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Multiplication

The product of two complex numbers is computed as follows:

For example, (3 + 2 i) (4 − i) = 3 ⋅ 4 − (2 ⋅ (− 1)) + (3 ⋅ (− 1) + 2 ⋅ 4) i = 14 + 5 i. {\displaystyle (3+2i)(4-i)=3\cdot 4-(2\cdot (-1))+(3\cdot (-1)+2\cdot 4)i=14+5i.} In particular, this includes as a special case the fundamental formula

This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number x always satisfies x 2 ≥ 0 {\displaystyle x^{2}\geq 0}.

With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property, the commutative properties (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a field, the same way as the rational or real numbers do.