Dividing 1/infinity does not exist because infinity is not a real number. However, we can find a way to target this problem that is valid and acceptable. Read this complete guide to find out the solution to this problem.
How To Solve 1/Infinity?
Solving $1/\infty$ is the same as solving for the limit of $1/x$ as $x$ approaches infinity, so using the definition of limit, 1 divided by infinity is equal to $0$. Now, we want to know the answer when we divide 1 by infinity, denoted as $1/\infty$, which we know does not exist since there exists no number that is largest among all. However, if we will use the definition of a limit of a function and evaluate the function $1/x$, where $x$ becomes larger and larger, we will see that the function $1/x$ approaches a particular number.
The following table, Table 1, shows the value of $1/x$ as $x$ gets larger and larger.
Table 1 shows that as $x$ gets larger and larger or as $x$ gets closer and closer to infinity, $1/x$ becomes closer to the value of $0$. We can verify this behavior using the graph of the function of $1/x$.
We can see from the graph of $1/x$ that as $x$ approaches infinity, $f(x)=1/x$ approaches $0$. Therefore, solving $1/\infty$ is the same as solving for the limit of $1/x$ as $x$ approaches infinity. Thus, using the definition of limit, 1 divided by infinity is equal to $0$.
Henceforth, we will consider infinity not as a real number where usual mathematical operations can be normally performed. Instead, when we are working with ∞, we make use of this as a representation of a number that increases without bound. Thus, we interpret it as how a certain function will behave when the value of x approaches infinity or increases without bound. We will study some other operations or expressions that work around infinity.
What Is Infinity?
Infinity is a mathematical concept or term used to represent a very large real number since we cannot find the largest real number. Note that real numbers are infinite. In mathematics, they use infinity to represent the largest number among the set of real numbers, which we know does not exist. The symbol for infinity is $\infty$.
Importance in Mathematics
When we are talking about the largest number, we can notice that we can’t find a specific number or a natural number that is greater than all of the natural numbers.
- $1,000,000$ is a large number, but we can find a larger number than this, which is $1,000,001$.
- $1,000,000,000$ is also a large number, but we can, again, find a number larger than this, which is $1,000,000,001$.
- $10^{100000000000000000}$ is a very large number, still, we can find another larger number than this, we just need to add 1 to it, and we already have one.
So, no matter how big the number that we have, there always exists a bigger number. Since we can never locate the largest real number, we use infinity instead to represent these very large numbers. Hence, infinity is not a real number since we will never find the largest real number.
Dividing by Infinity
We already know that $1/\infty$ is zero Now, for the case of $2/\infty$, $0/\infty$, $-10/\infty$, or $\infty/\infty$, will we still get zero? When the numerator is greater than 1 or less than 1, will the expression be still equal to zero? For the first three expressions, the answer is yes. However, the last expression, $\infty/\infty$, has a different answer, which we will tackle later.
Now, let’s try to solve $2/\infty$. Note that we can express this as the limit of $2/x$ as $x$ approaches infinity. So we have:
\begin{align*}
\dfrac{2}{\infty}&=\lim_{x\to\infty}\dfrac{2}{x}\\
&=\lim_{x\to\infty}\dfrac{2\cdot1}{x}\\
&=2\cdot\lim_{x\to\infty}\dfrac{1}{x}.
\end{align*}
We use the earlier information we gathered that $\lim_{x\to\infty}\dfrac{1}{x}$ is equal to zero. Thus, we have:
\begin{align*}
\dfrac{2}{\infty}=2\cdot0=0.
\end{align*}
Therefore, $2/\infty$ is also zero.
Similarly, since:
\begin{align*}
\dfrac{0}{\infty}&=0\cdot\left(\dfrac{1}{\infty}\right)\\
-\dfrac{10}{\infty}&=-10\cdot\left(\dfrac{1}{\infty}\right),
\end{align*}
then we get that both $0/\infty$ and $-10/\infty$ are equal also to zero. In general, for any real number $c$,
\begin{align*}
\dfrac{c}{\infty}=0.
\end{align*}
Take note that in this generalization, we mentioned that $c$ should be a real number so that $c/\infty$ is zero. Thus, since infinity is not a real number, then $\infty/\infty$ is not equal to zero.
Addition, Subtraction, and Multiplication of Infinities
We can now start to use the term “extremely large number” when referring to infinity so that we can understand better how to perform these operations with infinities.
Note that adding to infinities is like adding to very extremely large numbers. So what happens when we add two extremely large numbers? We still get an extremely large number. Thus,
\begin{align*}
\infty +\infty =\infty.
\end{align*}
Moreover, multiplying two infinities can similarly also be put in this way. If we already have a very large number and we take another very large number, and multiply it with the first very large number, then the product will also be a very large number. Thus, in the same way,
\begin{align*}
\infty \times\infty =\infty
\end{align*}
Now, looking at the difference between two infinities, we have two very extremely large numbers. Since these very large numbers are undefined or just a representation of a very large number, then we will never know if the two very large numbers are equal or if one of the very large numbers exceeds the other. Thus, infinity minus infinity is undefined.
\begin{align*}
\infty – \infty = \text{undefined}
\end{align*}
What Is Infinity Divided by Infinity?
Infinity divided by infinity is undefined, meaning it is not equal to any real number. Since infinity divided by infinity is definitely not equal to zero, we can answer right away that it is equal to 1 because the numerator and denominator are the same. In fundamental operations, we know that any number, except 0, when divided by itself, equals one. That is, whenever a is a nonzero real number, we have:
\begin{align*}
\dfrac{a}{a}=1.
\end{align*}
However, this rule does not apply in the case of $\infty/\infty$ because infinity is not a real number. So we find another way to show that infinity divided by infinity is indeed undefined. We use the information we obtained in the previous section.
We assume that $\infty/\infty=1$. Then, we use the fact that $\infty+\infty=\infty$. So, we have:
\begin{align*}
\dfrac{\infty}{\infty}&=\dfrac{\left(\infty+\infty\right)}{\infty}\\
&=\dfrac{\infty}{\infty}+\dfrac{\infty}{\infty}\\
\end{align*}
Since $\infty/\infty=1$, then this should be true:
\begin{align*}
\dfrac{\infty}{\infty}&=\dfrac{\infty}{\infty}+\dfrac{\infty}{\infty}\\
1&=1+1\\
1&=2.
\end{align*}
This is a contradiction because 1 will never be equal to 2. Thus,$\infty/\infty$ is undefined.
Infinity Divided by a Real Number
In the case where the numerator is infinity and the denominator is a real number, say $c$, then
\begin{align*}
\dfrac{\infty}{c}=\infty.
\end{align*}
Note that this only holds for nonzero real numbers. Consider a very large number divided into finite parts. Then, each part or share is still a large number since the initial number is extremely large.
What is 1/infinity^2?
The expression $1/\infty^2$ is equal to zero. This is because
\begin{align*}
\dfrac{1}{\infty^2}&=\dfrac{1}{\left(\infty\times\infty\right)}\\
&=\dfrac{1}{\infty}\\
&=0.
\end{align*}
Is 1^Infinity = e?
The answer to this question is not always. The expression $1^{\infty}$ is considered one of the indeterminate forms, meaning that it will have different answers depending on which situation it was used. Take note that expressions with infinity can be taken as an expression to represent a limit of a certain function where $x$ approaches infinity.
Thus, in the case of limits that will give a $1^{\infty}$, different methods can be used to move forward from this indeterminate form and derive a limit for the function as $x$ increases without bound.
Evaluate e^Infinity
In solving for $e^{\infty}$, we get that this expression is also equal to infinity. Here’s how we arrived at that answer. Note that $e$ is a real number greater than one. Thus, expanding $e^{\infty}$, we have:\begin{align*}e^{\infty} = e\times e\times e\times\dots\times e\times e\times \dots.\end{align*}This means that $e^{\infty}$ we multiply $e$ by itself infinitely many times. Since $e$ is greater than 1, then the powers of $e$ will just increase without bound as the powers of $e$ get multiplied by e many more times. Therefore, $e^{\infty}$ is equal to infinity.
Conclusion
Infinity is a mathematical term, concept, or symbol that is oftentimes carelessly utilized in mathematical solutions, especially in limit-finding problems. Let’s recall the important notes we learned in this discussion.
- Infinity is not a real number and is only used as a representation for an extremely large real number.
- Dividing 1 by infinity is equal to zero.
- In general, any real number divided by infinity is zero, and the quotient of nonzero real numbers that divide infinity is infinity.
- The sum and product of two infinities are equal to infinity, while the difference and quotient of two infinities are undefined.
- $1^{\infty}$ is an indeterminate form.
In this article, we defined infinity in a clearer manner and used it to perform operations and evaluate expressions with infinities.
FAQs
What is the answer of 1 divided by infinity? ›
Also, in some calculators such as the TI-Nspire, 1 divided by infinity can be evaluated as 0.
Why is 1 divided by infinity equal to 0? ›Infinity is a concept, not a number; therefore, the expression 1/infinity is actually undefined.
How do you divide a number by infinity? ›Answer and Explanation: Any number divided by infinity is equal to 0.
What is negative 1 divided by infinity? ›Answer and Explanation: −1∞ is 0. Any value divided by infinity is 0 except infinity divided by infinity, which is undefined.
Is 1 divided by 0 infinity or undefined? ›In mathematics, anything divided by zero is not defined (and not infinity).
What is the value of 1 divided by 0 infinity? ›We can say that the division by the number 0 is undefined among the set of real numbers. ∴ The result of 1 divided by 0 is undefined. Note: We must remember that the value of 1 divided by 0 is infinity only in the case of limits. The word infinity signifies the length of the number.
When a number is divided by infinity What is zero? ›He argued that a quantity divided by zero becomes an infinite quantity. This idea persisted for centuries, for example, in 1656, the English mathematician John Wallis likewise argued that 24 ÷ 0 = ∞, introducing this curvy symbol for infinity.
Is 1 to infinity indeterminate? ›We first learned that 1^infinity is an indeterminate form, meaning that a limit can't be figured out only by looking at the limits of functions on their own.
What if zero is divided by infinity? ›If zero is divided by infinity, the result is 0.
Why is infinity divided by infinity not 1? ›The reason you cannot say that infinity divided by infinity is 1 is because infinities can be seen as having (potentially) different values (although I say that lightly) so we don't know the 'value' of one infinity with respect to the other.
Why is infinity divided by infinity 1? ›
Infinity divided by infinity is undefined, meaning it is not equal to any real number. Since infinity divided by infinity is definitely not equal to zero, we can answer right away that it is equal to 1 because the numerator and denominator are the same.
What is 1 raised to infinity? ›1 raised to infinity is 1 …. Infact 1 raised to any number is always 1. 1 raised to infinity means 1 is multiplied with 1 infinte times , which will be equal to 1 only.
What is the value of 1 minus infinite? ›Negative infinity means that it gets arbitrarily smaller than any number you can give. so 1 - infinity = -infinity and 1 + infinity = + infinity makes sense only when looked as in this sense.
What is divided by negative infinity? ›Negative infinity results in 0 when divided by any other number.
Can you do infinity minus 1? ›If you subtract 1 from infinity, answer will be infinity because infinity is very large number and subtracting 1 won't affect the value much.
Is divided by 0 negative infinity? ›In IEEE 754 arithmetic, a ÷ +0 is positive infinity when a is positive, negative infinity when a is negative, and NaN when a = ±0. The infinity signs change when dividing by −0 instead.
Is infinity divided by infinity 1? ›Therefore, infinity divided by infinity is NOT equal to one. Instead, we can get any real number to equal to one when we assume infinity divided by infinity is equal to one, so infinity divided by infinity is undefined.
What is infinity divided by 2? ›And the infinity is such a number that when we divide it by 2,then it remains infinity. The concept around infinity revolves around endlessness. Hence, any “fraction” of infinity is infinite.
What number is half of infinity? ›It's infinite. One way to look at it is to realize that if you added two finite things together, the answer is finite, so 1/2 of infinity cannot be finite, hence infinite.
Do numbers never end? ›The sequence of natural numbers never ends, and is infinite. OK, 1/3 is a finite number (it is not infinite). There's no reason why the 3s should ever stop: they repeat infinitely. So, when we see a number like "0.999..." (i.e. a decimal number with an infinite series of 9s), there is no end to the number of 9s.
Why is 0 divided by 0 not infinity? ›
The reason that the result of a division by zero is undefined is the fact that any attempt at a definition leads to a contradiction. a=r*b. r*0=a. (1) But r*0=0 for all numbers r, and so unless a=0 there is no solution of equation (1).
Is a number divided by infinity indeterminate? ›Infinity is not a number, so cannot be divided, mathematically, by ANY number. Division by zero is undefined.
What is infinity divided by infinity called? ›answered Nov 6, 2021 by I've just been taught this. Now, it is easy to think that any number divided by itself equals one, which is true. BUT in Mathematics infinity divided by infinity is actually undefined.
Can you use L Hopital's rule for 1 infinity? ›L'Hopitals can applied in case of indeterminate forms which are 0/0, 0^0,infinity×0, 1^infinity, infinity×0, infinity/infinity,infinity- infinity.
Is 1 infinity divergent? ›Therefore and your question is equivalent to asking if the integral of 1 from 1 to infinity is convergent or divergent ! Of course, it goes to infinity so it is divergent.
What is the value of e ∞? ›Hence, the value of e - ∞ is 0 .
What is bigger than infinity? ›Infinity is endless and therefore cannot be reached. The expressions “beyond infinity” or “to infinity and beyond" simply represent limitless possibilities.
What is square root of infinity? ›⇒∞ Ans (square root of infinity is infinity)
Is there such thing as infinity 1? ›Yet even this relatively modest version of infinity has many bizarre properties, including being so vast that it remains the same, no matter how big a number is added to it (including another infinity). So infinity plus one is still infinity.
What is 1 divided by infinity squared? ›That means as the number increases, 1/x decreases so fast and it nears zero. So, when when you take 1/infinity it nears zero, so we take it as zero. And so does 1/infinity^2.
Is the square root of 1 infinity? ›
Answer and Explanation: The square root of infinity is infinity. If you choose a number and multiply it by itself, you would have squared the number.
What is negative infinite? ›Similarly, there is a concept called negative infinity, which is less than any real number. The symbol “-∞” is used to denote negative infinity. Take a close look at this visual display of infinity — positive and negative infinity ride along an infinitely long zip line in opposite directions.
What is negative infinity on a calculator? ›To specify negative infinity, input -1E99. The "E" symbol is short for scientific notation and can be accessed by pressing [2nd] [EE]. -1E99 can be interpreted as "negative one times 10 to the ninety-ninth power".
What is 5 /- infinity? ›Hence, 5 divided by infinity is 0. Alternatively, we know that any number divided by 5 is equal to 0.
What is the value of 1 /- infinity? ›so 1 - infinity = -infinity and 1 + infinity = + infinity makes sense only when looked as in this sense. Addition and subtraction are operations that are only defined for real numbers (or some other algebraic structure) and infinity is not a real number.
Is infinite divided by infinite 1? ›Now, it is easy to think that any number divided by itself equals one, which is true. BUT in Mathematics infinity divided by infinity is actually undefined.
What is the answer of 0 divided by infinity? ›Regardless of what large number we're dividing by our answer is 0 and by letting this large number increase (as much as we please, tending to infinity) the answer is still 0.
What is 1 multiplied by infinity? ›If a number is multiplied by infinity, then the value of the product is also equal to infinity.
What happens if you add 1 to infinity? ›. If you add one to infinity, you still have infinity; you don't have a bigger number.
Can you have infinity 1? ›Yet even this relatively modest version of infinity has many bizarre properties, including being so vast that it remains the same, no matter how big a number is added to it (including another infinity). So infinity plus one is still infinity.
Does infinity divided by 0 exist? ›
Answer and Explanation: We cannot really divide infinity by zero because infinity is not a number and we do not divide by zero.
What is 5 divided infinity? ›Answer: Evaluate the value of 5 divided by infinity. Hence, 5 divided by infinity is 0. Alternatively, we know that any number divided by 5 is equal to 0.
Is infinity minus 1 still infinity? ›Infinity is a number that never ends, so subtracting it by one or any integral would still leave you with infinity.
What is the answer of infinity divided by 2? ›And the infinity is such a number that when we divide it by 2,then it remains infinity.
Is 1 infinity convergent or divergent? ›The infinite arithmetic series is divergent. This is true for all infinite arithmetic series!
Is infinity even or uneven? ›Non-integer numbers like 1/2, 4.201, or infinity are neither even nor odd.
What is divergence to infinity? ›A sequence is said to diverge to infinity if it diverges to either positive or negative infinity. In practice we want to think of |r| as a very large number. This definition says that a sequence diverges to infinity if it becomes arbitrarily large as n increases, and similarly for divergence to negative infinity.