$\dfrac{1}{0}$ is a new number?

Algebra Level 3

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.

Yes, but that would not be useful; we need some other way to fix the problem No

1 solution

Brian Charlesworth
Jan 17, 2017

Suppose we have some unique $x$ such that $0 * x = 1$ . Using the "standard" rules, this would then imply that

$(-1 + 1)* x = 1 \Longrightarrow -x + x = 1 \Longrightarrow x = x + 1$ ,

i.e., that $x$ is equal to it's successor, in which case $x$ would be unique only if $0 = 1$ , which is not the case in the standard system.

1 0 \dfrac{1}{0} 0 1 ​ is a new number?

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$\dfrac{1}{0}$ is a new number?