Why is it stipulated that negative results in positive operations Among the basic operational properties of multiplication and division methods, there is a "rule" that "positive and negative get negative, negative and positive get negative, positive and positive get positive, and negative get positive." The first three rules are relatively easy to explain and accept, but the fourth rule "Negative gets positive" is more difficult to understand. So why should we stipulate that "negative results in positive"?
This first requires clarifying the meaning of negative numbers. Negative numbers first appeared in the "Equation" chapter of the ancient arithmetic masterpiece "Nine Chapters Arithmetic" written in China around the 1st century AD. Here, the amount of money left is positive, and the amount of money short is negative; the number of cattle sold is positive, and the number of cattle bought is negative. The chapter "Equation" also gives the concept of absolute value and the algorithm for addition and subtraction of positive and negative numbers, which is called positive-negative surgery. India's Brahmagupta also introduced negative numbers around 628 AD, with a positive number of property owned and a negative number of debts. In short, positive and negative numbers are quantities with practical significance, while negative and positive numbers have opposite meanings. For example, the amount of money received (or increased) is positive, and the amount of money spent (or decreased) is negative. After the introduction of the number axis, negative numbers have their exact geometric meaning. Numbers to the right of the origin are positive, and numbers to the left of the origin are negative. Positive and negative numbers with the same absolute value are the inverse of each other, and are symmetrically located on both sides of the origin, and are at equal distances from the origin.
Based on this information, the rationality of "negative results in positive" can be explained from the following aspects. From the perspective of life example, if I spend 5 yuan each time and spend 4 times in total, then the amount of money will decrease by 5×4=20 (yuan), which is (-5) ×4=-20; but if I spend less twice (spend-2 times), then the amount of money will increase by 5×2=10 yuan, which is (-5) × (-2)=10. On the number axis, a positive number a multiplied by-1 will get its opposite number-a, which is (-1) ×a = -a; a negative number-b multiplied by-1 should also get its opposite number b, which is (-1) × (-b)=b. From an algebraic perspective, the distribution law has 3× (-2)+(-3) × (-2)=(3-3) × (-2)=0× (-2)=0, so-6 +(-3) × (-2)=0, and after transposition,(-3) × (-2)=6. From a logical point of view, multiplying a negative number is equal to negation; negative negative is equal to negation of negation; and negation of negation is equal to affirmation, which can also explain the truth that "negative leads to positive".