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除法

將20個蘋果平均分成四等分(左上),每份有5個蘋果(右下),即;亦可以說成,將20個蘋果每5個分成一份(右下),共可分成四等分(左上),此時可以表達為

数学中,尤其是在基本计算裏,除法可以看成是「乘法的反运算」,也可以理解为「重复的减法」。除法运算的本质就是「把参与运算的除数变为,得出同比的被除数的值」。

例如:,就好像減了兩次後,就變成了

如果

而且不等于,那么

其中,a称为商数,b称为除数,c称为被除数

如果除式的商數()必須是整數,则称为带餘除法相差的数值,称为餘數)。

這也意味著

高等数学(包括在科学工程学中)和计算机编程语言中,写成。如果我们不需要知道确切值或者留待以后引用,这种形式也常常是称之为分数的最终形式。其中尋找商數的函數,尋找餘數的函數則為

在大部分的非英语语言中,代表,讀做c比b;則代表比值。用法请参照比例

整除

整除数学中两个自然数之间的一种关系。自然数可以被自然数整除,是指因數,且a是b的整数倍数,也就是除以没有餘数

因數判別法可參照整除規則

表示法

表示整除,即的倍数,因数

举例

可以被整除,记作

不能被整除(因为餘数为),记作。在上加一条斜线即表示不整除。

除法计算

根据乘法表,两个整数可以用长除法(直式除法)笔算。如果被除数有分数部分(或者说时小数点),计算时将小数点带下来就可以;如果除数有小数点,将除数与被除数的小数点同时移位,直到除数没有小数点。

算盘也可以做除法运算。

長除法

長除法俗稱「長除」,適用於正式除法、小數除法、多項式除法(即因式分解)等較重視計算過程和商數的除法,過程中兼用了乘法減法

使用長除法計算的過程可以表示為:

的演算過程

短除法

短除法是長除法的簡化版本。在短除法裏,被除數放中央,旁以一L型符號表示除法,被除數左側為除數,下側為商,省去了長除法逐層計算的過程。

  • 使用短除法計算的近似值:

多項式的除法

整数之间的带余除法类似,一元多项式之间也可以进行带余除法。可以证明,设有多项式和非零多项式,则存在唯一的多项式,满足:

而多项式若非零多项式,則其冪次严格小于的冪次。

作为特例,如果要计算某个多项式除以一次多项式得到的餘多项式,可以直接将代入到多项式中。除以的餘多项式是

具体的计算可以使用类似直式除法的方式。例如,计算除以,列式如下:

因此,商式是,餘式是

重要性質

通常不定义除以零这种形式。亦即當除以0 或分數的分母為0 時,該式或該數無意義

参见

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