Definition: Subtraction of Complex Numbers

Let \(x\) and \(y\) be complex numbers represented by \[\begin{array}{rcl}x&:=&(a,b),\\ y&:=&(c,d).\end{array}\]

The subtraction of complex numbers, written \(x-y\), is defined as the addition of the first complex number \(x\) with the inverse of the second complex number with respect to addition \((-y)\), formally \[x-y:=(a,b)+(-c,-d)=(a-c,b-d).\]

The result of the subtraction is called difference. Geometrically, the difference of two complex numbers can be interpreted as the vector pointing from \(y\) to \(x\), which has been shifted to start at the origin:

Proofs: 1 2
Propositions: 3

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