Are These Properties of Zero Correct?

1. Sep 9, 2014

bballwaterboy

x + 0 = x
x - 0 = x

x(0) = 0
x(0) = undefined ...is this how you call it or is it no solution?

x^0 = ??? What is x raised to the zero power?

If anyone knows of any other properties of zero I'm leaving out, please feel free to post them. It's been over a year since I last did this stuff and have to know it all for my class. I'm already a bit behind, so having to spend my own personal time catching up with the basics. Thanks for your help.

2. Sep 9, 2014

HallsofIvy

Yes, the first is pretty much the definition of "zero" and the second follows from the first.

If you mean "x times 0", yes, that is true. It can be show by considering that since a+ 0= 0 (above) the b(a+ 0)= ba+ b0= ba

??? This is the same as above! Did you mean "x/0"? If x itself is non-zero, then, yes, it is "undefined". I would not say "no solution" because I see no problem to be solved.

Note that I said "if x itself is non-zero". If x= 0, we say that "0/0" is "undetermined". We still "can't divide by 0" but the difference is that saying that a/0= b is the same as saying that a= b0= 0. If a is not 0 that is not true for any b. If a= 0 then it is true for all b. In either case, we cannot determine a specific value for b but the reason is different.

If x is not 0, then x^0= 1. For positive integer m and n, it is easy to show that x^(m+ n)= x^m x^n. If we want to extend that to 0, we must have x^(m+0)= x^m= x^mx^0 so must have x^0= 1. If x= 0, so we have 0^0, that is, again, "undefined". We have two basic rules: x^0= 1 for x non-zero and 0^x= 0 for x non-zero. At x= 0 those two cannot be reconciled.

Nothing special about that. The square root of 0 is 0 because 0^2= 0(0)= 0.

3. Sep 9, 2014

bballwaterboy

Ooops! Yes! That was supposed to be x divided by 0. Must have been a typo slip. Thanks!

But wait. Quick question. You say it could be solved? How's that? How do you divide by zero?

4. Sep 9, 2014

bballwaterboy

Hey, you're pretty good with this stuff. Are you an Ivy-leaguer by chance?

re: sq. rt. of 0. That makes sense!

re: x^0. I'm going to have to reread that slowly sometime, because it's a little over my head right now. Thanks so much though! Very helpful. I love explanations too. That's actually what I feel is lacking so much in my class sometimes. So thanks!

5. Sep 9, 2014

SteamKing

Staff Emeritus
This stuff used to be taught in arithmetic class, and you didn't need a fancy college education to understand it.

6. Sep 9, 2014

WWGD

In a sense, division by zero is possible in some situations, like when doing a quotient of a vector space by a zero subspace.

7. Sep 9, 2014

Staff: Mentor

The OP was talking about division with real numbers. Quotient spaces really have nothing to do with division in the sense intended here.

Last edited: Sep 9, 2014
8. Sep 10, 2014

HallsofIvy

I did NOT say it could be solved. I said that you had not posted a problem to be solved!

9. Sep 10, 2014

Matterwave

I think maybe OP was referring to the fact that "HallsofIvy" might be referring to the halls of an ivy league university?

10. Sep 10, 2014

WWGD

I know; please note the qualifying statements " In a sense" , and where I make specific references to quotient spaces. I am just trying to provide a broader view the Op may not be aware of. I would be glad if others did the same for me in areas I am not familiar with.

11. Sep 11, 2014

HallsofIvy

Actually, no. There used to be a radio program, about a college professor called "halls of ivy" and my father put a sign outside our house saying "halls of ivy"- it was a play on his name.

No, I did NOT go to a "fancy" ivy league college, I went to M.I.T. where, as SteamKing points out, I was taught arithmetic.