 pm Post subject:

I was just asked in a Private Message what ^ means. Here is the explanation I gave.

MaxxRonin · Posted: Tue May 06, 2003 2:45 pm Post subject:

(x+1)^2=x^2+1. this is true because 1 squared is 1 and x squared is x squared. (x+1)^2 = x^2 +1. x^2 + 1 is just the simplified version of (x+1)^2. that is why this is true.

MaxxRonin

Daniel · Posted: Tue May 06, 2003 2:50 pm Post subject:

No that is not correct. x^2+1 is not a simplified version of (x+1)^2. This simplification is invalid.

If you don't believe me, just put in a number. Ie x=10. Then you get 10^2+1 = 101, (10+1)^2 = 11^2 = 121. 101 doesn't equal 121. So you are wrong.

MaxxRonin · Posted: Tue May 06, 2003 2:58 pm Post subject:

then technically didn't you just prove your problem wrong?

Dom · Posted: Tue May 06, 2003 9:58 pm Post subject:

I may be way off the mark here, as I'm no mathematician.

The only number that x can be is 0 for the equation to work.
I suppose that x=0 would have to be my assumption.

Proof: I made an Excel spreadsheet to test my assumption.

This answer is probably complete rubbish, but I have no idea how to correctly answer these sort of questions.

Daniel · Posted: Tue May 06, 2003 11:59 pm Post subject:

No the problem is not wrong it is totally valid and has a perfectly workable solution.

Yes x=0 is one solution to the equation. I will give you a hint there is a second solution. What is it and why is it valid?

MaxxRonin · Posted: Wed May 07, 2003 12:09 am Post subject:

i did it incorrectly, i'm sorry. (x+1)^2=x^2+1 therefore x must be negative 1(on my calculator a negative number squared, i have also heard that a negative number squared is positive) but in any case if my calculator is right then x should equal negative 1. because -1+1 =0 square that and you get 0. -1 squared should be -1 according to my calculator. but i thing im wrong, but if not -1 +1 again equals 0

Daniel · Posted: Wed May 07, 2003 1:18 am Post subject:

Yes you have done it incorrectly. If you enter into your calculator - 1 then push squared, it will usually give you - (1) ^2 which is - 1squared which is -1.

But (-1) squared is 1 since -1 * -1 = 1.

So (-1 + 1)^2 = 0^2 = 0
But (-1)^2 + 1 = 1 + 1 = 2

These are not equal.

Daniel · Posted: Thu May 08, 2003 10:33 am Post subject:

OK, noone seems to be going anywhere with this. So I'll give one hint.

You can expand (x + 1)^2 to get x^2 + 2x + 1

And so you have

(x + 1)^2 = x^2 + 2x + 1 = x^2 + 1

That's enough of a hint for the time being. If noone has replied again within a day or so I will give the answer.

Daniel · Posted: Mon May 12, 2003 1:23 pm Post subject:

Noone seems to be bothering to try this anymore, so I'll give the solution.
----------------------------------

If we make the assumption that x is from the field Z subscript 2 then we have the following:

(x+1)^2 = x^2 + 2x + 1 = x^2 + 0x + 1 since 2 = 0 under Z2, and thus is = x^2 + 1.

Thus we have (x+1)^2 = x^2 + 1 for all x in Z2.

To check we can just try putting in all the possible x's from Z2.

x=0:
(0+1)^2=1^2=1=0^2+1 so true for x=0

x=1:
(1+1)^2=2^2=4=0=2=1+1=1^2+1 so true for x=1

For people who don't know any better I will give a simple explanation of Z2, in Z2 if you have a number k then you need to calculate k mod 2 or k%2 in a language like C++. This will give you the modulus of the number k. Basically if k is even you'll get a 0 and if its odd you get a 1. That is why you see things above liek 4 = 0 = 2, or 1 = 3 = 5 etc.

Anyway, that was all you had to say. That x is from the field Z2, or give an explanation like above for those who don't know about the more complex stuff. (I wouldn't have assumed that anyone here knows about fields, but people here may have encountered mod and % etc)

MaxxRonin · Posted: Mon May 12, 2003 4:14 pm Post subject:

x can't = 1. bacause (1+1)^2=4 and 1^2 +1 =2 therefore 1 can't equal 1. i have tried this solution many times. i have also asked many people, and they agree that it is an invalid solution.

Dom · Posted: Mon May 12, 2003 4:59 pm Post subject:

I remember now why I hate maths!

Daniel · Posted: Mon May 12, 2003 11:00 pm Post subject:

Did you even read it. Its in field Z2. This means you only have the numbers 0 and 1. And 0=2=4=6=8=10=12, 1=3=5=7=9 etc.

MaxxRonin · Posted: Wed May 14, 2003 3:04 pm Post subject:

i have never heard of field z2 right now, i am only in algebra 1. it might be a little advanced. (for me) Very Happy