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#1 2018-08-17 06:56:25

Justin Logic

Does 1 = .999...?

The website demonstrates that to find whether or not .999... = 1, you simply put solve this equation:

                                                                           10x = 9.999...

                In reality, though, the only thing this equation is telling us is that 9.999... = 9.999..., and by dividing .999... ends up with the equation 1 = 1. I'm not sure if they were just confused or tired, but this doesn't exactly tell us anything at all. This is more of a logic problem than a math problem.

               The way I like to think of this is the further the numbers go, the smaller the number between them gets, but it never is exactly 1. The best way to write the difference would be this:

               So now we are left with the equation: 0.999... + .00...1 = 1

But that doesn't make sense if .999... = 1. You can't simply say 0.00...1 has absolutely no value, whatsoever, no matter how infinitely small it is.

              In other words, you can't make numbers disappear, no matter how unimaginably small they really are.

              With this in mind, if they are exactly the same, why don't they both make linear equations? A single number can't make more than one line on a graph. There isn't really much math to it. Just logic.

#2 2018-08-24 20:41:28

Registered: 2006-03-12
Posts: 1,337

Re: Does 1 = .999...?

I had a long dispute with a major in mathematics.

So far I can tell you that mathematicians made a cunning update to cover this bug as far as redefining numbers.



#3 2018-08-25 19:42:08

bob bundy
Registered: 2010-06-20
Posts: 8,508

Re: Does 1 = .999...?

The difficulty arises because it is a property of the real number system that between any two real numbers there is another real number.  For all other reals this works ok but it fails whenever the decimal representation 'ends' with an infinite string of 9s.  I have encountered two 'solutions' to the problem and both allow a consistent set of axioms.

(1) Allow that for example 0.99999999.... is the same as 1   I can understand why some people dislike this but, if you remember that the reals exist irrespective of the decimal representations, then why not?  Lots of mathematical topics work perfectly using this solution.  For example, you can convert an infinite, and recurring, decimal representation into a fraction.

(2) You can forbid the existence of such numbers from the real number system.  That also works and is the basis of the approach used by Georgi E Shilov in his book Elementary Real and Complex Analysis.

It might help if you stop thinking that numbers have a concrete existence and accept that they are just convenient abstract ideas that help us to do certain types of mathematics. (You can hold 3 apples or even a wooden shaped '3' object but you cannot hold a 3.)  So we can make numbers do what we wish according to the model we are making.  Not everything obeys the rules for real numbers.  For example add one pile of sand to another pile of sand and you have one pile of sand. You cannot have half a stick of chalk.

In summary, decide what mathematical model you are building; invent a consistent set of axioms for your model; and then use it to discover new things.


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile


#4 2018-08-26 00:56:59

From: Aleppo-Syria
Registered: 2018-08-10
Posts: 14

Re: Does 1 = .999...?

Usually, mathematics is a tool, not a target.
For example, if I get a number in the range {9K5 to 10K5} for the value of a resistor, I would simply select the 10K resistor. Obviously, the designed circuit should be made to work properly even if the exact value of the resistor in use is also a bit outside this range.

On the other hand, solving math problems could be a target if it is for fun... as testing the level of one's logic (logical reasoning).


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