It is impossible to convince someone of something if they are ignorant of the base fundamentals regarding what you're talking about.
This is true. It is also going to be impossible to convince someone of something if they don't agree with your interpretation of the base fundamentals regarding what you are talking about.
Knowing all the facts, pro gun folks will argue that guns are responsible for saving many more lives than taking of lives and that for any given situation, a gun in the hands of a good guy are more likely to bring positive closure to the situation more often than not. The risk of injury or death to non-combatants or unintended actors (the good guy or people the good guy is attempting to protect) is miniscule.
Anti gun folks can see the same information and rationalize that anytime a person is wrongfully/unintentionally shot with a gun (as by the good guy shooting himself, bystander, or other unintended actor) that such actions could have been prevented. What the antis have argued indirectly is that we can't control the actions of law breakers who take lives, but we can control the actions of law abiders and as such we can keep law abiders from having guns and this will make many of the unintended shootings by the law abiders go away as the law abiders won't have any guns.
Pro gun folks will argue that the numbers are on the side. This is an attrition concept where it is believed that guns are valid for use because in the final analysis, more bad guys than good guys are going to be killed by the good guys and hence the gun is a rightful tool. I personally like the attritional concept and believe the odds are in my favor over that of the bad guy. Anti gun folks do not think such reasoning validates the use of guns.
Both sides can be completely aware of the raw data and of the fundatmentals of the argument and still reach differing conclusions on what is right or wrong. The classic interpretational example is the glass of water with equal parts of water and air. One side claims the glass is half empty. The other side, seeing the same glass, claims the glass is half full. Half empty is the negative connotation and half full is the positive connotation. Both interpretations are correct and shot an understand of the water available but also shows mutually exclusive positions on the status of the water. This is a simplified example, but sufficient for the point that both sides can arrive at different and seemingly valid interpretations based on full knowledge of the data under consideration.
You really have to understand that there comes a point in discussions where you have to agree to disagree and leave it alone or you will likely come out of the deal looking like a reactionary hothead. When you reach that point, you have come to a culmination of disparity between the pro gun belief system and the anti gun belief system. You might as well be arguing that your god is better than their god as both sides of the argument can see the same information and arrive at differnet conclusions based on their own beliefs.
Getting mad or upset with your opposition does ZILCH to help convince and promote your position. Believing that your opposition is ignornat or unintelligent because they don't agree with you or behave as you belive is best indicates that you have established and US and THEM categorization and anyone who isn't US is inferior because they are ignorant.