整数与小数的思维导图

# 《整数与小数的思维导图》

## 一、整数 (Integers)

### 1.1 定义 (Definition)
*   **概念**: 没有小数部分，是正数、负数和零的统称。 (Whole numbers, including positive, negative, and zero.)
*   **性质**: 具有离散性，每一个整数都有确定的位置。 (Discrete nature; each integer occupies a specific position.)

### 1.2 分类 (Classification)

*   **正整数 (Positive Integers)**: 大于零的整数。 (Integers greater than zero.)
    *   **例子**: 1, 2, 3, 4, ...
*   **零 (Zero)**: 既不是正数也不是负数。 (Neither positive nor negative.)
    *   **性质**: 是偶数，是自然数。 (Even number, natural number.)
*   **负整数 (Negative Integers)**: 小于零的整数。 (Integers less than zero.)
    *   **例子**: -1, -2, -3, -4, ...

### 1.3 运算 (Operations)

*   **加法 (Addition)**:  相同符号相加，绝对值相加，符号不变；不同符号相加，绝对值大的减去绝对值小的，符号与绝对值大的相同。 (Same sign: add absolute values, keep sign. Different signs: subtract smaller absolute value from larger, keep sign of larger absolute value.)
    *   **性质**: 交换律 (Commutative Property: a + b = b + a), 结合律 (Associative Property: (a + b) + c = a + (b + c))
*   **减法 (Subtraction)**: 减去一个数等于加上这个数的相反数。 (Subtracting a number is the same as adding its opposite.)
    *   **性质**:  a - b = a + (-b)
*   **乘法 (Multiplication)**:  相同符号相乘为正，不同符号相乘为负，零乘以任何数都为零。 (Same sign: positive product. Different signs: negative product. Zero multiplied by any number is zero.)
    *   **性质**: 交换律 (Commutative Property: a * b = b * a), 结合律 (Associative Property: (a * b) * c = a * (b * c)), 分配律 (Distributive Property: a * (b + c) = a * b + a * c)
*   **除法 (Division)**:  除以一个数等于乘以这个数的倒数。 (Dividing by a number is the same as multiplying by its reciprocal.)
    *   **性质**: 只有在除数不为零时才有意义。 (Only defined when the divisor is not zero.)
    *   **余数**:  除不尽时，会产生余数。(Remainder: when division is not exact.)

### 1.4 整除性 (Divisibility)

*   **因数 (Factors)**:  能整除给定整数的整数。 (Integers that divide evenly into a given integer.)
    *   **例子**: 12 的因数: 1, 2, 3, 4, 6, 12
*   **倍数 (Multiples)**:  给定整数乘以任何整数的结果。 (The result of multiplying a given integer by any integer.)
    *   **例子**: 3 的倍数: 3, 6, 9, 12, ...
*   **质数 (Prime Numbers)**:  只有 1 和自身两个因数的整数。 (Integers that have only two factors: 1 and themselves.)
    *   **例子**: 2, 3, 5, 7, 11, 13, ...
*   **合数 (Composite Numbers)**:  除了 1 和自身外，还有其他因数的整数。 (Integers that have more than two factors.)
    *   **例子**: 4, 6, 8, 9, 10, 12, ...
*   **公因数和公倍数 (Common Factors and Multiples)**: 两个或多个整数共有的因数和倍数。 (Factors and multiples shared by two or more integers.)
    *   **最大公因数 (Greatest Common Factor, GCF)**: 最大的公因数。 (The largest common factor.)
    *   **最小公倍数 (Least Common Multiple, LCM)**: 最小的公倍数。 (The smallest common multiple.)

## 二、小数 (Decimals)

### 2.1 定义 (Definition)

*   **概念**: 由整数部分、小数点和小数部分组成。 (Consists of an integer part, a decimal point, and a fractional part.)
*   **表示**: 用于表示比 1 小的数，是分数的一种特殊形式。 (Used to represent numbers less than 1, a special form of fractions.)

### 2.2 分类 (Classification)

*   **有限小数 (Terminating Decimals)**:  小数部分位数有限。 (Decimal part has a finite number of digits.)
    *   **例子**: 0.25, 1.5, 3.14
*   **无限小数 (Infinite Decimals)**: 小数部分位数无限。 (Decimal part has an infinite number of digits.)
    *   **无限循环小数 (Repeating Decimals)**:  小数部分从某一位开始循环出现。 (Decimal part repeats from a certain digit.)
        *   **例子**: 1/3 = 0.333..., 2/7 = 0.285714285714...
    *   **无限不循环小数 (Non-Repeating Decimals)**:  小数部分不循环出现。 (Decimal part does not repeat.)
        *   **例子**: π (pi) ≈ 3.1415926...,  √2 ≈ 1.4142135...

### 2.3 运算 (Operations)

*   **加法和减法 (Addition and Subtraction)**:  小数点对齐，然后按照整数的加减法规则进行计算。 (Align decimal points, then perform addition or subtraction as with integers.)
*   **乘法 (Multiplication)**:  先忽略小数点，按照整数乘法计算，再根据两个因数的小数位数之和确定积的小数位数。 (Ignore decimal points, multiply as integers, then place the decimal point in the product based on the sum of the decimal places in the factors.)
*   **除法 (Division)**:  将除数的小数点移动到整数位，同时将被除数的小数点也移动相同的位数，然后按照整数除法计算。 (Move the decimal point in the divisor to make it an integer, move the decimal point in the dividend the same number of places, then perform division as with integers.)

### 2.4 小数与分数的互化 (Conversion between Decimals and Fractions)

*   **小数化分数 (Decimal to Fraction)**:  根据小数的位数，将小数写成分母为 10、100、1000 等的分数，然后化简。 (Write the decimal as a fraction with a denominator of 10, 100, 1000, etc., based on the number of decimal places, then simplify.)
*   **分数化小数 (Fraction to Decimal)**:  将分数的分母化为 10、100、1000 等，或者用分子除以分母。 (Convert the denominator to 10, 100, 1000, etc., or divide the numerator by the denominator.)

### 2.5 近似数 (Approximation)

*   **四舍五入法 (Rounding)**:  根据要保留的位数，观察下一位的数字，大于等于 5 则进 1，小于 5 则舍去。 (Observe the digit to the right of the desired place value; round up if it's 5 or greater, round down if it's less than 5.)
*   **有效数字 (Significant Digits)**:  从左边第一个非零数字开始的所有数字。 (All digits from the first non-zero digit on the left.)

## 三、整数与小数的应用 (Applications of Integers and Decimals)

*   **日常生活 (Daily Life)**: 购物、测量、时间计算等。 (Shopping, measurement, time calculation, etc.)
*   **科学研究 (Scientific Research)**: 数据分析、实验计算等。 (Data analysis, experimental calculations, etc.)
*   **工程技术 (Engineering Technology)**: 设计、建模、计算等。 (Design, modeling, calculation, etc.)
*   **金融经济 (Finance and Economics)**: 财务报表、利率计算等。 (Financial statements, interest rate calculation, etc.)

《整数与小数的思维导图》

一、整数 (Integers)

1.1 定义 (Definition)

概念: 没有小数部分，是正数、负数和零的统称。 (Whole numbers, including positive, negative, and zero.)
性质: 具有离散性，每一个整数都有确定的位置。 (Discrete nature; each integer occupies a specific position.)

1.2 分类 (Classification)

正整数 (Positive Integers): 大于零的整数。 (Integers greater than zero.)
- 例子: 1, 2, 3, 4, ...
零 (Zero): 既不是正数也不是负数。 (Neither positive nor negative.)
- 性质: 是偶数，是自然数。 (Even number, natural number.)
负整数 (Negative Integers): 小于零的整数。 (Integers less than zero.)
- 例子: -1, -2, -3, -4, ...

1.3 运算 (Operations)

加法 (Addition): 相同符号相加，绝对值相加，符号不变；不同符号相加，绝对值大的减去绝对值小的，符号与绝对值大的相同。 (Same sign: add absolute values, keep sign. Different signs: subtract smaller absolute value from larger, keep sign of larger absolute value.)
- 性质: 交换律 (Commutative Property: a + b = b + a), 结合律 (Associative Property: (a + b) + c = a + (b + c))
减法 (Subtraction): 减去一个数等于加上这个数的相反数。 (Subtracting a number is the same as adding its opposite.)
- 性质: a - b = a + (-b)
乘法 (Multiplication): 相同符号相乘为正，不同符号相乘为负，零乘以任何数都为零。 (Same sign: positive product. Different signs: negative product. Zero multiplied by any number is zero.)
- 性质: 交换律 (Commutative Property: a b = b a), 结合律 (Associative Property: (a b) c = a (b c)), 分配律 (Distributive Property: a (b + c) = a b + a * c)
除法 (Division): 除以一个数等于乘以这个数的倒数。 (Dividing by a number is the same as multiplying by its reciprocal.)
- 性质: 只有在除数不为零时才有意义。 (Only defined when the divisor is not zero.)
- 余数: 除不尽时，会产生余数。(Remainder: when division is not exact.)

1.4 整除性 (Divisibility)

因数 (Factors): 能整除给定整数的整数。 (Integers that divide evenly into a given integer.)
- 例子: 12 的因数: 1, 2, 3, 4, 6, 12
倍数 (Multiples): 给定整数乘以任何整数的结果。 (The result of multiplying a given integer by any integer.)
- 例子: 3 的倍数: 3, 6, 9, 12, ...
质数 (Prime Numbers): 只有 1 和自身两个因数的整数。 (Integers that have only two factors: 1 and themselves.)
- 例子: 2, 3, 5, 7, 11, 13, ...
合数 (Composite Numbers): 除了 1 和自身外，还有其他因数的整数。 (Integers that have more than two factors.)
- 例子: 4, 6, 8, 9, 10, 12, ...
公因数和公倍数 (Common Factors and Multiples): 两个或多个整数共有的因数和倍数。 (Factors and multiples shared by two or more integers.)
- 最大公因数 (Greatest Common Factor, GCF): 最大的公因数。 (The largest common factor.)
- 最小公倍数 (Least Common Multiple, LCM): 最小的公倍数。 (The smallest common multiple.)

二、小数 (Decimals)

2.1 定义 (Definition)

概念: 由整数部分、小数点和小数部分组成。 (Consists of an integer part, a decimal point, and a fractional part.)
表示: 用于表示比 1 小的数，是分数的一种特殊形式。 (Used to represent numbers less than 1, a special form of fractions.)

2.2 分类 (Classification)

有限小数 (Terminating Decimals): 小数部分位数有限。 (Decimal part has a finite number of digits.)
- 例子: 0.25, 1.5, 3.14
无限小数 (Infinite Decimals): 小数部分位数无限。 (Decimal part has an infinite number of digits.)
- 无限循环小数 (Repeating Decimals): 小数部分从某一位开始循环出现。 (Decimal part repeats from a certain digit.)
  - 例子: 1/3 = 0.333..., 2/7 = 0.285714285714...
- 无限不循环小数 (Non-Repeating Decimals): 小数部分不循环出现。 (Decimal part does not repeat.)
  - 例子: π (pi) ≈ 3.1415926..., √2 ≈ 1.4142135...

2.3 运算 (Operations)

加法和减法 (Addition and Subtraction): 小数点对齐，然后按照整数的加减法规则进行计算。 (Align decimal points, then perform addition or subtraction as with integers.)
乘法 (Multiplication): 先忽略小数点，按照整数乘法计算，再根据两个因数的小数位数之和确定积的小数位数。 (Ignore decimal points, multiply as integers, then place the decimal point in the product based on the sum of the decimal places in the factors.)
除法 (Division): 将除数的小数点移动到整数位，同时将被除数的小数点也移动相同的位数，然后按照整数除法计算。 (Move the decimal point in the divisor to make it an integer, move the decimal point in the dividend the same number of places, then perform division as with integers.)

2.4 小数与分数的互化 (Conversion between Decimals and Fractions)

小数化分数 (Decimal to Fraction): 根据小数的位数，将小数写成分母为 10、100、1000 等的分数，然后化简。 (Write the decimal as a fraction with a denominator of 10, 100, 1000, etc., based on the number of decimal places, then simplify.)
分数化小数 (Fraction to Decimal): 将分数的分母化为 10、100、1000 等，或者用分子除以分母。 (Convert the denominator to 10, 100, 1000, etc., or divide the numerator by the denominator.)

2.5 近似数 (Approximation)

四舍五入法 (Rounding): 根据要保留的位数，观察下一位的数字，大于等于 5 则进 1，小于 5 则舍去。 (Observe the digit to the right of the desired place value; round up if it's 5 or greater, round down if it's less than 5.)
有效数字 (Significant Digits): 从左边第一个非零数字开始的所有数字。 (All digits from the first non-zero digit on the left.)

三、整数与小数的应用 (Applications of Integers and Decimals)

日常生活 (Daily Life): 购物、测量、时间计算等。 (Shopping, measurement, time calculation, etc.)
科学研究 (Scientific Research): 数据分析、实验计算等。 (Data analysis, experimental calculations, etc.)
工程技术 (Engineering Technology): 设计、建模、计算等。 (Design, modeling, calculation, etc.)
金融经济 (Finance and Economics): 财务报表、利率计算等。 (Financial statements, interest rate calculation, etc.)

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