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Does 0.9999 repeating 9 equal 1?

greenspandan

By greenspandan
in The Tailgate

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jnhay

jnhay

Posted
Posted

Being that they're two different numbers, they don't equal each other. I can't think of a situation where .9 repeating actually exists, but it's still a number. I think it's probably used most often to specifically say that a function will never equal 1.

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jpillian

jpillian

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But is .250 equal .2501 and .2499?

Didn't know that they were.

I took from it that .9 repeating == 1.

Therefore, .2509 9's repeating would equal .2501, and .249 9's repeating would equal .250.

Of course, I also agree with Zhouse on the concept of infinity. Seems like Man's attempt to simplify deal with horrendously large numbers. Doesn't seem to have a basis in reality.

And then you'll get Mathematicians talking about how some infinite sets are BIGGER than other infinite sets. Seems to me that it's a good way for us to wrap our mind around certain concepts, but I'm not convinced that anything in existence actually exhibits the property of belonging to an infinite set.

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greenspandan

greenspandan

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How do you figure?

the mathematical concept in trigonometry of a "point" (which is itself *infinitely* small) is nonetheless absolutely fundamental, necessary, valid, etc.

i'd like to see someone write a modern video game, for example, without being able to use coordinates, or indeed any other abstract math concepts.

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PeterMP

PeterMP

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I took from it that .9 repeating == 1.

Therefore, .2509 9's repeating would equal .2501, and .249 9's repeating would equal .250.

Of course, I also agree with Zhouse on the concept of infinity. Seems like Man's attempt to simplify deal with horrendously large numbers. Doesn't seem to have a basis in reality.

And then you'll get Mathematicians talking about how some infinite sets are BIGGER than other infinite sets. Seems to me that it's a good way for us to wrap our mind around certain concepts, but I'm not convinced that anything in existence actually exhibits the property of belonging to an infinite set.

What is bigger than infinity? infinity+1

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Ignatius J.

Ignatius J.

Posted
Posted

This isn't a philosophical question as some here make it out to be. You have to know what the definition of a real number is, and then it is as clear as day that .999999 repeating does in fact equal one.

So, for beginners, how do we construct the real numbers?

1) Start with the counting numbers, 1,2,3,4,5,6,7

2) add in zero and the negative numbers to get the integers.

3) Define the rational numbers. These are formed by the set of all combinations (a,B) where a and b are integers subject to the condition that (a,B) = (c,d) if there exists an integer g such that (ga =c and gb = d) or (gc = a and gd = B). So (2,4) = (1,2) just as you would expect. I'll assume you know how to multiply, divide add and subtract these numbers.

4) Define cauchy sequences. Given a sequence of rational numbers (a_i, b_i), the sequence is cauchy if for every rational number epsilon there exists a counting number n such that (a_i, b_i) - (a_(i+1), b_(i+1)) is less than epsilon for all i larger than n.

5) Given two cauchy sequences of rational numbers (a_i, b_i) and (c_i, d_i) , we say that two sequences are equivalent if and only if the difference between the two sequences approaches zero. Formally, (a_i, b_i) = (c_i, d_i) if and only if:

for every epsilon in the rational numbers there exists an n in the counting numbers such that

(a_i d_i - c_i b_i, b_i d_i) is less than epsilon for all i greater than n.

6.) the real numbers are defined to be the set of all cauchy sequences subject to the above equivalence.

*********************

So, with that in mind, what is the real number 1? it is the cauchy sequence (1,1,1,1,1,1,1.....) etc.

What about .999999999.... ? It's cuachy sequence is

.9, .99, .999, .999, .9999, .99999 etc

It is quite clear that the difference between the elements of these two cauchy sequences approach 0, therefor they are the same real number.

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Zhouse

Zhouse

Posted
Posted
This isn't a philosophical question as some here make it out to be. You have to know what the definition of a real number is, and then it is as clear as day that .999999 repeating does in fact equal one.

So, for beginners, how do we construct the real numbers?

1) Start with the counting numbers, 1,2,3,4,5,6,7

2) add in zero and the negative numbers to get the integers.

3) Define the rational numbers. These are formed by the set of all combinations (a,B) where a and b are integers subject to the condition that (a,B) = (c,d) if there exists an integer g such that (ga =c and gb = d) or (gc = a and gd = B). So (2,4) = (1,2) just as you would expect. I'll assume you know how to multiply, divide add and subtract these numbers.

4) Define cauchy sequences. Given a sequence of rational numbers (a_i, b_i), the sequence is cauchy if for every rational number epsilon there exists a counting number n such that (a_i, b_i) - (a_(i+1), b_(i+1)) is less than epsilon for all i larger than n.

5) Given two cauchy sequences of rational numbers (a_i, b_i) and (c_i, d_i) , we say that two sequences are equivalent if and only if the difference between the two sequences approaches zero. Formally, (a_i, b_i) = (c_i, d_i) if and only if:

for every epsilon in the rational numbers there exists an n in the counting numbers such that

(a_i d_i - c_i b_i, b_i d_i) is less than epsilon for all i greater than n.

6.) the real numbers are defined to be the set of all cauchy sequences subject to the above equivalence.

*********************

So, with that in mind, what is the real number 1? it is the cauchy sequence (1,1,1,1,1,1,1.....) etc.

What about .999999999.... ? It's cuachy sequence is

.9, .99, .999, .999, .9999, .99999 etc

It is quite clear that the difference between the elements of these two cauchy sequences approach 0, therefor they are the same real number.

:gus:

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PokerPacker

PokerPacker

Posted
Posted

the difference between y<1 and y <=1 is the same as the difference between .99999 repeating. The difference is infintessimally small and cannot be recorded in a discreet manner, but it the distinction does exist.

.9999999 repeating to infinity is a solution for the inequality y<1. 1, however is not a solution for such inequality, it is a solution for the inequality of y<=1. Since they cannot be used interchangeably, they are not the same number.

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Spec138

Spec138

Posted
Posted
the difference between y<1 and y <=1 is the same as the difference between .99999 repeating. The difference is infintessimally small and cannot be recorded in a discreet manner, but it the distinction does exist.

.9999999 repeating to infinity is a solution for the inequality y<1. 1, however is not a solution for such inequality, it is a solution for the inequality of y<=1. Since they cannot be used interchangeably, they are not the same number.

.999... is not a solution for that inequality because it does, in fact, equal 1. No one really comprehends infinity, there is no end, which we can't imagine.

x=.999...

10x = 9.999...

10x-x= 9.999... - .999...

9x=9

x= 1

.999...= 1

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