Why Can't the Denominator of a Fraction Be Zero?

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In summary, the bottom fraction, or denominator, cannot equal zero because division by zero is undefined. This applies to both constants and variables, regardless of whether numbers are plugged in or not. Other operations such as taking the square root of a negative number or the logarithm of a non-positive number are also not allowed if the result is intended to be a real number. It is important to have a strong foundation in algebra and trigonometry before tackling more advanced topics such as derivatives.
  • #1
member 731016
For this problem,
1677526334788.png

I don’t understand why the bottom fraction cannot equal zero since we are not plugging in any numbers it just in terms of variables. Would anybody else be able to provide with an explanation?

Many thanks!
 
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  • #2
Callumnc1 said:
For this problem,
View attachment 322953
I don’t understand why the bottom fraction cannot equal zero since we are not plugging in any numbers it just in terms of variables. Would anybody else be able to provide with an explanation?

Many thanks!
The "bottom fraction", really the bottom of the fraction, or denominator, can't be zero because division by zero is undefined. So the subsequent work is to determine the values of the x and y variables that make the fraction, and hence y', undefined.

This is very basic stuff.
 
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  • #3
Mark44 said:
The "bottom fraction", really the bottom of the fraction, or denominator, can't be zero because division by zero is undefined. So the subsequent work is to determine the values of the x and y variables that make the fraction, and hence y', undefined.

This is very basic stuff.
Thank you for your help @Mark44 !
 
  • #4
Callumnc1 said:
I don’t understand why the bottom fraction cannot equal zero since we are not plugging in any numbers it just in terms of variables.
It's really immaterial whether we're dealing with constants or variables, and in the latter case, whether we're "plugging in numbers" or not. There are certain arithmetic and algebraic operations that aren't allowed, among which the most prominent is division by zero. Other operations that aren't allowed, if the result is intended to be a real number, are taking the square root or other even root of a negative number, taking the logarithm or a non-positive number, and so on.

These ideas normally are presented before derivatives are presented, but it looks like you might have skipped over or forgotten them.
 
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  • #5
Mark44 said:
It's really immaterial whether we're dealing with constants or variables, and in the latter case, whether we're "plugging in numbers" or not. There are certain arithmetic and algebraic operations that aren't allowed, among which the most prominent is division by zero. Other operations that aren't allowed, if the result is intended to be a real number, are taking the square root or other even root of a negative number, taking the logarithm or a non-positive number, and so on.

These ideas normally are presented before derivatives are presented, but it looks like you might have skipped over or forgotten them.
Thank you for your reply @Mark44!

They did not really teach me too much about those ideas sadly. I got taught cacl before conics and alot of the other precacl topics.

Many thanks!
 
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  • #6
Callumnc1 said:
They did not really teach me too much about those ideas sadly. I got taught cacl before conics and alot of the other precacl topics.
Then it would be a very good idea to spend some time going over the precalculus topics, either from a precalc textbook (which you can get fairly cheaply) or via internet sites such as khanacademy.org. Not having a solid foundation in algebra and trig is really going to hold back your understanding in the kinds of problems you've been posting.
 
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Mark44 said:
Then it would be a very good idea to spend some time going over the precalculus topics, either from a precalc textbook (which you can get fairly cheaply) or via internet sites such as khanacademy.org. Not having a solid foundation in algebra and trig is really going to hold back your understanding in the kinds of problems you've been posting.
Thank you @Mark44, I will have a look!
 

FAQ: Why Can't the Denominator of a Fraction Be Zero?

1. Why is division by zero undefined?

Division by zero is undefined because there is no number that, when multiplied by zero, will yield a non-zero number. For example, if we try to divide 5 by 0, we are essentially asking what number multiplied by 0 gives us 5, which is impossible. Thus, the operation does not have a meaningful result.

2. What happens if you try to divide a number by zero?

If you attempt to divide a number by zero, such as 10 ÷ 0, you would encounter an undefined result. In mathematical terms, this means that there is no solution or value that satisfies the equation, leading to a breakdown in the arithmetic operations.

3. Can you explain the concept of limits in relation to division by zero?

In calculus, limits can help us understand the behavior of functions as they approach division by zero. For example, as a denominator approaches zero, the value of the fraction can grow very large or very small, depending on the direction from which it approaches zero. However, this does not provide a defined value when the denominator is exactly zero; it merely illustrates the discontinuity at that point.

4. Are there any exceptions to the rule that the denominator cannot be zero?

No, there are no exceptions in standard arithmetic and algebra. The rule that the denominator cannot be zero holds true universally. In certain mathematical contexts, such as limits or certain algebraic manipulations, we might explore behavior around zero, but division by zero itself remains undefined.

5. How does division by zero affect equations and functions?

Division by zero can lead to undefined points in equations and functions, which can create vertical asymptotes in graphs. This means that the function does not have a value at that point and can lead to discontinuities. Understanding where these points occur is crucial for analyzing the behavior of functions and ensuring accurate mathematical modeling.

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