This depends on what properties you want your number system to satisfy. Usually you want for any three numbers a,b,c to satisfy
Associativity of addition: a+(b+c)=(a+b)+c
This is quite useful, so we don’t want to give this up
Commutativity of addition: a+b=b+a
Also useful but you could get around that if you really want to, but for our purposes let’s keep it
An additive identity (or zero): 0+a=a=a+0
You want a zero, so this is needed
Additive inverses: There exists x such that a+x=0 (here x=-a); you also want this
Associativity of multiplication: a*(bc)=(ab)*c
Same as above, you want this property
Commutativity of multiplication: Useful but not necessary
A multiplicative identity (or one): 1a=a=a1
Usually with 1=/=0, also useful
Multplicative inverses for nonzero elements: Not that necessary, there are useful number systems without this (like the integers …,-1,0,1,…)
Distributivity: a(b+c)=ab+ac, (a+b)c=ac+bc
You ant this, as it links addition and multiplication and this is quite desirable.
If you assume 4. and 9., you get 0a = (0+0)a=0a+0a, hence 0=0a. This means that you would have to give up distributivity wihin your number system, however distributivity is what links addition and multiplication together, hence your question would just be “what if we have two binary operations that don’t really interact with each other?” and the answer is: Maybe there are useful cases?
Edit: I forgot about losing property 4, in which case some examples are found in the following math stackexchange post
This depends on what properties you want your number system to satisfy. Usually you want for any three numbers a,b,c to satisfy
Associativity of addition: a+(b+c)=(a+b)+c This is quite useful, so we don’t want to give this up
Commutativity of addition: a+b=b+a Also useful but you could get around that if you really want to, but for our purposes let’s keep it
An additive identity (or zero): 0+a=a=a+0 You want a zero, so this is needed
Additive inverses: There exists x such that a+x=0 (here x=-a); you also want this
Associativity of multiplication: a*(bc)=(ab)*c Same as above, you want this property
Commutativity of multiplication: Useful but not necessary
A multiplicative identity (or one): 1a=a=a1 Usually with 1=/=0, also useful
Multplicative inverses for nonzero elements: Not that necessary, there are useful number systems without this (like the integers …,-1,0,1,…)
Distributivity: a(b+c)=ab+ac, (a+b)c=ac+bc You ant this, as it links addition and multiplication and this is quite desirable.
If you assume 4. and 9., you get 0a = (0+0)a=0a+0a, hence 0=0a. This means that you would have to give up distributivity wihin your number system, however distributivity is what links addition and multiplication together, hence your question would just be “what if we have two binary operations that don’t really interact with each other?” and the answer is: Maybe there are useful cases?
Edit: I forgot about losing property 4, in which case some examples are found in the following math stackexchange post