Since school times, everyone is familiar with this rule of arithmetic: multiplication of numbers with different signs gives a negative result, and with the same — positive. And many wondered: why does a minus for a minus give a plus? In many schools, this fact was left without explanation and the students took it for granted. Today we will figure out why a minus for a minus gives a plus.
Let’s go through the entire logical chain from start to finish. In ancient times, when arithmetic was just being formed, negative numbers did not exist, and all arithmetic operations appeared to simplify with the development of trade relations. Let’s imagine this situation: you worked for 5 days, earning $ 10 a day, it is obvious that if you add up all the income you get: 10 + 10 + 10 + 10 + 10 = 50 or 10 * 5 = 50 dollars, thus , the multiplication operation appeared as an abbreviated notation for the sum.
Let’s say that later you decided to go to the store and it turned out that you collected $ 60 worth of products, you agree with the seller that you will then have to give him the missing ten and he gives you a note where your debt is indicated. In total, you have 50-60 = -10 dollars on your balance. Thus, negative numbers appeared, denoting something that we give or owe. If the same situation repeats next week, then our debt will be 2 * (- 10) = — $ 20, and from this it is already clear that multiplying a positive number by a negative one gives a minus, because several debts are actually summed up.
Let a week later, when you have worked for another five days and have -10 + 5 * 10 = 40 dollars on your account, you met this seller, and he asks you for a certain service, and instead of paying for it, he offers to forgive the debt , this situation is described by the following expression: 40-1 * (- 10) because we give (therefore minus) one piece of paper with -10 dollars. It is logical that after that you will have 50 dollars on your balance, then the result of the above expression should be 50, which means that -1 * (- 10) should equal +10. It was on such ordinary experience that people once came to the conclusion that the product of two negative numbers should be equal to positive.
Let’s take another look at this issue from the point of view of pure mathematics: when negative numbers were introduced into it, scientists wanted the same mathematical operations to be valid for them as for natural numbers. One of them is parenthesis expansion: c * (a + b) = c * a + c * b is valid for any natural numbers. Now let’s check how this works for integers (positive integers + 0 + negative). Suppose we have the expression -3 * (6 + (- 6)), expand the brackets: we get -3 * 6 + (- 3) * (- 6), we already know that multiplying negative by positive gives a minus, so it will come out -18 + (- 3) * (- 6). Now let’s calculate the value of the same expression without opening the brackets: -3 (6 + (- 6)) = — 3 (6-6), it is clear that 6-6 = 0 and then -3 * 0 = 0, which means , and -18 + (- 3) * (- 6) = 0, then the product (-3) * (- 6) must equal 18 to restore balance.
Thus, the statement that minus by minus gives plus is not just an agreement between mathematicians, but has logical grounds.
Author: Vladislav Kigim. Edited by Fedor Karasenko.
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