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.
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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
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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.
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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
Solving 1 Divided by Infinity? ›
Dividing 1/infinity does not exist because infinity is not a real number.
What is the answer of 1 divided by infinity? ›Infinity is a concept, not a number; therefore, the expression 1/infinity is actually undefined.
What is the answer if any number divided by infinity? ›Answer and Explanation: Any number divided by infinity is equal to 0.
Is 1 divided by 0 infinity or undefined? ›In mathematics, expressions like 1/0 are undefined. But the limit of the expression 1/x as x tends to zero is infinity. Similarly, expressions like 0/0 are undefined. But the limit of some expressions may take such forms when the variable takes a certain value and these are called indeterminate.
What is negative 1 divided by infinity? ›− 1 ∞ is 0. Any value divided by infinity is 0 except infinity divided by infinity, which is undefined.
Why anything divided by 0 is infinity? ›As much as we would like to have an answer for "what's 1 divided by 0?" it's sadly impossible to have an answer. The reason, in short, is that whatever we may answer, we will then have to agree that that answer times 0 equals to 1, and that cannot be true, because anything times 0 is 0. Created by Sal Khan.
Can you have infinity plus 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.
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 1 raised to infinity? ›Scaling by 1 does not change the number, so 1∞=1.
What if infinity is divided by 2? ›Infinity divided by anything that is finite and non-zero is infinity.
Is 1 divided by 0 infinity or 0? ›
We can say that the division by the number 0 is undefined among the set of real numbers. $\therefore$ 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.
Can you do 0 divided by infinity? ›Zero over Infinity
If zero is divided by infinity, the result is 0.
Just say that it equals "undefined." In summary with all of this, we can say that zero over 1 equals zero. We can say that zero over zero equals "undefined." And of course, last but not least, that we're a lot of times faced with, is 1 divided by zero, which is still undefined.
Is 1 over 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 is infinity divided by 5? ›Hence, 5 divided by infinity is 0. Alternatively, we know that any number divided by 5 is equal to 0.
Is Infinite a real number? ›Infinity is a "real" and useful concept. However, infinity is not a member of the mathematically defined set of "real numbers" and, therefore, it is not a number on the real number line.
Why is 4 divided by 0 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).
Why is 6 divided by 0 infinity? ›Let us take the denominator of LHS to RHS. The above form is not possible because 6 is not equal to 0. Therefore, any number when divided by 0 is undefined except for 0. Therefore, the result when 6 is divided by 0 is undefined.
Why is 9 divided by 0 infinity? ›Mathematicians say that "division by 0 is undefined", meaning there is no way to define an answer to the question in any reasonable or consistent manner.
Is Ω the biggest number? ›The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite.
What is the biggest number ever? ›
A googolplex is 10 raised to the power of a googol, that is it's one followed by a googol of zeroes." A googolplex is so large, there is not enough matter in existence to write it longhand. As numbers increase towards infinity, mathematicians know less and less about them.
What is the last number on earth? ›There is no biggest, last number … except infinity. Except infinity isn't a number. But some infinities are literally bigger than others.
What is bigger than infinity? ›Mathematically, if we see infinity is the unimaginable end of the number line. As no number is imagined beyond it(no real number is larger than infinity).
Is there a 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.
Can you split infinity in half? ›If you say half infinity, you are wrong. Infinity is not a number. It is a concept of numbers without end. Thus, there can be no half way point without an end.
What is infinity 0? ›Answer: Infinity to the power of zero is equal to one.
Let's understand the solution in detail. Explanation: ∞0 is an indeterminate form, that is, the value can't be determined exactly.
The number 'e' is known as Euler's number which is an irrational number. It is a numerical constant having a value of 2.718281828459045..so on, or you can say e∞ is equal to ( 2.71…) ∞.
What is the e in math? ›An irrational number represented by the letter e, Euler's number is 2.71828..., where the digits go on forever in a series that never ends or repeats (similar to pi). Euler's number is used in everything from explaining exponential growth to radioactive decay.
Is half of infinity still infinity? ›In other words, you can place all the even numbers and all the natural numbers side by side in two columns and both columns will go to infinity, but they are the same "length" of infinity. That means that half of countable infinity is still infinity.
What is tan 1 infinity? ›What is the Value of tan-1 Infinity? To calculate the value of the tan inverse of infinity(∞), we have to check the trigonometry table. From the table we know, the tangent of angle π/2 or 90° is equal to infinity, i.e., tan 90° = ∞ or tan π/2 = ∞ Therefore, tan-1 (∞) = π/2 or tan-1 (∞) = 90°
Who discovered infinity? ›
Srinivasa Ramanujan (1887-1920), the man who reshaped twentieth-century mathematics with his various contributions in several mathematical domains, including mathematical analysis, infinite series, continued fractions, number theory, and game theory is recognized as one of history's greatest mathematicians.
Why can't we divide by 0? ›The short answer is that 0 has no multiplicative inverse, and any attempt to define a real number as the multiplicative inverse of 0 would result in the contradiction 0 = 1.
Why is 1.0 0 infinity? ›Mathematically, division by zero is undefined, although it can be loosely be regarded as being infinity. (With a little more rigour, it's a number that is greater than x for any value of x.) An IEEE754 floating point double (used by Java) has a representation of infinity. That is the result of 1.0 / 0.0.
How is 5 divided by 0 infinity? ›We can say that the division by the number 0 is undefined among the set of real numbers. Therefore, the result of 5 divided by 0 is undefined. Note: We must remember that the value of 5 divided by 0 is infinity only in the case of Limits. The word infinity signifies the length of the number.
Can you add infinity to infinity? ›Never. There are an infinite number of different values for infinity, and some are infinitely larger than others, and some are infinitely smaller than others.
Does 1 infinity equal zero? ›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.
Does everything divide 0? ›Every integer divides 0. Proof. Let n be an integer. Then 0 = 0 · n, so that n divides 0.
When 0 is divided by no? ›Zero Divided by a Number:
Dividing 0 by any number will give us a zero. Zero will never change when you multiply or divide any number by it.
One to the power infinity is unknown because infinity itself is endless. Take a look at some examples of indeterminate forms. When we plug infinity into this function, we see that it takes on the indeterminate form of one to the power infinity.
What is the power of infinity? ›Therefore, e to the power of infinity is infinity (∞).
(infinite number of times). We have e = 2.71828 > 1. When we multiply this number by itself an infinite number of times, we can't even imagine how big a number we will obtain and hence e to the power of infinity results in ∞.
What is 2 raised to infinity? ›
Answer: the answer is infinity.
What is infinity 3? ›Infinity 3 is the world's most musical looping pedal. Simple to operate, yet tremendously powerful and flexible, the Infinity 3 guarantees latency-free looping.
What is 1 2 3 4 5 all the way to infinity called? ›For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12.
What is 0.5 times infinity? ›The expression "0.5 * infinite" is not mathematically well-defined. In mathematics, infinity is not a number in the usual sense, but rather a concept that represents an unbounded quantity or a limit.
Why is 1729 a magic number? ›Discovered by mathemagician Srinivas Ramanujan, 1729 is said to be the magic number because it is the sole number which can be expressed as the sum of the cubes of two different sets of numbers. Ramanujan’s conclusions are summed up as under: 1) 10 3 + 9 3 = 1729 and 2) 12 3 + 1 3 = 1729.
Is there negative infinity? ›About Infinity
Similarly, there is a concept called negative infinity, which is less than any real number. The symbol “-∞” is used to denote negative infinity.
It had to be a number 9. It also means that everything either always goes back to its original value or if multiplied by infinity becomes infinity. 9 = ∞.
Why infinity divided by infinity is not equal to 1? ›Conclusion. Since infinity is just a concept and is not a fixed number, hence the operations like addition, subtraction, multiplication, or division do not give similar results as obtained by other numbers when operated by these operations. Infinity over infinity is Undefined.
When 1 is divided by 0 What is the answer? ›Just say that it equals "undefined." In summary with all of this, we can say that zero over 1 equals zero. We can say that zero over zero equals "undefined." And of course, last but not least, that we're a lot of times faced with, is 1 divided by zero, which is still undefined.
Why is 1 divided by 3 infinite? ›No, it just means that decimal notation is unable to represent the value exactly in a finite number of digits. This has nothing to do with the ability to divide one by three, it's just an inconvenience of our notation. You can represent 1/3 in an infinite amount of decimal digits. The threes just go on forever.
Why is infinity 1 0? ›
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. In the case of limits, we only assume that the value of limit x tends to something and not equal to something. So, we consider it infinity.
Why is 1 ∞ not equal to 1? ›Infinite exponentiation means that you take 1 and multiply it by a scaler, a, infitnite number of times. Scaling by a > 1 yields greater and greater number. So, (a>1)∞=∞. Downscaling infinitly by a<1 yields 0: (a<1)∞=0.
What happens if you half 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.
What is 1 infinite? ›We use the terms infinity and - infinity not as a number but to say that it gets arbitrarily large. 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.
Is infinite a real number? ›Infinity is a "real" and useful concept. However, infinity is not a member of the mathematically defined set of "real numbers" and, therefore, it is not a number on the real number line.
Who invented 1 to infinity? ›infinity, the concept of something that is unlimited, endless, without bound. The common symbol for infinity, ∞, was invented by the English mathematician John Wallis in 1655.
Is infinity times 0? ›Any number times 0 equals 0 and any number times infinity equals infinity.