Are These Properties of Zero Correct?

[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?
square root of 0 = ? Not sure about this one either.

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.

 

[HallsofIvy]

x + 0 = x
x - 0 = x

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

x(0) = 0

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

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

? 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.

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

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.

square root of 0 = ? Not sure about this one either.

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

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.

 

[bballwaterboy]

? 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.

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?

 

[bballwaterboy]

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.

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!

 

[SteamKing]

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

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

 

[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.

 

[Mark44]

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

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

 

[HallsofIvy]

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?

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

 

[Matterwave]

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

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

 

[WWGD]

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

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.

 

[HallsofIvy]

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

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.