The German mathematician Kronecker has a famous saying: "God created the natural numbers, and the rest are man-made." Human beings have tied knots to count since the primitive times. With the need to distribute prey, the addition, subtraction, multiplication and division of numbers also began to be used "naturally", so there were natural numbers. Later, negative numbers appeared due to the need for subtraction, fractions were introduced due to the closed nature of division, and real numbers and complex numbers appeared due to the unimpeded access of square roots. These are no longer "natural".
The first "artificial" number is a positive fraction. Logically, there should be negative integers first, then fractions, but the historical order is exactly the opposite. Negative numbers first appeared in China's "Nine Chapters of Arithmetic" (written about the 1st century BC), while historically recorded fractions appeared in ancient Egyptian papyri, about 4,000 years ago. "Nine Chapters on Arithmetic" also describes complete fraction knowledge. The Chinese mathematical terms "one-third" and "fraction" are indeed precise and expressive, and are much easier to understand than the English "one-third (one and third)". In many countries and regions in East Asia that use Chinese characters, students' scores in learning fractions are generally better than those in European and American countries, and it is said to be related to this.
To this day, fraction knowledge is one of the components of ordinary people’s mathematical literacy. Students all over the world, without exception, need to learn fractions. In mathematics courses in European and American countries, fractions are mostly taught in middle schools (sixth to seventh grade). In my country, fractions are taught earlier. In the 1960s, fractions were taught in the fifth grade, but now they start learning in the third or fourth grade. .
1. Fractions are our “new friends” after we get to know natural numbers.
Fraction teaching in various countries mostly starts with "cutting the pie" or "dividing the cake". For example, cut a round flatbread into four equal pieces, each piece being 1/4 of the entire flatbread, pronounced as a quarter. Generally, a unit is divided equally into several parts, and the number representing such one or several parts is called a fraction. This kind of fraction defined by "parts" is easy to understand and easy to learn. However, it can be used as an entry point for teaching, but its connotation is very limited, especially it cannot form a fixed mindset.
The real source of fractions lies in the promotion of division of natural numbers. A pie is divided equally among four people, resulting in a pie of a certain size. For this objectively existing quantity, according to the meaning of division, it should be the quotient obtained by 1÷4. However, this kind of division in which the divisor is greater than the dividend cannot be divided before, so there is no "quotient". Then, the opportunity for “innovation” came. We regard the natural numbers we already know as old friends, and the quotient of 1÷4 as a new friend, and its name is one quarter. After getting to know such a "new friend", division between any two natural numbers can be performed. So there is this definition: A fraction is the quotient of dividing two natural numbers a and b (b≠0). The quotient of a÷b is the new number a/b , read b divided by a. When b=1, the fraction is a natural number.
In short, by "number of copies"