Can You Divide By Zero in Math?

  • Context: High School 
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Discussion Overview

The discussion centers around the question of whether division by zero is permissible in mathematics, particularly in the context of an expression involving terms that include division by zero. Participants explore the implications of such expressions and the nature of indeterminacy versus undefined terms.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant presents an expression involving division by zero and questions whether the terms can be divided out.
  • Another participant asserts that the expression is indeterminate, not undefined, and argues against dividing the terms.
  • A different participant challenges the clarity of the original expression, suggesting it is poorly written and lacks meaning.
  • Some participants clarify that division by zero is not well-defined in the real number system, indicating that such manipulations are not valid.
  • One participant attempts to illustrate the concept of division by increasingly small numbers approaching zero, suggesting a connection to infinity, but acknowledges that this does not justify division by zero.
  • Another participant emphasizes that informal statements about division by infinity cannot be treated as mathematically rigorous without proper context.
  • A note is made about a separate discussion regarding the foundations of limits, indicating a divergence in the conversation.

Areas of Agreement / Disagreement

Participants express differing views on the validity of dividing terms that involve division by zero. There is no consensus on whether such expressions can be manipulated or what the implications of doing so are.

Contextual Notes

There are unresolved issues regarding the definitions and implications of indeterminate forms versus undefined expressions, as well as the limitations of informal reasoning about infinity.

ShawnD
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Say I had an expression that went like this

[tex]\frac{ 5 \frac{x}{0} }{3 \frac{x}{0} }[/tex]

Can I divide those [itex]\frac{x}{0}[/itex] terms or do they make the expression undefined?
 
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Nope, you can't divide them out. The entire expression is indeterminate (NOT undefined).

- Warren
 
I wouldn't say you can't divide them out so much as what you wrote is just plain wrong, being polite about it - the symbols make no sense as written.
 
I think he means

[tex]\frac{ 5 \cdot \frac{x}{0} }{3 \cdot \frac{x}{0} }[/tex]

to be read "5 times x over 0...", not "5 and x zeroths..."

- Warren
 
But is still makes no sense. x/0 is not a well-defined symbol in the real number system that one can manipulate like this.
 
I think that was part of his question.
 
1/0.1 is tha same as 1*10
1/0.01 is tha same as 1*100
1/0.001 is tha same as 1*1000
and so on ...

1/0 is the same as 1*oo and in both cases we are no longer in a finite system.

oo is a general notation for infinity therefore 1/0 is also a general notation for infinity.

Please look at: http://mathworld.wolfram.com/Infinity.html
 
Organic said:
1/0.1 is tha same as 1*10
1/0.01 is tha same as 1*100
1/0.001 is tha same as 1*1000
and so on ...

1/0 is the same as 1*oo and in both cases we are no longer in a finite system.

oo is a general notation for infinity therefore 1/0 is also a general notation for infinity.

Please look at: http://mathworld.wolfram.com/Infinity.html
Note for others, in the link given it goes points out:

"Informally,[itex]1 / \infty = 0[/tex] , a statement which can be made <b>rigorous using the limit concept</b>"<br /> <br /> You can't just say: <br /> <br /> [tex]\frac{1}{\infty} = 0[/tex] <br /> <br /> or any manipulation of that as and think it is mathematically true.[/itex]
 
Discussion over the foundations of limits split to new thread. Please stop hijacking threads.
 

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