This book records the development and changes of elementary mathematics in the world. It can be roughly divided into seven items: the appearance of numbers, the origin and development of numbers and symbols, fractions, algebra and equations, geometry, number theory and description of names, which span thousands of years. It can let readers know the glorious history and development of mathematics. This is an interesting encyclopedia reading, which combines history and mathematics.

The emergence of numbers

First of all, the concept of number appeared.

People are born with the concept of "number". Since primitive people, people can distinguish one, two and three, so they have an understanding of logarithm. In order to represent numbers, primitive people created and used an ancient but clumsy and impractical method-the knot number method. The number of objects is represented by tying a knot on the rope. In order to identify the number, an important counting method appears. This method seems clumsy now, but it is a key step for people to understand mathematics from zero to one. From this clumsy step, people also realize that the explanation of mathematics must be as concise and clear as possible. This is the first understanding of mathematics that has influenced human beings since then, and it is also a key step for human beings to understand mathematics.

The Origin and Development of Numbers and Symbols

First of all, the emergence of numbers

Soon, mankind took another big step. With the appearance of words, the most primitive numbers appeared. What is more gratifying is that people integrate their knowledge into the design. They thought of the method of "taking one generation as the smallest", which is the "carry system" in character representation. Among the numerous numbers, there are binary numbers of ancient Babylon and ancient Roman characters, but the Arabic numerals that have been passed down to this day are universal. They told us that simplicity is the best.

Now there are low-order decimal numbers such as "binary number" and "ternary number". Sometimes people think it is too concise, which makes the data too long and inconvenient to write, and the conversion of decimal Arabic numerals is also very troublesome. In fact, people are higher animals and have a strong understanding ability. Since ancient times, ten has been taken as a whole, so it is customary to use decimals. However, not everything has IQ, and it is impossible to clearly distinguish 1- 10, but two numbers can be expressed in obviously opposite ways. As a result, human beings have created "binary numbers", but they are not convenient to write, and are only suitable for computers and some intelligent machines. But it is undeniable that it has created a new form of digital expression.

Second, the emergence of symbols.

Mathematical symbols such as addition, subtraction, multiplication and division are the most familiar symbols for each of us, because we can't do without them not only in mathematics learning, but also in almost every day's daily life. Don't treat them so simply.

Single, it was not until the middle of17th century that it was fully formed.

French mathematician Soso used some symbols in his three arithmetic papers written in 1484, such as D for addition and M for subtraction. These two symbols first appeared in the commercial speed algorithm written by German mathematician Weidemann. He used "+"for excess and "-"for deficiency.

1, plus sign (+) and minus sign (-)

Addition and subtraction symbols "+","-",1489 German mathematician Weidemann first used these two symbols in his works, but they were officially recognized by everyone from the Dutch mathematician Heuck in 15 14. At 15 14, Heck of the Netherlands used "+"for addition and "-"for subtraction for the first time. 1544, German mathematician Steefel formally used "+"and "-"to express addition and subtraction in integer arithmetic, and these two symbols were gradually recognized as real arithmetic symbols and widely used.

2. Multiplication symbol (×, ...)

163 1 year, the British mathematician orcutt proposed the multiplication with "x". British mathematician Oughtred introduced this symbol in Key to Mathematics published in 163 1. The reason why it is derived from the addition symbol+is because the multiplication operation is developed from the addition operation of the same number. Another multiplication symbol, heriott, was invented by mathematicians. Later, Leibniz thought that "×" was easily confused with "x" and suggested that "×" be used to represent the multiplication sign, so "×" was also recognized.

3. Division (÷)

Division and division symbol "∫" was first popular in continental Europe as a negative sign, and Orkut used ":"to indicate division or ratio. Some people also used fractional lines to express proportions, and later some people combined them into "∫". In the works of Swiss mathematician Laha, "6" was officially used as a division symbol. The symbol "⊙" was first used in Varis, England, and was later popularized in England. In addition to the original meaning of fen, the horizontal line in the middle of the symbol "⊙" separates the upper and lower parts, which vividly represents fen.

At this point, the four major combat symbols have been completed, which is far from being widely adopted by various countries.

4. Equal sign (=)

The equal sign "=" was first used in 1540 by Professor Richter of Oxford University. 159 1 year, the French mathematician Veda was widely used in his works and gradually accepted by people.

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First, the generation and definition of scores

The earliest number in human history is a natural number (positive integer). When measuring the average in the future, it is often impossible to get accurate integer results, which leads to scores.

An object, a figure and a unit of measurement can all be regarded as the unit "1". Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction. In the fraction, the denominator indicates how many shares the unit "1" is divided into, and the numerator indicates how many shares there are; One of them is called fractional unit.

The numerator and denominator are multiplied or divided by the same number (except 0) at the same time, and the size of the fraction remains the same. This is the basic nature of fractions.

Scores generally include: true scores, false scores, and scores.

The true score is less than 1.

False score is greater than 1 or equal to 1.

Band score is greater than 1, which is the simplest score. A fraction consists of an integer and a real fraction.

note:

① There cannot be 0 in denominator and numerator, otherwise it is meaningless.

(2) The numerator or denominator in a fraction cannot have irrational numbers (such as the square root of 2), otherwise it is not a fraction.

③ Only two prime factors (2 and 5) in the denominator of the simplest fraction can be converted into finite decimals; If the denominator of simplest fraction only contains prime factors other than 2 and 5, it can become a pure cyclic decimal; If the denominator of simplest fraction contains both prime factors of 2 or 5 and prime factors other than 2 and 5, it can be converted into mixed cyclic decimals. (Note: If it is not simplest fraction, it must be transformed into simplest fraction to judge; The simplest fraction with denominator of 2 or 5 can be converted into finite decimal, and the simplest fraction with denominator of other prime numbers can be converted into pure cyclic decimal.

Second, the history and evolution of scores

Fractions have a long history in China, and the original forms of fractions are different from the present ones. Later, India appeared a score representative similar to China's. Later, the Arabs invented the fractional line, and the expression of the score became like this.

In history, fractions are almost as old as natural numbers. As early as the early days of the invention of human culture, scores were introduced and used because of the need of measurement and average score.

There are records of scores and various scoring systems in ancient documents of many nationalities. As early as 2 100 BC, the ancient Babylonians (present-day Iraq) used fractions with a denominator of 60.

Fractions were also used in Egyptian mathematical literature around 1850 BC.

More than 200 years ago, the Swiss mathematician Euler said in his book General Arithmetic that it is impossible to divide a 7-meter-long rope into three equal parts because there is no suitable number to represent it. If we divide it into three equal parts, each part is 3/7 meters. Like 3/7 is a new number, which we call a fraction.

Why is it called a score? The name "fraction" visually represents the characteristics of this number. For example, if a watermelon is shared equally by four people, shouldn't it be divided into four equal parts? From this example, we can see that fraction is the need of measurement and the need of mathematics itself-the need of division operation.

China was the first country to use scores. In the Spring and Autumn Period (770-476 BC), Zuo Zhuan stipulated that the size of the vassal's capital should not exceed one third, one fifth and one ninth of that of Zhou Wenwang. The calendar of Qin Shihuang's time stipulated that the number of days in a year was 365 and a quarter days. This shows that scores appeared very early in China and were used in social production and life.

Nine Chapters Arithmetic is a mathematical monograph written by China 1800 years ago. The first chapter, Square Domain, talks about four algorithms of fractions.

In ancient times, China used scores 1000 years earlier than other countries. So China has a long history and splendid culture.