This article is first published on Read.cash as a response to an article called "Why "Negative x Negative" Should Not Be Taught At School Anymore" by a user of there called wrabbiter.
Wrabbiter ponders about the Mathematical rule and why it doesn't make sense. Below I'll say why I believe they are wrong:
Sometimes, we follow the rules because: "The rules say so." Sometimes nothing is wrong with that, most people don't have the time or energy to find the rule's root of existence. As long as rules don't become a hindrance to life, it's okay to follow them. Only if they become a hindrance we'll need new rules will come in their place.
One of the Mathematical rules is: "Whenever you multiply Negative by Negative, the result will be Positive."
Wrabbiter's article questions the Negative x Negative rule and provides examples to prove that the rule can't be applied in real life. Below are some quotes of the article:
"A negative times a negative is positive, it is just like that by definition, stop asking and just get over it!" All my friends tell me that. But here's the problem: I SIMPLY CAN'T GET OVER IT.
It is quite understandable that the concept of negative numbers is established because it is meant to represent debts, deficits, lack(s), or anything that someone should fill out for or replace once he is capable of doing so. I told them, I don't really think that such a question can ever be applied in reality.
If there is no real-world application to the concept of negative times negative, then it shouldn't be taught anymore, in my opinion. I'm not saying the entire subject of Algebra should be stopped though, for we can tell that it's a very important field of discipline. I'm just referring to the concept of multiplying negative numbers.
I talked with Wrabbiter, and I'm pretty sure they don't believe what they wrote. I'm posting this because I believe many people think that way & have the same question. Most of them are afraid of asking because everyone else is following the rules because "the rules say so!"
So my comments on 's article were challenging their idea.
I loved the passion present in Wrabbiter's article. I could follow their logic. I believe that the logic itself is flawed though. It's based on some misconceptions on Math/Algebra.
First Problem: Applying Math Where It Shouldn't Be Applied
Negative x Negative as mathematical concept, should only be applied to math, not another field!
You can't say that Negative represents the color black if positive white, because a negative number can only represent a negative number. The whole thing becomes wrong if you applied this to anything else.
Second Problem: Wrong Examples Lead to Wrong Conclusions
The examples are provided without taking in mind the fact that Mathematical problems are abstractions of real life problems. You have to abstract these problems with certain rules for them to work with Math!
Let's see the first example:
Juan has a debt of three pesos. ( -3 )
Later, he doubled it ( -3) * (-2 )
According to Algebra, this would result into positive 6. ( -3 )* ( -2) = 6
Now let's ask: how can that be?
If we are going to bring these situations into reality, it should actually make Juan a debtor of six pesos. (-6) But instead, Juan would actually have his debt vanished.
In this example Juan's debt (-3) is doubled by multiplying it by (-2.) I can see that, of course when you double something you want two of it... Except that we're talking mathematics here. We don't take the sign (negative/positive) in consideration, because Double means Multiplication by 2. (Only 2... You can't ever replace it with -2.)
So, Juan debt is doubled: (-3) * (2) which equals (-6.)
Another thing you should keep in mind, and I think the reader already noticed it:
When we convert the debt problem in mathematics, we don't say Juan has (-6) in debt because it means he doesn't have any debt. We don't say he has (-6) in money because negative money doesn't necessarily mean debt. We say he has (6) debt. The value of debt should be a positive number, the human mind understands when that means a negative value in money.
Mathematics is an abstraction. Numbers don't human mind, so we must always abstract Real Life problems before solving them with numbers.
Let's go over another example:
Let's take a look at it from a different perspective. Let's say negative is equal to a "lie" while positive is equal to a "truth."
We would then have a statement that goes like this: "3 lies", multiplied by "2 lies", is equal to "6 truths" -- Really? How? Why?
For this one I'll just paraphrase my own comment:
A Lie is not a negative truth, they are different things. Sometimes a Lie can be a negative Truth in abstraction, but that depends on the Lie itself... It's like you're saying "X = - Y" but that only works as long as the X you are referring to, is the one that equals "-Y."
As long your abstraction works, you can continue using it, but the moment it stops working you have to replace it with another abstraction.
Also, in your "3 Lie x 2 Lie" abstraction: what does the multiplication stand for? You can arguably say that Lies can be added on top of each other, but how does one multiply lies? If that's possible, could it work with Mathematical multiplication or do we need to abstract that to something else?
Third Problem: We Don't Follow Rules Because They Say So, But Because They Never Failed Us
We don't change the mathematical rules because they are proven again and again to work with Real Life Physics.
Using this "Negative x Negative" rule correctly, we solved many equations that apply to real life like Velocity, Force, Electricity... Almost everything that can be turned into Mathematical equation won't work without this "Negative x Negative" rule.
Finally: Keep in Mind:
If some mathematician said "because the rules said so" he either doesn't know or doesn't want to explain the science behind it because it is high level. Normal students won't understand it anyway. (Not that I know it myself.)
We still use these rules because they work for REAL LIFE equations and solved many problems in the last few centuries.
The Mathematical rules and numerical systems we use are NOT the only ones in existence. If you want to spend your life making a new system that works better you might succeed, but that's unlikely and you will be just reinventing the wheel. In the end you're just making an alternative system to something that already works and is tested with real life physics.
There are things our mathematics can't solve yet, but we're solving new equations almost ever day using these same rules.
A new system might take more than your life-time to create. If you want to do it, I hope you succeed. Most people will think that's unnecessary effort though.
Finally, my note to @wrabbiter :
I really appreciate that you've written this post though, it means you're thinking for yourself and who knows, you might be right and all of us are wrong, but you can only prove that by providing a better system, and that's something most people won't want to try finding.
If you want to abstract a real life concept into its Mathematical variation, you have to create a equations that works first. That's why all Math concepts start as assumptions and/or theories. They weren't proven to work yet.
I don't think anyone should be blamed for not understanding Math. Especially when most Teachers don't have the capability to explain the reasoning behind Mathematics rules. At least try to understand the reason why such rules exist before blindly following them or dismissing the entire thing.
What do you think?
I hope this article was useful for you. If I made any mistake or if you have something to add, please tell me in comments!