In order to give a satisfactory response to the above question we need to understand the meanings of two words used in the above question – “before” and “infinity”. Once the meanings of these two terms are clarified we may attempt to answer the above question.
What does it mean to find a number “before” another number?
Suppose we are asked the following question – What is the number before 4? Most of us would answer the question by saying that the number before 4 is equal to 3.
Technically speaking this answer may not be completely correct. This is because even fractions are considered to be numbers and 3.5 also occurs before 4. So, another person might answer that the number before 4 is 3.5 or even 3.9!!!
Therefore, in order to make this question precise, we will stipulate that whenever we ask for a number before “x” we wish to find the number “x-1”?
What is infinity?
The usual numbers such as integers, fractions, negative numbers, etc. collectively form a set. These numbers that we are familiar with form the set of real numbers.
The number infinity is defined to be the number that is greater than every real number. It is denoted by the symbol \infty. Since \infty is itself not a real number it is considered to be a part of the set of extended real numbers.
The number \infty obeys the following laws pertaining to addition, subtraction, multiplication, and division:
- \infty + \infty = \infty
- \infty \times \infty = \infty
- \infty + n = \infty for any finite number n.
- \infty - n = \infty for any finite number n. This is because subtracting a finite number from infinity does not make any difference to infinity itself.
What is the number before infinity?
We can now answer the above question as follows. By the number before \infty we mean the number \infty - 1. But as we can see from the rules above that subtracting a single finite number, that is 1, from the much bigger infinity does not make any difference in its value. We conclude that, \infty - 1 = \infty
Therefore from the above equation, we can say that the number before infinity is actually infinity itself. From another perspective, we may say that the notions of “before” and “after” make sense only for real numbers and not the extended real numbers. So from this perspective, it would also be correct to say that there is no number before infinity since the question does not make sense.