# Is infinity equal to undefined?

## Undefined

**Undefined** is a term used when a mathematical result has no meaning.

More precisely, undefined «values» occur when an expression is evaluated for input values outside of its domain.

- − 9 >> (If no complex numbers)
- ln ( − 4 ) (If no complex numbers)
- tan ( π / 2 ) (Units in radians, no complex infinity)
- n 0 >> (If no complex infinity) Visit Division by zero for more info.
- x 0
> (This would mean to raise a number to 1 0 >> or find a number that when raised to 0, equals x . Even if we were to include complex infinity, any number raised to complex infinity is undefined.) Visit Zeroth Root for more details.

In the above two examples, -4 and -9 are both outside of the domain over which the square root and natural logarithm functions are defined. Therefore, the result is «undefined».

Upon extending the functions definition to include a domain and range containing the complex numbers, we find that undefined only occurs upon a division by zero — or zero to the power of zero — because a logical and mathematical contradiction results. It must necessarily be excluded from possibility. See division by zero for more information and examples on this. No contradiction results if one allows for complex values, thus much of the domain and range we excluded earlier in simpler algebra can now be re-included.

Allowing for complex values, we find that the above example expressions equal the complex values 3 i or − 3 i and ln ( 4 ) + i ⋅ π .

Thus, until complex numbers are introduced and functions extended to include them (in higher levels of algebra), most functions are restricted to their real number domains which yield a real number range, and any number outside of that restriction is summarily regarded as «undefined».

In truth, however, all mathematical concepts included, division by zero is one of the few cases where a true **undefined** occurs. In such instances, the input value which results in a division by zero should have been left out of the domain. Thus, even division by zero is a special case of the evaluation of an expression outside of its domain.

If **x=0** then 1 x = 1 0 >=>> . This is undefined in the truest sense of the word because, even if you allow for complex values, the expression still has no meaning if one doesn’t allow complex infinity.

## Contents

- 1 Indeterminate Values
- 2 Calculus and the Evaluation of Limits
- 3 Graphical Properties of Undefined and Indeterminate
- 4 Expressions with an undefined subexpression

## Indeterminate Values

Now suppose that **a** was zero. We now have 0 0 = b >=b> . Algebra will now show that 0 = 0 ⋅ b . The same zero identity guarantees that **0b** equals zero regardless of what b is. The equation is not a contradiction any longer, its a theorem and is true for all values of **b**. Thus, no value of **b** is *determinable* from the equation as all values of **b** hold true.

## Calculus and the Evaluation of Limits

Though the arithmetic evaluation of divisions by zero are mathematically impossible, producing either an indeterminate or undefined value, the study of calculus has necessitated their treatment with the use of limits.

Limits allow us to study the properties and trends of expressions at infinite and infinitesimal values. Whereas we cannot evaluate an expression that divides by zero, we may analyze the expression for how it operates «near» a division by zero.

Division by zero is always undefined or indeterminate, such as 3 ( 1 − 1 ) 2 = 3 0 >>=>> . However, upon introducing the limit, we find that lim x → 1 3 ( x − 1 ) 2 >>> is evaluable and in fact the expression «*approaches*» infinity.

Limits do not say *what is* at a given value. Limits only describe how things are *trending* **near** a given value.

Thus, even in calculus and the implementation of limits, division by zero is still not permitted. The function may be described, however, near certain values. Limits never technically **equal** anything. They **tend toward**.

A common misconception in amateurs is that the simple arithmetic operation 1 0 >> is evaluable as «infinity», or otherwise explainable with limits. They fail to realize that introducing limits changes the question entirely.

*How does this section pertain to indeterminate and undefined expressions?*

In the case of undefined expressions, limits that evaluate them will tend to infinity or negative infinity. In the case of indeterminate expressions, however, the limits that evaluate them will tend toward a finite real value.

- An undefined a/0 form: lim x → 1 3 ( x − 1 ) 2 = ∞ >>=infty >
- An indeterminate 0/0 form: lim x → 1 3 ( x − 1 ) x − 1 = 3 >=3>

## Graphical Properties of Undefined and Indeterminate

The graphs of functions which have undefined and indeterminate values equally illustrate the distinctions between the two.

As stated in the previous section, undefined values *tend* toward infinity. Consequently, a graphical illustration of undefined shows a vertical asymptote, on either one or both sides of which the function increases in magnitude without bound. The function never crosses or touches the asymptote (by definition) and thus there is no value — not even infinite — **at** the asymptote.

Indeterminate values are a bit different. On a graph they will not have a vertical asymptote at all as the indeterminate value. Instead, there will be a point-gap: a single point which is excluded from the domain and has no range- or output-value. The function on either side of the point-gap will approach one or two distinct finite values.

In some cases, a function may be asymptotic and thus undefined when approaching a domain-value from one side, but when approached from the opposite side it is indeterminate and thus a finite real value is approached. These functions tend to be Expressions with an undefined subexpression

As a general rule, if any part of an expression is undefined (for example, 3 + 1 0 >> ), then the entire expression is undefined. There can be arguments for there to be exceptions to this rule, such as 1 0 / 0 > being equal to 1 , since 1 raised to any exponent equals 1 , and 0 ( | 0 / 0 | ) + 1 > , since 1 + the absolute value of anything is always at least 1 , and 0 raised to any positive power always equals 0 .

## Is a number divided by 0 infinity?

No, a number divided by 0 is not infinity. Infinity is a concept of something that is boundless and unending. A number divided by 0 is undefined, not infinite. It is impossible to divide by 0, so when we do, the result does not exist and cannot be defined.

When a number is divided by 0, it is represented as either undefined or not a number (NaN). The expression of any number divided by 0 is undefined and cannot be described as infinite.

Table of Contents

## Why division by zero is not infinity?

Division by zero is not equal to infinity because infinity is not a real number; it is an idea of something that has no end or beginning. When you divide a number by zero, it produces a form of undefined result because the denominator in a mathematical equation represents how many of the same unit you have—in this case, zero.

Because there is no number that can exist in the denominator of a fraction, it means that there is no number that you can divide something by to get infinity. Furthermore, some mathematicians and scientists consider division by zero to be “undefined,” meaning that it is not possible to meaningfully define what such an operation would mean.

## Is divide by 0 infinity or undefined?

Divide by zero is neither infinity nor undefined. Dividing by zero does not yield an answer that can be represented on the number line. It is not possible to divide a number by zero, as it would result in an infinite quantity.

It is, therefore, impossible to assign a definite value to the result of dividing a number by zero. So, it is neither infinity nor undefined.

## Is divided by 0 possible?

No, it is not possible to divide by zero. Division is defined by taking a given number and splitting it into smaller equal groups. Since zero is an absence of value and can’t be split into any groups, it is impossible to divide by zero.

In mathematics, any attempt to divide by zero is undefined and will lead to an error. This is because division operations utilize multiplying and dividing by the same number to return the original number.

However, when that number is zero, the operation becomes impossible because any number multiplied by zero is still zero.

## What happens to anything divided by 0?

Dividing by 0 is an operation that is not possible. When attempting to divide a number by 0, the result is undefined and not a real number. In some cases, a computer can interpret division by 0 as an error and will usually return an error message.

While some mathematicians believe that division by 0 is possible, results such as infinity or negative infinity can not be used in the same way that real numbers can be used. Therefore, division by 0 is not considered to be a valid mathematical operation.

## What kind of error is divide by 0?

Divide by 0 is an arithmetic error, specifically an ‘undefined result’ error. This occurs when a number is divided by 0 and the outcome is undefined due to the impossibility of dividing any number by 0.

Mathematically, it is impossible to divide any number by 0 as it will produce an indeterminate form (i. e. infinity) which is not a valid number. As a result, divide by 0 will most commonly lead to an error in programming languages which is usually termed “divide by zero” error.

Usually in programming languages, a divide by 0 will lead to a runtime (or logical) error, as the program will not be able to complete the given task with an undefined result.

## Is dividing by zero a fatal error?

No, dividing by zero is not a fatal error. A “fatal error” typically means that a program or hardware device has encountered a critical condition and is unable to continue functioning. Division by zero can be a logical error, which means that it produces an unintended result.

This can cause programs to become stuck in an infinite loop, but it doesn’t typically cause the entire program to crash. Depending on the programming language and environment, the program may throw an exception, output a warning, or simply output an undefined or unexpected result.

## What does Siri say when you say 0 0?

Siri responds to the command “0 0” with a simple response: “Everything is fine. ” This response is a reflection of the Siri’s helpful and friendly personality. Despite its AI nature, Siri is designed to provide a sense of humanity and comfort when encountering an unusual request from a user.

It acknowledges the request with a friendly response and reassures them that the command will not to cause any harm. Additionally, if the user wishes to dive deeper into the request, Siri will provide additional information about the command and what it means.

## Is 0 0 1 or infinity?

0 0 1 (i. e. , “000 1”) is not infinity. 0 0 1 is actually a numerical expression that evaluates to 1. It is simply a shorthand for writing 1 quickly. Mathematically speaking, 0 0 1 is equal to the expression “1*10^0”.

In other words, 0 0 1 is the same as writing out the number one in scientific notation (1*10^0). Since 1 is a finite number, it is not infinity.

The phrase “0 0 1” can also be a bit confusing, as it appears to have three parts (the zeros and 1 in the middle). However, the two zeros preceding the 1 actually do not add anything to the expression, and can be omitted.

In other words, 0 0 1 and 1 are the same thing.

## Do numbers never end?

No, numbers do not never end. While the concept of numbers is often thought to be limitless, mathematics actually uses something called natural numbers, which have a highest number. Natural numbers are the set of non-negative integers (whole numbers) starting with 1, 0, then counting up by one (1, 2, 3 etc).

So while numbers may seem to go on forever, they actually reach a maximum point. There is an ongoing debate in the mathematics world on whether numbers can truly be infinite, but the consensus is that there is an end to the set of natural numbers.

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