What is the biggest number smaller than the square of itself?

Answer 1

There is no biggest number.
Any number greater than one is smaller than the square of itself.
For example:
5 < 25
100 < 10000

On the other hand, any number between 0 and 1 is greater than the square of itself:
.5 > .25
.6 > .36

As we get closer and closer to 1, such as .9999 square which is =.99801, we note that the square get closer to the number and of course bigger too.
So if we look at .99999992 we see it is very close to 1. The more 9's we add the closer we get to 1 and in fact and in fact an infinite number of 9's will be 1 one itself.

But there is no "biggest number" that is less than 1. For any number you can think of that is between 0 and 1 (not including 1), it is possible to name a bigger number. This has to do with something called the density of the reals. Actually in this case it is the density of the rationals in the reals. Say you give me a number with a decimal and a million 9's to the left of the decimal, I just make it a million and one and my number is bigger and closer to the square.