While you cannot divide by zero, division by zero has numerous mathematical meanings depending on the scenario. For example, in calculus, division by zero results in discontinuities in a graph. This wikiHow will show you how to recognize and interpret division by zero.
Steps
Method 1
Method 1 of 2:
Understanding Division by Zero
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Acknowledge that dividing by zero produces an undefined result. When reaching a result that is any nonzero number divided by zero in algebra, this value is considered undefined, as division by zero is not possible. Alternatively, dividing zero by itself results in an “indeterminate” value, since, while any other number divided by itself equals one, any number times zero yields zero, but functions cannot map the same input to multiple outputs.
- Indeterminate form does not just result from performing , it can also result if any of your expressions evaluate to , , , , , or .
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Know when division by zero is inevitable. Division by zero occurs when an expression in the denominator simplifies to zero. Since there is no number that, when multiplied by zero, will yield the numerator for all numbers except zero, division by zero is undefined except when the numerator is zero.Advertisement
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Understand the different ways division by zero can be expressed. There are several ways that division by zero can be expressed, not just by writing zero in the denominator. For example, zero can be expressed as any number subtracted from itself, or it can be the result of zero times something else. Many complicated expressions will evaluate to zero for specific values. Regardless, division by zero is not always obvious.
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Check the numerator if you are unsure what division by zero means for a particular function. If a function is undefined for one real input, then it is said to be discontinuous . Discontinuous functions will appear to have a hole or will appear as two separate lines rather than one connected line when drawn.
- If the numerator is nonzero for an input, then there will be an asymptote at that particular location. An asymptote will have the line approach positive or negative infinity as the particular value gets closer to the asymptote location from either side. The approached value of this function is referred to as the limit of the function.
- If the numerator is zero for an input, then there will be a hole at that particular location. The precise location of this hole can be determined by finding particular factors of the numerator and denominator.
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Method 2
Method 2 of 2:
Avoiding Division by Zero
In some circumstances resulting in division by zero using one method, it may be possible to manipulate an equation or expression to resolve this.
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1Factor the numerator and denominator to attempt to find cancelling terms. You may find specific expressions that cancel each other out from the numerator and denominator. If you do, then you may be able to resolve discontinuities in the function.
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2Set the location of a hole as the input to determine what the value would be at this particular point. This is the precise location of the hole.
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3Attempt different methods when finding limits . When looking for the limit of a function, and you find that direct substitution results in the indeterminate form of zero divided by itself, use methods such as factoring, multiplying radical expressions by their conjugate, transforming trigonometric functions, or finding the derivative with L'Hôpital's rule. These methods allow you to manipulate the expression in such a way that shifts away from division by zero while still remaining equivalent to the initial function.Advertisement
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