Why is less for less more? The mathematical answer to this question can be summarized with “because if it weren’t so things wouldn’t add up“. This, in fact, is a typical (for mathematics) example of a rule formulated in such a way as not to clash with other pre-existing rules. It is therefore a rule that cannot be provenhowever there is solid reasoning that leads to this conclusion. Let’s see it, accompanied by two examples to intuitively visualize the reasons for this rule of signs.
The mathematical answer to why less times less is more
THE negative numbers they were invented after the positive numbers and they arise as an extension of positive numbersan extension that allows you to do operations, such as 3 – 5which otherwise could not be done.
In mathematics one of the main things you do with numbers are operations, and when you invent new numbers you also need to figure out how to calculate the operations. In this case, since it is an extension of positive numbers, it is desirable that the results of the operations do not clash with the rules and results of the operations of the positive numbers.
Regarding in particular the “less for less is more” it’s about whether it is better that the result of –1 × –1 is –1 or 1.
Let’s try to see what would happen if the result was –1so let’s test the hypothesis –1 × –1 = –1. To do this, let’s calculate –1 × (–1+1) and let’s see if everything goes smoothly. This calculation can be done in two ways, the first is first calculating the sum in brackets and then the multiplication:
–1 × (–1+1) = –1 × 0 = 0
However, there is another way to do this calculation, as the distributive propertywhich is written
a × (b+c) = a × b + a × c
Applying this property to our operation we obtain:
–1 × (–1+1) = –1 × –1 + (–1) × 1
At this point, since we are testing the hypothesis (–1) × (–1) = –1, we can write:
–1 × –1 + (–1) × 1 = –1 + (–1) = –2
In practice the result of –1 × (1–1) And both 0 and –2at the same time, and this is very strange.
Now let’s test the hypothesis instead –1 × –1 = 1. Our calculation becomes:
–1 × (–1+1) = –1 × –1 + (–1) × 1 = 1 + (–1) = 0
which is in perfect agreement with the result obtained by first carrying out the sum in brackets.
Basically the reason why –1 × –1 = 1 and therefore “less for less is more” is that if this were not the case, strange things would happen such as the number -2 which becomes equal to the number 0…or we should give up the distributive property, but that would make mathematicians very sad.
The economic explanation of the rule of signs
Let’s imagine we have a bank account and we need to make a deposit installment of €10 every month. Since it represents an exit for us, we can indicate this figure with a negative number, –10 € (if instead we took money from the bank the installment would be positive, +€10).
If we have just paid we are even, but if with a time machine we leap forward three months we find ourselves having to pay a single installment from –30 € which takes into account the 3 installments of –10 € that we have not yet paid. Paying 3 installments of –10 € is equivalent to paying one installment of –30 €, in other words –10 € × 3 times = –30 €which is written in mathematicalese –10 × 3 = –30. Now, with the time machine, let’s go back to the present and then jump back 3 months. This time we are in credit with the bank for €30 because we have already paid the 3 installments of –€10, and since we are going back in time we can say that we have paid the installment –3 times, in other words –€10 × –3 times = 30 €which is written in mathematicalese –10 × –3 = 30.
This type of explanation, of which various versions can be found on social media, cannot be considered a demonstration of the rule, but only a way to illustrate it with an economic example.
An imaginative example: Gardner’s room of goodness
Let us now see the imaginative explanation developed by the well-known American mathematician Martin Gardner.
Let’s imagine a room full of two types of people, the good people and the bad peoplewhere the goodness of the room is given by the balance between good and bad people, so if the room contains 10 good and 7 bad people then it has a goodness equal to 3.
Good people are the positive numberswhile bad people are the negative numbers. At this point we can add people to the room by letting them in, or we can subtract people to the room by making them go outside. Adding a positive number will therefore mean letting in good people, while adding a negative number will mean letting in bad people.
Multiplications are repeated additions, so calculate –10 × 3 means to do enter 10 bad people For 3 times, but if we do go out 10 people per 3 times then we are calculating –10 × –3 and the goodness of the room will increase by 30, therefore also in this case we conclude that –10 × –3 = 30.
