Thread: Is 0.999... equal to one?

Someone posted this question at BAUT. The question is whether 0.999... (that is, the number represented by an infinite stream of the digit 9 after the decimal point) is exactly equal to one. Not close, not good enough for government work, but exact.

Let's see if this board does better than BAUT. The bar is not set very high.

I posted a spoof thread in the Fun-n-Games section there, but a humourless moderator locked it immediately. No rule violation was cited, but BAUT does have the notorious Rule 0, which says the rules are whatever the mods say the rules are.

i do not think it does and here's why.if you had to use fractions on 1 to start,0.000...1,then youy need those incriments to increase to 0.999... so you would still need 0.0000...1 to reach 1.plus 1 is a whole number not a division.1=1.at least thats my take.

I think it would be equal. Lets forget about integers for a second. How could we measure the difference between 1 and 1.00000000000...1 it can not be measured by any imaginable way other than by pure math and even in pure math it approaches 1 so would be assumed to be 1 in any way that it is useful ... any ratio would be 1 etc.

Originally Posted by tom

Lets forget about integers for a second.

Can't. 1 is a counting number which is a subset of integers. .999.... is a real number which is a super set of integers. No matter how hard you try, there is not a one to one mapping of real numbers to counting numbers. There is a mapping of 1 (counting number) to 1 (real number) but there is no mapping of 1 (counting number) to .999..... (real number).

Originally Posted by tom

How could we measure the difference between 1 and 1.00000000000...1 it can not be measured by any imaginable way other than by pure math

The only way this question can be answered is by pure math. There are no measurements in mathematics only logical consistencies.

Originally Posted by tom

and even in pure math it approaches 1 so would be assumed to be 1 in any way that it is useful ... any ratio would be 1 etc.

Only functions can be described as "approaching" such as the limit of 1-1/x as x approaches positive infinity =1 but since the question is asking for the comparison of 2 discrete numbers, this statement does not fly. Although it may be close enough, there is no way to logically say 1 = .999....

Originally Posted by William E. Davenport

Only functions can be described as "approaching" such as the limit of 1-1/x as x approaches positive infinity =1

I think the question then is, do you define the symbol 0.999.... as the limit of the sequence, 0.9, 0.99, 0.999, 0.9999, 0.99999, and so on, or as something different from that? If something different, then what?

Originally Posted by William E. Davenport

but since the question is asking for the comparison of 2 discrete numbers

Correct.

Originally Posted by William E. Davenport

this statement does not fly. Although it may be close enough, there is no way to logically say 1 = .999....

This I cannot agree with. The standard construction of the real numbers is by limits of convergent sequences, and the the numbers constructed by this method obey the exact same rules of arithmetic as the rational numbers, in a logically consistent fashion. There is no approximation being made, and no violence against logic committed, to say that 1 = 0.999....

Originally Posted by tom

I think it would be equal.

I agree with you

Originally Posted by tom

How could we measure the difference between 1 and 1.00000000000...1

I would say the second symbol has no meaning. This question was inspired by a nearly six year old thread at BAUT, and in that thread, there are numerous arguments involving symbols like 1.0000...1 or 0.000...1. Problem is, nobody ever defines what they mean by these symbols.

For convenience, I would like to deal with the symbol 0.000...1 rather than 0.000...1. So if someone says there is a number 0.000...1 (this symbol has no meaning in my maths), I would like to ask what this number divided by 10 is? And is it equal to itself divided by 10, or not?

Even though

Originally Posted by Homo bibiens

The standard construction of the real numbers is by limits of convergent sequences, and the the numbers constructed by this method obey the exact same rules of arithmetic as the rational numbers, in a logically consistent fashion.

.999.... is not a counting number while 1 is. If equality is only concerned about the property of value then your statement is true, but for complete equality it is false.

"Is 0.999... equal to one?" is too vague for a definitive answer.

Originally Posted by William E. Davenport

Can't. 1 is a counting number which is a subset of integers.

But what is an integer. I mean 1 apple plus a an infinitely small nonexisting part of an apple is still one apple. I get the point that is the definition but 1 + 0 still equals 1. 1/infinity = 0 right?

Originally Posted by Homo bibiens

For convenience, I would like to deal with the symbol 0.000...1 rather than 0.000...1. So if someone says there is a number 0.000...1 (this symbol has no meaning in my maths), I would like to ask what this number divided by 10 is? And is it equal to itself divided by 10, or not?

1 + 1/infinity = 1 + 0 = 1 right?

Originally Posted by William E. Davenport

Even though .999.... is not a counting number

In my maths, it is a counting number, because it is defined as the limit of the sequence 0, 0.9, 0.99, 0.999, 0.9999, etc., which is equal to 1.

Originally Posted by tom

1 + 1/infinity = 1 + 0 = 1 right?

I would avoid this sort of construction. It can be shown that it is impossible to extend the real number system to include an "infinity" number, and still have the infinity number obey the same rules of arithmetic as the other numbers. You can have an "infinity" number and have it obey some of the rules of arithmetic, but then you have to be extra careful to make sure you are not performing an illegal operation on infinity.

If the idea is to define things like 0.000....1 as the limit of the sequence 0.1, 0.01, 0.001, 0.00001, etc., then it would be equal to zero. Which is certainly consistent with what you have said earlier. 0.000...1 just seems like very odd notation to me - how can there be a "1" after an infinite stream of "0" digits, since the infinite stream of "0" digits has no end? Many of the BAUTians seem to have some alternate arithmetic system with a number 0.000...1 that is not equal to zero. I am glad you do not do this