Question:
is 1.9 repeating the same as 2?
explain.

Hi Andrew,

One way to look at this is to see what the difference between the two numbers is. If the difference is 0, then we will say that the two numbers ARE the same.

Now try subtracting:
2-1.9 = .1
2-1.99 = .01
2-1.999= .001
2-1.9999 = .0001
2 -1. 99999 = .00001

If you go out, say 20 places, the difference will be .000 000 000 000 000 000 01
So whatever the real error is (when you don't approximate but go on repeating), it is SMALLER that any positive number you can name. What number is smaller than any positive real number? 0.

A SECOND point of view is a little less direct:

Let S = 1.999999999 (repeating)
Then 10 S = 19.99999999 .... (repeating)
Subtract the first from the second:
9 S = 18.00000 ( repeating) = 18

Divide by 9
S = 2

This is actually the same reasoning (in disguise). However, this principle for figuring out what any repeating decimals are as fractions is widely used as you continue in mathematics. You will see it again around sequences (adding geometric sequences in calculus).

Walter Whiteley