# 国中数学/国中数学七年级/1-3 正负数的乘除

 1-2 正负数的加减 ◄ 国中数学七年级1-3 正负数的乘除 ► 1-4 指数记法与科学记号

## 正负数的乘法

${\displaystyle 3}$
${\displaystyle 15}$万元
${\displaystyle 2}$
${\displaystyle 10}$万元
${\displaystyle 1}$
${\displaystyle 5}$万元

${\displaystyle 0}$万元(不赚不赔)
${\displaystyle 1}$
${\displaystyle 5}$万元
${\displaystyle 2}$
${\displaystyle 10}$万元
${\displaystyle 3}$
${\displaystyle 15}$万元

${\displaystyle 3}$
${\displaystyle 15}$万元
${\displaystyle ({\color {blue}+}5)\times ({\color {blue}+}3)={\color {blue}+}15}$
${\displaystyle 2}$
${\displaystyle 10}$万元
${\displaystyle ({\color {blue}+}5)\times ({\color {blue}+}2)={\color {blue}+}10}$
${\displaystyle 1}$
${\displaystyle 5}$万元
${\displaystyle ({\color {blue}+}5)\times ({\color {blue}+}1)={\color {blue}+}5}$

${\displaystyle 0}$万元(不赚不赔)
${\displaystyle ({\color {blue}+}5)\times 0=0}$
${\displaystyle 1}$
${\displaystyle 5}$万元
${\displaystyle ({\color {blue}+}5)\times ({\color {red}-}1)={\color {red}-}5}$
${\displaystyle 2}$
${\displaystyle 10}$万元
${\displaystyle ({\color {blue}+}5)\times ({\color {red}-}2)={\color {red}-}10}$
${\displaystyle 3}$
${\displaystyle 15}$万元
${\displaystyle ({\color {blue}+}5)\times ({\color {red}-}3)={\color {red}-}15}$

 ${\displaystyle a,b}$為正數，則${\displaystyle (+a)\times (-b)=-(a\times b)}$


 例题${\displaystyle 1}$计算以下各式： ${\displaystyle (1)5\times (-4)}$ ${\displaystyle (2)0.3\times (-1.2)}$
 解${\displaystyle (1)5\times (-4)=-(5\times 4)=-20}$ ${\displaystyle (2)0.3\times (-1.2)=-(0.3\times 1.2)=-0.36}$

 答对加上的分数： 答错的分数： 忽略问题的系数：

1 ${\displaystyle 3\times (-7)=?}$

 ${\displaystyle 21}$ ${\displaystyle -21}$

2 ${\displaystyle 1.4\times (-5)=?}$

 ${\displaystyle 7}$ ${\displaystyle -7}$

${\displaystyle 3}$
${\displaystyle 15}$万元
${\displaystyle 2}$
${\displaystyle 10}$万元
${\displaystyle 1}$
${\displaystyle 5}$万元

${\displaystyle 0}$万元(不赚不赔)
${\displaystyle 1}$
${\displaystyle 5}$万元
${\displaystyle 2}$
${\displaystyle 10}$万元
${\displaystyle 3}$
${\displaystyle 15}$万元

${\displaystyle 3}$
${\displaystyle 15}$万元
${\displaystyle ({\color {red}-}5)\times ({\color {blue}+}3)={\color {red}-}15}$
${\displaystyle 2}$
${\displaystyle 10}$万元
${\displaystyle ({\color {red}-}5)\times ({\color {blue}+}2)={\color {red}-}10}$
${\displaystyle 1}$
${\displaystyle 5}$万元
${\displaystyle ({\color {red}-}5)\times ({\color {blue}+}1)={\color {red}-}5}$

${\displaystyle 0}$万元(不赚不赔)
${\displaystyle ({\color {red}-}5)\times 0=0}$
${\displaystyle 1}$
${\displaystyle 5}$万元
${\displaystyle ({\color {red}-}5)\times ({\color {red}-}1)={\color {blue}+}5}$
${\displaystyle 2}$
${\displaystyle 10}$万元
${\displaystyle ({\color {red}-}5)\times ({\color {red}-}2)={\color {blue}+}10}$
${\displaystyle 3}$
${\displaystyle 15}$万元
${\displaystyle ({\color {red}-}5)\times ({\color {red}-}3)={\color {blue}+}15}$

 ${\displaystyle a,b}$為正數，則${\displaystyle (-a)\times (+b)=-(a\times b)}$；${\displaystyle (-a)\times (-b)=a\times b}$


 例题${\displaystyle 2}$计算以下各式： ${\displaystyle (1)(-3)\times (-9)}$ ${\displaystyle (2)(-3.6)\times 1.2}$
 解${\displaystyle (1)(-3)\times (-9)=3\times 9=27}$ ${\displaystyle (2)(-3.6)\times 1.2=-(3.6\times 1.2)=-4.32}$

 答对加上的分数： 答错的分数： 忽略问题的系数：

1 ${\displaystyle (-3)\times (-7)=?}$

 ${\displaystyle 21}$ ${\displaystyle -21}$

2 ${\displaystyle (-2.9)\times 10=?}$

 ${\displaystyle 29}$ ${\displaystyle -29}$

3 ${\displaystyle (-11)\times 7=?}$

 ${\displaystyle 77}$ ${\displaystyle -77}$

4 ${\displaystyle (-1.4)\times (-1.6)=?}$

 ${\displaystyle 2.24}$ ${\displaystyle -2.24}$

### 正负数乘法的口诀

 正正得正，正負得負，負正得負，負負得正


 同號數相乘，其值為正；異號數相乘，其值為負。

 例题${\displaystyle 3}$计算下列各式的值。 ${\displaystyle (1)(-9)\times (-3)}$ ${\displaystyle (2)(-1.2)\times 0.3}$ ${\displaystyle (3)1.2\times (-3.6)}$
 解${\displaystyle (1)}$因为负负得正的关系，所以${\displaystyle (-9)\times (-3)=9\times 3=27}$ ${\displaystyle (2)}$因为负正得负的关系，所以${\displaystyle (-1.2)\times 0.3=-(1.2\times 0.3)=-0.36}$ ${\displaystyle (3)}$因为正负得负的关系，所以${\displaystyle 1.2\times (-3.6)=-(1.2\times 3.6)=-4.32}$

### 乘法的交换律

${\displaystyle (-9)\times (-3)}$
${\displaystyle (-3)\times (-9)}$

${\displaystyle 27}$
${\displaystyle 27}$

${\displaystyle 1.}$比较例题${\displaystyle 3}$${\displaystyle (2)}$题与例题${\displaystyle 1}$${\displaystyle (2)}$题，是否相同？[习题解答 1]
${\displaystyle 2.}$比较例题${\displaystyle 3}$${\displaystyle (3)}$题与例题${\displaystyle 2}$${\displaystyle (2)}$题，是否相同？[习题解答 2]

 若${\displaystyle a}$、${\displaystyle b}$為兩數，則${\displaystyle a\times b=b\times a}$。


### 乘法的结合律

 例题${\displaystyle 4}$计算下列各式的值。 ${\displaystyle (1)[(-9)\times (-10)]\times 8}$ ${\displaystyle (2)(-9)\times [(-10)\times 8]}$
 解${\displaystyle (1)[(-9)\times (-10)]\times 8}$ ${\displaystyle =(9\times 10)\times 8}$(负负得正) ${\displaystyle =90\times 8}$ ${\displaystyle =720}$ ${\displaystyle (2)}$${\displaystyle (-9)\times [(-10)\times 8]}$ ${\displaystyle =(-9)\times [-(10\times 8)]}$(负正得负) ${\displaystyle =(-9)\times (-80)}$ ${\displaystyle =9\times 80}$(负负得正) ${\displaystyle =720}$

 答对加上的分数： 答错的分数： 忽略问题的系数：

1 计算${\displaystyle [3\times (-5)]\times 8}$${\displaystyle 3\times [(-5)\times 8]}$的结果，答案是否相同？

 相同 不相同

2 计算${\displaystyle [(-0.6)\times 1.5]\times (-3)}$${\displaystyle (-0.6)\times [1.5\times (-3)]}$的结果，答案是否相同？

 相同 不相同

3 计算${\displaystyle (10\times 12)\times (-0.4)}$${\displaystyle 10\times [12\times (-0.4)]}$的结果，答案是否相同？

 相同 不相同

4 计算${\displaystyle [(-0.3)\times (-0.4)]\times (-0.5)}$${\displaystyle (-0.3)\times [(-0.4)\times (-0.5)]}$的结果，答案是否相同？

 相同 不相同

 若${\displaystyle a}$、${\displaystyle b}$、${\displaystyle c}$為三數，則${\displaystyle (a\times b)\times c=a\times (b\times c)}$。


### 1、0、-1的乘法

1. 任何数乘以${\displaystyle 1}$之后皆等于自己本身。即若${\displaystyle a}$是任意数，则${\displaystyle a\times 1=1\times a=a}$
2. 任何数乘以${\displaystyle 0}$之后皆等于${\displaystyle 0}$。即若${\displaystyle a}$是任意数，则${\displaystyle a\times 0=0\times a=0}$
3. 任何数乘以${\displaystyle -1}$之后皆等于该数的相反数。即若${\displaystyle a}$是任意数，则${\displaystyle a\times (-1)=(-1)\times a=-a}$

 答对加上的分数： 答错的分数： 忽略问题的系数：

1 计算${\displaystyle (-1313)\times 1=?}$

 ${\displaystyle 1313}$ ${\displaystyle 0}$ ${\displaystyle -1313}$

2 计算${\displaystyle (-1313)\times 0=?}$

 ${\displaystyle 1313}$ ${\displaystyle 0}$ ${\displaystyle -1313}$

3 计算${\displaystyle (-1313)\times (-1)=?}$

 ${\displaystyle 1313}$ ${\displaystyle 0}$ ${\displaystyle -1313}$

4 计算${\displaystyle 1\times (-0.734)=?}$

 ${\displaystyle 0.734}$ ${\displaystyle 0}$ ${\displaystyle -0.734}$

5 计算${\displaystyle 0\times (-0.734)=?}$

 ${\displaystyle 0.734}$ ${\displaystyle 0}$ ${\displaystyle -0.734}$

6 计算${\displaystyle (-1)\times 0.734=?}$

 ${\displaystyle 0.734}$ ${\displaystyle 0}$ ${\displaystyle -0.734}$

### 连续数的乘法

1. 有括号要先算。
2. 从左而右计算。
3. 可以利用交换律与结合律简化计算。
 例题${\displaystyle 5}$计算${\displaystyle [(-25)\times (-17)]\times (4\times 3)}$的值。
 解${\displaystyle [(-25)\times (-17)]\times (4\times 3)}$ ${\displaystyle =(-25)\times (-17)\times 4\times 3}$(拿掉括号) ${\displaystyle =(-25)\times [(-17)\times 4]\times 3}$(括中间) ${\displaystyle =(-25)\times [4\times (-17)]\times 3}$(交换律) ${\displaystyle =[(-25)\times 4]\times (-17)\times 3}$(结合律) ${\displaystyle =(-100)\times [(-17)\times 3]}$(结合律) ${\displaystyle =(-100)\times (-51)}$ ${\displaystyle =5100}$

${\displaystyle 3.}$计算下列各式的值：

• ${\displaystyle (1)125\times [37\times (-8)]}$[习题解答 3]
• ${\displaystyle (2)(-13)\times (-4)\times [(-5)\times (-7)]}$[习题解答 4]

#### 连续多数的乘法之正负性

1. 如果连乘算式当中有奇数个负数时，答案为负数
2. 如果连乘算式当中有偶数个负数时，答案为正数
3. 如果连乘算式当中有${\displaystyle {\color {green}0}}$，答案为${\displaystyle {\color {green}0}}$
 例题${\displaystyle 6}$判断以下各式计算的结果为正数、负数或${\displaystyle 0}$？ ${\displaystyle (1)(-333)\times (-4444)\times (-55555)\times (-666666)}$ ${\displaystyle (2)(-2)\times 3\times (-4)\times 5\times (-6)\times 7}$
 解 ${\displaystyle (1)}$式子中没有${\displaystyle 0}$，有${\displaystyle 4}$个负数，为偶数，所以计算结果为正数。 ${\displaystyle (2)}$式子中没有${\displaystyle 0}$，有${\displaystyle 3}$个负数，为奇数，所以计算结果为负数。

 答对加上的分数： 答错的分数： 忽略问题的系数：

1 连乘算式${\displaystyle 134\times 73\times 0\times (-32)\times (-79)\times (-137)}$的值为正数、负数或${\displaystyle 0}$

 正数 ${\displaystyle 0}$ 负数

2 连乘算式${\displaystyle (-2)\times (-4)\times (-6)\times \cdots \times (-50)}$的值为正数、负数或${\displaystyle 0}$

 正数 ${\displaystyle 0}$ 负数

## 数的除法

${\displaystyle (-3)\times 5=-15}$
${\displaystyle (-15)\div (-3)=5}$
${\displaystyle (-15)\div 5=-3}$
${\displaystyle 4\times (-7)=-28}$
${\displaystyle (-28)\div 4=-7}$
${\displaystyle (-28)\div (-7)=4}$
${\displaystyle (-6)\times (-9)=54}$
${\displaystyle 54\div (-6)=-9}$
${\displaystyle 54\div (-9)=-6}$

 同號數相除，其值為正；異號數相除，其值為負。


正正得正，正負得負，負正得負，負負得正。


### 关于1的除法

1. ${\displaystyle a}$是任意数，则${\displaystyle a\div 1=a}$
2. ${\displaystyle a}$是任意非${\displaystyle 0}$的数，则${\displaystyle 1\div a={\frac {1}{a}}}$，其中${\displaystyle {\frac {1}{a}}}$${\displaystyle a}$倒数[注 3]

### 关于-1的除法

1. ${\displaystyle a}$是任意数，则${\displaystyle a\div (-1)=-a}$，即${\displaystyle a}$相反数
2. ${\displaystyle a}$是任意非${\displaystyle 0}$的数，则${\displaystyle (-1)\div a=-{\frac {1}{a}}}$

### 关于0的除法

1. ${\displaystyle a}$是任意数，则${\displaystyle 0\div a=0}$
2. 我们不定义任何数除以${\displaystyle 0}$的结果。即${\displaystyle 0}$不能是除数
 例题${\displaystyle 7}$计算下列各式的值： ${\displaystyle (1)(-98)\div (-7)}$ ${\displaystyle (2)(-130)\div 13}$ ${\displaystyle (3)87\div (-29)}$
 解 ${\displaystyle (1)}$因为负负得正，所以${\displaystyle (-98)\div (-7)=98\div 7=14}$。 ${\displaystyle (2)}$因为负正得负，所以${\displaystyle (-130)\div 13=-(130\div 13)=-10}$。 ${\displaystyle (3)}$因为正负得负，所以${\displaystyle 87\div (-29)=-(87\div 29)=-3}$。

 答对加上的分数： 答错的分数： 忽略问题的系数：

1 ${\displaystyle (-45)\div (-9)=?}$

 ${\displaystyle 5}$ ${\displaystyle -5}$

2 ${\displaystyle 84\div (-12)=?}$

 ${\displaystyle 7}$ ${\displaystyle -7}$

3 ${\displaystyle (-72)\div 9=?}$

 ${\displaystyle 8}$ ${\displaystyle -8}$

4 ${\displaystyle (-121)\div a=11}$，则${\displaystyle a=?}$

 ${\displaystyle 11}$ ${\displaystyle -11}$

5 哪一个算式没有意义？

 ${\displaystyle 1\div (-12)}$ ${\displaystyle 13\div (-1)}$ ${\displaystyle 0\div (-14)}$ ${\displaystyle (-15)\div 0}$

## 正负数乘除混合运算

1. 有绝对值要先算
2. 有括号要先算，顺序依序为${\displaystyle ()\rightarrow []\rightarrow }${}[注 4]
3. 没括号时，从左而右计算

${\displaystyle a}$${\displaystyle b}$${\displaystyle c}$为三个任意数(但不能除以${\displaystyle 0}$)，则：

1. ${\displaystyle a\times b\div c=a\div c\times b}$
2. ${\displaystyle a\div b\times c=a\times c\div b}$[注 7]
3. ${\displaystyle a\div b\div c=a\div c\div b}$

 例题${\displaystyle 8}$计算下列各式的值： ${\displaystyle (1)(-96)\div (-2)\div 8}$ ${\displaystyle (2)(-390)\div (13\times 5)}$ ${\displaystyle (3)805\div |(-23)\times 5|}$ ${\displaystyle (4)139\times (-97)\div (-139)}$
 解 ${\displaystyle (1)}$因为没有括号，所以需要从左而右计算。 ${\displaystyle (-96)\div (-2)\div 8}$ ${\displaystyle =48\div 8}$ ${\displaystyle =6}$ ${\displaystyle (2)}$因为有括号要先算，所以 ${\displaystyle (-390)\div (13\times 5)}$ ${\displaystyle =(-390)\div 65}$ ${\displaystyle =-6}$ ${\displaystyle (3)}$因为有绝对值要先算，所以 ${\displaystyle 805\div |(-23)\times 5|}$ ${\displaystyle =805\div |-115|}$ ${\displaystyle =805\div 115}$ ${\displaystyle =7}$ ${\displaystyle (4)}$观察${\displaystyle 139\div (-139)}$比较好算，利用交换性，所以 ${\displaystyle 139\times (-97)\div (-139)}$ ${\displaystyle =139\div (-139)\times (-97)}$ ${\displaystyle =(-1)\times (-97)}$ ${\displaystyle =97}$

${\displaystyle (1)12\div (-3)\times 13}$[习题解答 5]
${\displaystyle (2)(-168)\div [(-3)\times 7]}$[习题解答 6]
${\displaystyle (3)123\div |-41|\times (-9)}$[习题解答 7]
${\displaystyle (4)(-9600)\div (-25)\div 96}$[习题解答 8]

## 正负数四则运算

1. 有绝对值要先算。
2. 有括号要先算，顺序依序为${\displaystyle ()\rightarrow []\rightarrow }${}。
3. 先乘除后加减
4. 从左而右计算。
 例题${\displaystyle 9}$计算下列各式的值： ${\displaystyle (1)54-(-27)\div (-9)}$ ${\displaystyle (2)(-91)\div 13-|-12|\times (-4)}$ ${\displaystyle (3)(-39)\div [24+7\times (-3)]}$
 解 ${\displaystyle (1)54-(-27)\div (-9)}$ ${\displaystyle =54-3}$(先乘除后加减) ${\displaystyle =51}$ ${\displaystyle (2)(-91)\div 13-|-12|\times (-4)}$ ${\displaystyle =(-91)\div 13-12\times (-4)}$(有绝对值要先算) ${\displaystyle =(-7)-(-48)}$(先乘除后加减) ${\displaystyle =41}$ ${\displaystyle (3)(-39)\div [24+7\times (-3)]}$ ${\displaystyle =(-39)\div [24+(-21)]}$(有括号要先算、先乘除后加减) ${\displaystyle =(-39)\div 3}$ ${\displaystyle =-13}$

${\displaystyle (1)5\times (-3)+66\div (-11)}$[习题解答 9]
${\displaystyle (2)(-450)\div |29-(-3)\times 7|}$[习题解答 10]
${\displaystyle (3)6\times [|-21|+3\times (-9)]}$[习题解答 11]
${\displaystyle (4)(-12)-260\div (-5)\div (-13)}$[习题解答 12]

## 注解

1. 因为无论几个正数相乘都还是正数，但是乘以一个负数会变成负数，再乘一个负数的话因为负负得正的关系又变成正数，故只要两两一组的负数相乘就会变成正数，所以判断连乘算式的正负性，只要查看负数的数量。至于因为乘以${\displaystyle 0}$就会变成${\displaystyle 0}$，所以只要连乘算式里有乘以${\displaystyle 0}$，答案必为${\displaystyle 0}$
2. 除法运算符合乘法运算口诀之因为“除以一个数，等于乘以这个数的倒数”。
3. 倒数会在分数的乘除单元介绍。
4. 这是大括号符号，通常加在中括号外面。
5. 正负数加减混合运算也是，只是我们将减法改成加法运算，所以这里就没有细谈。
6. 原因依旧是除以一个数，就等于乘以这个数的倒数，再利用乘法交换律得到这样的结果。
7. 只是第1条式子${\displaystyle b}$改成${\displaystyle c}$${\displaystyle c}$改成${\displaystyle b}$然后等号两边颠倒而已。

## 习题解答

1. 习题${\displaystyle 1.}$
比较项目
例题${\displaystyle 3}$${\displaystyle (2)}$
例题${\displaystyle 1}$${\displaystyle (2)}$
题目
${\displaystyle (-1.2)\times 0.3}$
${\displaystyle 0.3\times (-1.2)}$
答案
${\displaystyle -0.36}$
${\displaystyle -0.36}$

故一样。

2. 习题${\displaystyle 2.}$
比较项目
例题${\displaystyle 3}$${\displaystyle (3)}$
例题${\displaystyle 2}$${\displaystyle (2)}$
题目
${\displaystyle 1.2\times (-3.6)}$
${\displaystyle (-3.6)\times 1.2}$
答案
${\displaystyle -4.32}$
${\displaystyle -4.32}$

故一样。

3. 习题${\displaystyle 3.(1)-37000}$
4. 习题${\displaystyle 3.(2)1820}$
5. 习题${\displaystyle 4.(1)-52}$
6. 习题${\displaystyle 4.(2)8}$
7. 习题${\displaystyle 4.(3)-27}$
8. 习题${\displaystyle 4.(4)4}$
9. 习题${\displaystyle 5.(1)-21}$
10. 习题${\displaystyle 5.(2)-9}$
11. 习题${\displaystyle 5.(3)-36}$
12. 习题${\displaystyle 5.(4)-16}$