what happened when ∞/∞ = 1 ? : Real Analysis
what heppened when ∞/∞ =1?
" Infinity " , In my earlier post , we have discussed about infinity and its properties.
Come to our question " what happened when ∞ / ∞ = 1 ? "
Assume ∞/∞ = 1 that implies ∞ = ∞.
Let the ∞ in the L.H.S. is the number of water drops in a riverand the ∞ in the R.H.S. is the
number of water drops in a sea. It is impossible to count the number of water drops in a sea as
well as in the river. That why we consider both having infinite water drops. Now as our
assumption if ∞ = ∞ then the number of water drops in a river must equal to the number of
water drops in a sea it results the river and sea must be of equal in size.
Is it true? No. Hence ∞ ≠ ∞ i.e. ∞/∞ ≠ 1.
First of all this question may arise only if ∞ is a number.
Now my question " Is ∞ a mumber ? "
Answer is ∞ is not a number. It is a symbolic representation of non - terminating existance or
commencement of numbers or decimals etc. in a sequence of numbers, things, people etc .
For example : Consider set of natural numbers N = { 1, 2 , 3 , ...∞ } . Basically here no need to
show the symbol ∞ . But it represents the commencement of natural numbers is non - terminating
Hence we can not find the highest natural number ( in other sense upper bound ).
Because of the same reason we can not conclude 1^∞ = 1.
For clarification What is the result of 1^10 ?
1 is multiplying itself 10 times. Here the the process of multiplication is ended after 10 steps and
hence we conclude 1 ^ 10 = 1.
But in case of 1^∞, the multiplication process is not terminating . Thats why we are not came to a
conclusion in the result of 1^∞ as 1.
So in Real Analysis, ∞/∞,1^∞, ∞-∞,∞^∞ , 0^∞ etc are all called the indeterminate forms.
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* what is an interior of a set
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