Why o.999... if represented in form of p/q (q is not =0) the answer is 1?
Answers
Well, in higher classes you will be introduced to calculus (in maths, not the physics part.) , it holds the answer you are asking for. Till that time, what i can tell you is that 0.999999.....till infinity is never exactly = 1,but at every instant the distance between this number and 1 becomes smaller.
For example,
Lets take 0.9
0.9 is 0.01 away from 1
Now 0.99
It is 0.001 away from one
Similary, 0.999999999999999
It is 0.0000000000000001 away from 1
Just like this, if we include INFINITE 9s
We would also get infinitely many 0s in 0.0001
This would make this distance of 1/10^n INFINITLY SMALL. And that's one of the concepts in calculus you would be taught elaborately.
Since the distance betwenn 0.999.... And 1 is infinitely small, this implies that 0.9999... APROACHES 1 and the distance between them keeps getting smaller and smaller and more and more negligible.
In calculus, APROACHES is the same as EQUALS TO.
EDIT
There's one thing i would like to add, 0.999... In p/q form is 1 because 1/3 in p/q form is 0.33333....
And as you know
3 x 1/3 = 3/3 = 1
Similarly,
3 x (0.333.....) = 0.9999....
You may be wondering why exactly this happens, well this is because our base 10 system lacks versatility.
1/3 is not a whole number
Whereas if you use a duodecimal system (base 12.) is has a more clean solution because 1/3 in this system doesnt equal to 0.3333
Rather it equals to 0.4
And in this system 3 x 1/3
= 3 x 0.4
= 0.1 easily
Yet in base 10 it gives an ugly 0.99999...
Are you wondering what these base number systems even mean???
Ever heard of binary number system?.
The computer's language huh?
It is base two, which means, only 2 characters 0 and 1
Similarly base 10 uses 10 numbers as individual characters, and base 12 uses 12 numbers as individual characters
If you are interested about these bases, go check out some youtube videos, it would be a fun journey
The brainliest answer would be appreciated, thank you