Four rules

Japanese: 四則 - しそく（英語表記）four rules

The four arithmetic methods of addition, subtraction, multiplication, and division are collectively called the four arithmetic operations. If any real numbers are a and b, combining a and b with a + is called addition, and the result of the addition, a + b, is called the sum of a and b. Finding an x that satisfies b + x = a is called subtraction (subtracting b from a), and this x is written as ab and is called the difference between a and b. Multiplication of a and b is combining a and b with × or ・ (meaning multiplication), and the result of the multiplication, a × b or a・b (also written simply as ab), is called the product of a and b. Finding an x that satisfies b × x = a (b ≠ 0) is called division (dividing a by b), and this x is written as a÷b or a/b and is called the quotient of a and b. However, in addition to the meaning of multiplication, in general algebraic systems, the arithmetic method of combining two elements can also be called multiplication. The commutative, associative, distributive, and reduction laws apply to addition and multiplication. Subtraction and division are the inverse operations of addition and multiplication, respectively.

(1) Commutative law a+b=b+a
a×b＝b×a
(2) Associative law a+ (b+c) = (a+b) +c
a×(b×c)＝(a×b)×c
(3) Distributive law (a+b) ×c=a×c+b×c
(4) Simplification rule: if a + c = a + b, then c = b
If a×c＝a×b, then c＝b (a≠0)

Source: Encyclopaedia Britannica Concise Encyclopedia About Encyclopaedia Britannica Concise Encyclopedia Information

Japanese:

加法，減法，乗法，除法の4つの算法をまとめて，四則あるいは四則計算という。任意の実数をa，bとするとき，a，bを，＋で結合することを加法といい，加えた結果a＋bを，aとbの和という。また，b＋x＝aを満足するxを求めることを減法 (aからbを引く) といい，このxをa-bと書いて，これをaとbの差という。a，bの乗法とは，×または・ (掛け算の意味) でa，bを結合することで，掛けた結果a×bあるいはa・b (単に abとも書く) をaとbの積という。またb×x＝a (b≠0) を満足するxを求めることを除法 (aをbで割る) といい，このxをa÷bまたはa/bと書いて，これをaとbの商という。ただし，このような掛け算の意味以外に，一般の代数系において，2元の結合をつくる算法を乗法ということもある。加法，乗法については交換法則，結合法則，分配法則，簡約法則が成り立つ。また減法と除法は，それぞれ加法と乗法の逆算である。

(1) 交換法則
　　a＋b＝b＋a
　　a×b＝b×a
(2) 結合法則
　　a＋ (b＋c) ＝ (a＋b) ＋c
　　a× (b×c) ＝ (a×b) ×c
(3) 分配法則
　　 (a＋b) ×c＝a×c＋b×c
(4) 簡約法則
　　a＋c＝a＋bならばc＝b
　　a×c＝a×bならばc＝b (a≠0)

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