How Can x*0=0 Be Proven Without Using m(-1)=-m?

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The discussion centers on proving that x*0=0 for any integer x without relying on the proposition m(-1)=-m. The user initially struggled with the proof but ultimately recognized that multiplication can be defined as repeated addition. By applying this definition, they demonstrated that x*0 equals the sum of zero added x times, which results in 0. This approach effectively establishes the validity of the equation without circular reasoning.

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I have to prove that x*0= 0 where x is any integer. I can do this pretty easily using the proposition that m(-1)=-m but I am not allowed to use this. In addition, I am unable to prove m(-1)=-m unless I accept the fact that anything times zero equals zero. Can anyone give me a hint or push on how to show this? I am just going around in circles.
 
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0.m=(0+0).m =0.m+0.m

subtract m.0 from both sides.
 
Thanks, I am an idiot. I kept trying to represent 0 as (m + -m)
 
When x is an integer, multiplication can be defined as:

x*a = \sum^x_{n=1} a

When a = 0:

x*0 = \sum^x_{n=1} 0 = \underbrace{0 + 0 + ... + 0}_x = 0
 
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