Two days back I had a heated argument with my wife. The good part is, like every other time we quickly patched up things and life was back to normal. Later, we looked back and laughed as to how stupid we were just fighting on a “small thing” that didn’t even matter anymore!

Next day, when I was introspecting myself (like I usually do when I’m stuck in an hour long jam on I-880) and replaying the whole situation in my head, it struck upon me, how did that “one small thing” become such a “big thing” after all? Big enough to shake the entire relationship, even if it were for a moment? And why generally there’s always “that one thing” that unleashes the “beast” in us? When people have quarrels, why don’t they generally say “It all started because he or she did so, and so, and so.”; Or may be “It all started because he said this and this and that to me”? It’s generally always “just one stupid reason”. Why?

I’m sure many people would try to answer this from a psychological point of view. Some would perhaps throw a biological angle to it. And few would choose the path of *Mathematics*. I thought to go that direction and see if I can come up with some conclusion, *mathematically*, behind it all. This post is about my quest of understanding a part of the human psychology using *Mathematics. *Here’s my take:

**Assumptions**

- A quarrel can only begin if there are at least two people involved. This is kind of “obvious”, but since we are under the shelter of
*Mathematics*, let’s call it out loud.
- A quarrel is always initiated by one person. While many quarrels may seem as if they started with both the people taking action at the same time, if you take a careful look at it, you’ll notice that it is always initiated by one person. After that it may continue back and forth.
- Quarrel can be described as
*inverse *of Understanding. In other words, when someone quarrels with someone, his/her “understanding” about the other person depletes. This again is one of those “obvious” things, but it’s important to call it out here.

**Formulation**

To make our equations easy to discuss, let’s assume there two individuals, and . And let’s assume started the quarrel. With these assumptions, let’s define Quarrel as and understanding of a person about a person as . So our equation of quarrel between and becomes:

Now let’s dig deeper into the function because that seems to be the center theme here. signifies the “understanding” person has for person . This can be further broken down into function of traits that that shows and how much weight gives to each of those traits. Hence each “weighted trait” can be represented as:

Here is the weight person assigns to the trait of person .

Since traits can be positive or negative we can safely assume that the best way to combine traits in order to form the understanding will be by performing a root mean square of all the . This brings us to our third equation:

Combining and gives us:

In case of a quarrel, understanding becomes Zero. Let’s represent that as . Hence becomes:

Further simplification of this equation gives us:

Since will always be a positive number, the only way equation can resolve is if all the values are significantly small. In all practical terms this is going to be harder to achieve as the value of increases. This is because the higher the number becomes, the more traits are being evaluated; And the more traits are evaluated, the harder it is going to be to sum them all up to Zero! Hence, the least possible value of will give us highest probability of quarrel. That means, for highest probability of quarrel, . In other words person only considers one trait of person . Hence, we have:

Assuming traits in general don’t vary drastically, however weights may change based on the “heat of the situation”, we can combine equations and and reach to this:

In other words, equations and tell us that the possibility of a quarrel becomes extremely high when person “judges” person on only one trait that highlights during the argument and applies a very low weight on that trait. While doing so person completely disregards all other traits of person that otherwise would have created a healthy understanding between both of them, just enough to not lead to any quarrel.

**Conclusion**

When you are in a heated debate with someone, while you may get “tempted” to just pick that “one trait” that highlights at that moment, try to look at as many other traits as possible. This will help you “judge” the person in a much more “neutral” way and will possibly avoid quarrels at the first place! It’s common sense! It works! And *Mathematics* has just proved it!

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