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Arithmetic Operations


The basic arithmetic operations are addition, subtraction, multiplication and division.

Operations with rational numbers:

\begin{align} -a+(-b) &= -(a+b)\\ \\ a+(−b) &= \left\{ \begin{matrix} a−b,\; if\; a \geq b \\ -(b-a),\; if\; a < b \end{matrix} \right.\\ \\ a–b&=a+(-b)\\ \\ -a-b &= -(a+b)\\ \\ -a \times b&=a \times (-b)\\ \\ -a \times (-b)&=a \times b\\ \\ -a \div b &= a \div (-b)\\ \\ -a \div (-b) &= a \div b\\ \end{align}


Relationships between operations:

\begin{align} x+a=b &\; \Rightarrow \; x=b-a. \\ \\ a+x=b &\; \Rightarrow \; x=b-a. \\ \\ x-a=b &\; \Rightarrow \; x=a+b. \\ \\ a-x=b &\; \Rightarrow \; x=a-b. \\ \\ a\times x=b &\; \Rightarrow \; x=b\div a. \\ \\ x\times a=b &\; \Rightarrow \; x=b\div a. \\ \\ a\div x=b &\; \Rightarrow \; x=a\div b.\\ \\ x\div a=b &\; \Rightarrow \; x=a\times b.\\ \end{align}