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Difference Between Infinity and Not Defined

July 12, 2022

Most of us use “infinity” and “not defined” interchangeably. But the fact is  these two have different meanings. These are more frequently used concepts in many fields, especially in Mathematics and Physics.

What is the Meaning of Infinity?

If someone asks you to think of a largest possible number. What is the largest number that you can imagine?

1000000000, or 999999999999, or even 9999999999999999999?

Whatever the number you pick, one can introduce a number larger than the number you pick by simply adding one to that.

1000000000 + 1 = 1000000001, so 1000000000 cannot be largest

Similarly,  999999999999 + 1 = 1000000000000

And, for 9999999999999999999 also, 9999999999999999999 + 1 = 10000000000000000000

So, we observe here that for any possible largest number there exists even a larger number. Then again back to the same question – “What is the largest number?”

The concept ‘infinity’ could have been introduced as an answer to this question. The term ‘infinity’ comes from the Latin word “infinitas”, which means “unboundedness”.

By definition, there is no number larger than infinity.  However, infinity is not the largest number, because it is not a number and it does not have an exact value.

To understand this, consider this example.

In set theory, the set of natural numbers, set of integers, and set of real numbers are said to be infinite sets, because all these sets contain infinitely many numbers. It is clear that the set of integers contains more numbers than the set of natural numbers and the real numbers contain more elements than in the set of integers. In other words, it is possible for one infinite set to contain more (or less) elements than another infinite set.

Therefore, it should be clearly understood that the concept of infinity varies depending on the subject area that we are talking about. Infinity has various applications in mathematics; in set theory, calculus and so many other fields.

If we consider distances at atomic or molecular level, then even a distance of a centimetre or a millimetre is infinite, whereas if we consider astronomical distances, then even distances of the order of a lakh or a million kilometre are  not infinite. It’s because infinity is just a concept and not an exact value.

What is Undefined?

What is the value when you divide any number by zero? Is it infinity? If you are a physicist, it can be zero depending on the theory you are applying it to. However, if you are a mathematician, it is undefined.

For another example; what would be the logarithm of a negative number? Since we cannot find a number $x$, such that $n^{x} = -r$, where $n$ and $r$ are integers; we would say the logarithm of a negative number is undefined.

Mathematically, “undefined” can be defined as an expression which is impossible, or an expression which does not have an exact definition, or an expression that cannot be interpreted. However, something that is undefined today, may be given a definition in the future. Then it does not remain undefined at that point of time.

For example, in lower grades you are taught that the square root of a negative number is undefined. But as you move on to higher grades, you are taught that the $\sqrt{-1}$ is not undefined, rather it  is an imaginary number $i$. $\left(i = \sqrt{-1} \right)$.

Conclusion

“Infinity” is just a concept and refers to something that is very large and “undefined” as something that cannot be evaluated or has a meaning under a given situation or application.