In elementary school, students learned natural numbers, zero-sum fractions, and now they learn negative numbers. These numbers are collectively called rational numbers. However, have you ever thought about how rational numbers are produced?
A long time ago, human ancestors lived in forests and caves, covered with skins and leaves, ate wild animals on mountains, wild fruits on trees and fish in water, and lived by hunting all the year round. At that time, although not much food was hunted every day, there was still a problem of counting. At first, people only distinguished between "more" and "less". Gradually, some people think that you can count with your fingers, because at that time, the results of hunting every day were only "a handful". Later, the hunting tools were improved and the level was improved. When there are more than ten prey, the "fingers" can't be counted, so I thought of tying a knot on the rope to count. Zhou dynasty (about 10 century BC), I ching? 6? 1 The "ancient ruling" recorded in the copula refers to that ancient time. I don't know how many years have passed, and people gradually feel that knotting is not only troublesome, but also often can't remember what these knots mean after a long time. Finally, they thought of using some symbols to represent different things and the number of things, and the earliest numbers appeared. For example, there are symbols such as ||||||× or X ∧ or a+8 +| in the pottery of Xi 'an Banpo site in 3000 BC and Erlitou site in nearly 2000 BC respectively.
1 2 3 4 5 6 7 8 70。
There are also many figures in Oracle Bone Inscriptions of Yin Ruins (see the article "The Best in China's Mathematical World"). Abroad, around the eighth century AD, an Indian number was introduced into Arabia. They are:
〡 ∧ ∨ 10。 And so on, representing L: 2, Gong 4, 5, 5, 7: 8, 9, 10 respectively. This number was later spread to Europe by Arabs and called Arabic Latin by Europeans. These digital symbols are constantly improved in the process of use, and finally evolved into the numbers we use now.
The appearance of numbers has brought great convenience to people's production and life. But how to represent so many numbers with as few numbers as possible? After China people first invented decimal notation, this problem was finally solved satisfactorily.
Hunting Sometimes two people cooperate to catch a rabbit, and sometimes five people cooperate to catch two sheep. How to distribute these foods? At first, people only knew "one China, two" and "five China, two"; Later, the concept of fractions was gradually formed, and it was recorded as "half", "two-fifths" and ..., which was also initiated by China people. Fractions have been widely used in Zhou Zhi than Jing, and Nine Chapters Arithmetic (about 100 ~ 50 BC) gives a fairly complete theory of fractions, which is about 1400 years earlier than similar works in Europe. The fractional division we are talking about now is "multiplying the divisor backwards", which is the original words of China ancient educator Liu Hui (3rd century BC).
Man's understanding of zero is relatively late. If you can't hit the beast, you will return empty-handed. This is the first impression of "zero"-emptiness, hunger and nothing. When recording this situation, most ethnic groups invariably use vacancy to express it. Later, the symbol "□" was used to indicate the vacancy (some people speculated that it was an empty livestock pen), and it gradually evolved into the current "0".
As the great teacher Engels summed up brilliantly, "the concepts of numbers and shapes come from the real world, not from anywhere else."
In primary school teaching, the formula "2-3" gives us the impression that reduction is not enough. But after learning the knowledge of rational education, we can solve this problem. Rational numbers include positive numbers, negative numbers and 0. The concept of positive and negative effects also comes from the actual needs of production. With the development of production, on the one hand, people's "wealth" has increased, and at the same time, people are encouraged to exchange needed goods. Therefore, people record private property as positive and debt as negative; Income is recorded as positive and expenditure as negative; Incoming records are positive and outgoing records are negative; People abstract the concepts of positive and negative numbers from these quantities with opposite meanings. Negative numbers are relative to positive numbers. Positive and negative numbers are both opposite and interdependent. Our ancestors not only recognized the existence of negative numbers at the earliest, but also summed up the addition and subtraction algorithms of positive and negative numbers (such as "Nine Chapters Arithmetic"), which was also a great creation with world significance at that time.
Due to the needs of production practice and the development of science and technology, the concept of number is expanding. At present, the relationship between the concepts of numbers that human beings have mastered can be summarized as follows:
Integer: positive number
Rational number: 0
Score: negative number