Recall that we defined \(\sqrt{-1}\) as a new number to come up with a solution to the equation \[x^2+1=0.\]

The resulting number system, namely the Complex Number System, preserves the rules of arithmetic\(^1\) that works for the Real Number System.

Similarly, can't we define \(\dfrac{1}{0}\) as a new number and thus come up with a number system that includes solution to \(0 \cdot x =1\) and preserves the rules of arithmetic for the Real Number System?

**Footnote**:

- Commutative, Distributive and Associative Laws.

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