Understanding of numbers

A number used to measure the number of things or to indicate the order of things. That is, the numbers represented by the numbers 0, 1, 2, 3, 4, ... natural numbers start from 0, one by one, forming an infinite set. There are addition and multiplication operations in the set of natural numbers. The result of addition or multiplication of two natural numbers is still a natural number, and subtraction or division can also be done, but the result of subtraction and division is not necessarily a natural number, so subtraction and division operations are not always effective in the set of natural numbers. Natural numbers are the most basic of all numbers that people know. In order to make the number system have a strict logical basis, mathematicians in the19th century established two equivalent theories of natural numbers, namely ordinal number theory and cardinal number theory, which made the concept, operation and related properties of natural numbers strictly discussed.

Ordinal theory was put forward by Italian mathematician G. piano. He summed up the nature of natural numbers and gave the following definition of natural numbers by axiomatic method.

The set n of natural numbers refers to a set that meets the following conditions: ① There is an element in n, which is recorded as 1. ② Every element in n can find an element in n as its successor. ③ 1 is not the successor of any element. ④ Different elements have different successors. ⑤ (inductive axiom) Any subset m of n, if 1∈M, and as long as X is in M, it can be deduced that the successor of X is also in M, then M = N. ..

Cardinality theory defines natural numbers as the cardinality of finite sets. This theory puts forward that two finite sets that can establish one-to-one correspondence between elements have the same quantitative characteristics, which are called cardinality. In this way, all the single element sets {x}, {y}, {a}, {b} and so on have the same cardinality, which is recorded as 1. Similarly, whenever two fingers can establish a one-to-one set, their cardinality is the same, recorded as 2, and so on. The addition and multiplication of natural numbers can be defined by ordinal number or cardinal number theory, and the operations under the two theories are consistent.

Natural numbers play a great role in daily life, and people use them widely.

Whether "0" is included in natural numbers is controversial. Some people think that natural numbers are positive integers, that is, counting from 1; Others think that natural numbers are non-negative integers, that is, counting from 0. There is no consensus on this issue at present. However, in number theory, the former is often used; In set theory, the latter is often used. At present, our primary and secondary school textbooks classify 0 as a natural number! See/,www. 1088.com.cn for details. Natural numbers are integers, but not all integers are natural numbers.

integer

Sequence …, -2,-1, 0, 1, 2, …

The numbers in are called integers. All the integers make up an integer set, which is a ring, marked Z (usually written as a hollow letter Z in modern times). The potential of ring z is Alef 0.

In an integer system, natural numbers are positive integers, 0 is zero,-1, -2, -3, …, -n, … are negative integers. Positive integers, zero integers and negative integers form an integer system.

Positive integers have been a tool for human counting since ancient times. It can be said that the process of abstracting from "one cow and two cows" or "five people and six people" into positive integers is quite natural. In fact, we sometimes call positive integers natural numbers.

Zero not only means "none", but also means a vacant symbol. In ancient China, a computer chip was used to calculate numbers. When performing operations, no space was placed in the computer chip. Although there is no vacancy sign, it still creates good conditions for digital counting and four operations. The word "zero" in Indian Arabic numerology comes from India, and its original meaning is also "empty" or "blank".

China first introduced negative numbers. "Positive Numbers and Negative Numbers" discussed in Nine Chapters of Arithmetic. Equation "is the addition and subtraction of integers." The need for subtraction also promotes the introduction of negative integers. The subtraction operation can be regarded as solving the equation A+X = B. If both A a+x=b are natural numbers, then the given equation may not have a natural number solution. In order to make it always have a solution, it is necessary to expand the natural number system into an integer system.

Positive integers, zero and negative integers are collectively referred to as integers. Integer is the most basic mathematical tool that human beings can master. /kloc-Kroneck, a great German mathematician in the 0 th and 9 th centuries, therefore said, "Only integers are created by God, and others are created by human beings themselves."

The given integer n can be negative (n∈Z-), nonnegative (n∈Z*), zero (n=0) or positive (n ∈ z+).

See: algebraic integer, complex number, countable, natural number set n, natural number, negative number, positive number, real number), z, Z-, Z+, Z*+, Z*, zero.

Decimal system:

When measuring objects, we often get numbers that are not integers, so the ancients invented decimals to supplement integers. Decimal is a special form of decimal. All fractions can be expressed as decimals, except infinite acyclic decimals, all decimals can express the number of components. Irrational numbers are infinitely cyclic decimals.

According to the decimal bit value principle, the decimal part is written in the form without denominator, which is called decimal. The point in the decimal is called the decimal point, which is the dividing line between the integer part and the decimal part of a decimal. The part to the left of the decimal point is an integer part, and the part to the right of the decimal point is a decimal part. Decimals with zero integer parts are called pure decimals, and decimals with non-zero integer parts are called decimals. For example, 0.3 is pure decimal, 3. 1.

Like integers, the counting units of decimals are arranged in a certain order, and their positions are called decimals.

Numbers. The numerical sequence is as follows:

There are two ways to read decimals: one is to read fractions, and the part with decimals is to read integers; The fractional part is read by the fraction. For example, 0.38 is pronounced as 38%, and 14.56 is pronounced as 14 and 56%. In another way, the integer part is still read as an integer, the decimal point is read as a "dot", and the decimal part reads the numbers on each digit in sequence. For example, 0.45 is read as 0.45; 56.032 is pronounced as 56.032.

The comparison method of decimal size is basically the same as that of integer, that is, the numbers on the same digit are compared in turn from the high position.

Therefore, to compare the sizes of two decimals, first look at their integer parts, and the larger the integer part, the larger the number; If the integer parts are the same, the one with the largest number in the tenth place is larger; If the deciles are the same, the percentile is larger;

Because decimals are decimal fractions, they have the following properties: ① Add or remove zero at the end of decimals, and the size of decimals.

No change. For example; 2.4 = 2.400, 0.060 = 0.06.② When the decimal point is shifted to the right by one, two and three places respectively, the decimal size will change, and the decimal value will be enlarged by 10 times, 1000 times and1000 times respectively. ...

Times; If the decimal point is shifted to the left by one place, two places and three places respectively, the decimal value will be reduced by 10 times, 100 times and 1000 times respectively. For example, expand 7.4 10 times to be 74, and expand 100 times to be 740. ..

Infinitely circulating decimals can only be expressed by decimals, not fractions. All finite decimals and infinite cyclic decimals can be expressed by fractions. Decimals can be divided into finite decimals and infinite decimals, such as 1/5. Infinite decimals include infinite acyclic decimals (such as 0.0 100 1 ...) and infinite cyclic decimals (such as 65438).

Rational number: a number that can be accurately expressed as the ratio of two integers.

For example, 3, -98. 1 1, 5.72 ... and 7/22 are rational numbers.

Integers and fractions are rational numbers. Rational numbers can also be divided into positive rational numbers, 0 and negative rational numbers.

In the decimal representation system of numbers, rational numbers can be expressed as finite fractions or infinite cyclic fractions. This definition also applies to other decimals (such as binary). Encyclopedia of China (Mathematics)

Therefore, there is no contradiction.

Add "0" or remove "0" at the end of the decimal, and the size of the decimal remains the same. This is the so-called nature of the decimal.

Decimal times integer:

Convert decimal multiplication into integer multiplication calculation.

First, the decimal is expanded into an integer, and the product will be reduced by multiplying the expansion factor by the integer.

The decimal places of the product are related to the decimal places of the multiplicand. If the multiplicand has several decimal places, so does the product. Because to convert decimal multiplication into integer multiplication, the product will be amplified as many times as the multiplicand. So how many times must the product be reduced?

Calculate decimal multiplication with integers. First, calculate the product according to the calculation method of integer multiplication, and then see how many decimal places the multiplicand has. Count a few from the right of the product and point to the decimal point.

A decimal, starting from somewhere in the decimal part, and one or several numbers are repeated in turn. This decimal is called a cyclic decimal.

Circular part: the decimal part of a circular decimal, which is a number that appears repeatedly in turn.

It is called the cyclic part of this cyclic decimal. For example: 0.33 ... The loop part is "3"

2. 14242 ... The cyclic part is "42"

Pure cyclic decimal: the cyclic part starts from the first position of the decimal part.

Mixed cycle decimal: the cycle part does not start from the first position of the decimal part. (For example:

Blackboard book)

Simple notation: When writing cyclic decimals, for simplicity, only the cyclic part of decimals is written.

The first cycle part. If there is only one number in the loop, add a dot to this number; If there are multiple numbers in the loop part, please add a dot to the first and last numbers in this loop part.

Score:

Fractional unit

Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction. The number representing this share is called the fractional unit.

definition

Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction. The denominator is to divide an object into several parts, and the numerator is to take several parts.

1 → molecule

-→ Fraction line

2 → denominator

The horizontal line in the middle of the score is called fractional line, the number above the fractional line is called numerator, and the number below the fractional line is called denominator.

origin

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

produce

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.

classify

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.

pay attention to

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

② Irrational numbers (such as the square root of 2) cannot appear in the score, otherwise it is not a score.

produce

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.

(3) To judge whether a fraction can be a finite decimal: First, we should look at whether it is a simplest fraction. Second, if the denominator is a multiple of 2 or 5 (excluding any other numbers), it can become a finite decimal.

history

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.

In the Spring and Autumn Period (770 BC-476 BC), Zuo Zhuan stipulated that the capital scale of vassals should not exceed one third of the capital of Zhou Wenwang, one fifth of the medium capital and one ninth of the small capital. 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.

Percentage:

A number indicating that one number is a percentage of another. Percentages are also called percentages or percentages. Percentages are usually not written in the form of fractions, but are represented by the symbol "%"(called percent sign). For example, write 4 1% and 1% respectively. Since the denominator of percentages is 100, they are all 65438+.

The formation of the concept of percentage should be based on students' real life or examples in industrial and agricultural production. For example, there are 100 students in senior one, among whom 47 are girls, accounting for 47% of the whole grade and 47% of the writing. For example, there are 200 senior two students, including girls 100, accounting for 50% of the whole grade. The number of students in two grades is "standard quantity", while the number of girls is "comparative quantity". In the teaching of percentage application problems, we should grasp the quantitative relationship = percentage (percentage) for analysis.

There are three calculation problems in the application of percentage: ① Find the percentage of one number to another, for example, find the percentage of 45 to 225, that is, = 20%; ② Find the percentage of a number; For example, find 75% of 2.2, that is, 2.2× 75% =1.65; 3 know one.

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

The difference between fraction and percentage: a number indicating that one number is a percentage of another. Percentages are also called percentages or percentages. Percentages are usually not written in the form of fractions, but are represented by the symbol "%"(called percent sign).

Relationship between fraction and percentage: both percentage and fraction are to find the fraction or percentage of one number or another, that is, to find the multiple relationship.