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Undefined and Imaginary Numbers


I found something strange with undefined and imaginary numbers.

First, please take this two mathematical definitions into consideration.

  1) The square root of a negative number is undefined.

  2) The square root of -1, or i, is defined as an imaginary number.

This is traditionally shown using these two equations.

i =  -1 

i2 = -1

So I have to ask, if  -1  or i is defined as imaginary, then how is it also undefined? Can something be both defined and undefined at the same time? I think not. The only possible explanation is that  -1  or i is both undefined and imaginary, and imaginary is just a mathematical representation of something undefined and not a definition in itself.






This got me thinking if undefined equations can also be imaginary too. Please take this into consideration:

If we draw the graph of: y = 1 / x

Graph of One over x

As you can see, when x approaches 0, then y approaches infinity and -infinity.

Thus, y is undefined when x=0.




Alternatively, if we draw the graph of: y = (1 / x)2

Graph of square of one over x

As you can see, when x approaches 0, y approaches infinity.

Therefore, we could write:

( 1 / 0 )2 = limit of x as x approaches infinity

Thus, we can define this imaginary equation or imaginary number:

b1 = 1 / 0

Then b1 can be usable, yet still undefined and imaginary.




Note this equation above is true when the numerator is equal to any constant, except for zero (i.e. zero divided by zero). In other words,

( c / 0 )2 = limit of x as x approaches infinity when c <> 0




Therefore, if we draw the graph of: y = 0 / x

graph of zero over x

As you can see, this results with a straight line or y=0, except when x=0. However when x=0, then y is approaching 0 too.

Therefore, we could write:

0 / 0 = limit of x as x approaches zero

Thus, we can define this imaginary equation or imaginary number:

b2 = 0 / 0

Then b2 can be usable, yet still undefined and imaginary too.

by Phil for Humanity
on 20170901

Related Articles
 » Proof: 0 * Undefined = -1
 » What does Infinity Minus Infinity Equal?
 » What does Infinity Divided by Infinity Equal?