What is the answer for 0.999999... _ 1.0000 ?
a) <
b) =
c) >
Many of you will be tempted to answer the obvious a) <
Right,
NO, it is the wrong answer.
Correct answer is b) =
To see it physically. Draw a rectangle, cut it into three parts. each eqaling 1/3 i.e., 0.3333... Add them up and it becomes 0.9999...
But, 1/3 + 1/3 + 1/3= 1/3 x 3 =1, Right, so 1= 0.999999...
Now, let us turn to the pure mathematical algebraic side.
we shall start with the obvious,
3x 1/3 =1 = RHS
Now,
LHS= 3x 0.3333... = 0.9999...
RHS = 1
So, 0.9999... = 1.00000
There is no approximation, no assumption its plain mathematical operation with all valid simple mathematical operation . Both numbers, apparently different are representing one and the same point on a number line!
In fact, any whole no., n can be represented as (n-1).9999... because you can break every no. as (n-1) +1 = (n-1).99999...
Meaning,
3=2.99999...
4=3.99999...
5=4.99999....
and so on...
Now let's revisit the equation:
3 x 1/3=1
Or .9999... = 1.0000...
or, 1.0000.. - 0.999999.. = 0.0000......1 (NOTE IT !)
Also, 1- 3x1/3= 1-1 = 0.0000000....
So, 0.0000...1 = 0.000000
Now, comparing place to place at the last point at infinity, If infinity is an exact point it has both the value 0 and 1. Did quantum physicist discovered the obvious!