The fact that minus times minus equals plus, i.e. [math(-1)*(-1)=1[/math, has no profound explanation, but is simply a DEFINITION.
The question now, of course, is why this is defined in this way, i.e. the real question is: Why DO you DEFINE that minus times minus equals plus?
And the answer is: This makes some things easier in mathematics:
Example 1: General Distributive Law
Only with the definition that minus times minus equals plus, the general distributive law applies:
If [matha[/math and [mathb[/math are positive real numbers, [math(a+(-a))*(-b)=0[/math.If you want to apply the Distributive Act here, you have to deal with how to define the multiplication of two negative numbers in a meaningful way. [math(-a)*(-b)[/math should then be defined as [matha*(-b)+(-a)*(-b)=0[/math and from this you can immediately see that one should define [math(-a)*(b)[/math as [matha*b[/math 🙂
Why do we need the general distributive law?
For example, we can now easily calculate that [math(x-1)*(x-2)[/math is the same function as [mathx-2-3x+2[/math.
This also has advantages for people who are not interested in multiplying two negative numbers, because: For [mathx>=1[/math, there are no multiplications of two negative numbers in the above expressions.These people could then at least easily show that both functions are the same for [mathx>=1[/math.However, they would also think that the first function is defined only for [mathx>=1[/math and the second function for [mathx>=0[/math.
Example 2 (requires knowledge about the polynomial division, students may have to settle for Example 1 ;-)): Help in resolving equations
Take, for example, the equation [mathx-2-9x-10[/math.Because we have defined how to multiply two negative numbers together, namely according to the rule “minus times minus equals plus”, suddenly [math-1[/math is a zero digit of the above polynomial and with the help of the polynomial division we can now use the term [math(x+1)[/math “split” and get: [mathx-2-9x-10=(x+1)*(x-10)[/math , and behold, we now also know that [math10[/math is another zero point.
The fact that [math10[/math is a zero digit is now also of interest to those people who are not interested in multiplying two negative numbers, because if you use the [math10[/math above, there is no way to multiply two negative figures.
By the way, the polynomial division is often based on the fact that two negative numbers can be multiplied by each other, because otherwise, for example, the two functions occurring above would be [mathx-2-9x-10[/math and [math(x+1)*(x-10)[/math for example in the case [mathx=-1[/math for the second function, but not for the first function, i.e. one could not use the comparison of the two functions for all [mathx[/math.For values of [mathx[/math for which the multiplication of two negative numbers occurs in both functions, this might not be too bad, but if at least one of these does not occur, you have a problem, even if, as I said, you do not how to multiply two negative numbers.
How did you come up with the definition of “minus minus equals plus”?
It is not known, as far as I know, who first gave this definition, but it may well be that several have considered this at about the same time or time, without knowing that others have already come to it.
In mathematics, one has often learned that it is useful to generalize things as much as possible, even if one does not know what it is good for, i.e. a motive might have been to think out of curiosity how to two negative numbers together (associated with the hope that this could still be useful in some way).
Another reason might have been that the definition simply imposed itself, e.g. by calculating with variables:
See.Example 1: [math(x-1)*(x-2)[/math, if you do not define how to multiply two negative numbers together, is defined for [mathx>=1[/math), while [mathx-2-3x+2[/math is defined for [mathx>=0[/math.
For [mathx>=1[/math, both expressions are identical, so you might think that you could define [math(x-1)*(x-2)[/math generally for [mathx>=0[/math).If you use [mathx=0[/math naively, for example, you get [math(-1)*(-2)[/math, which could lead you to consider whether you could define the multiplication of two negative numbers in general.
However, what now appears to be a simple idea was far from simple and it took a long time for someone to come up with this idea.
Even the introduction of variables, negative numbers, etc.are achievements that no one has ever come to, but which we now take for granted.
In mathematics, by the way, it is almost always the case that one could generally dispense with new terms and express everything with the old terms, e.g. one could do without the introduction of negative numbers and express everything with the natural numbers instead of saying: You own [math-1000鈧琜/math you could just say that you have [math1000鈧琜/math debt!At some point, however, it would become more and more complicated to express everything with the old terms, and new terms are becoming more and more enforced. The absence of new terms can even hinder the progress of a field of science, as it becomes very difficult to get into the area and it is becoming increasingly cumbersome to operate with the old terms.