Thread: Is 0.999... equal to one?

Originally Posted by David M W

Yes my bad! I obviously meant to write "10-20 = -10"

Nothing that you said about in your previous post is true. In business and finance, negative numbers have very definite meanings. Science and engineering, negative numbers also have very definite meanings. To be certain, there are physical quantities that cannot be negative. However, there are many, many others that can be.

Originally Posted by Wayne Bruinekool

It most certainly is. It is not, however, exactly measurable. It can only be approximated as the integral approaches infinity. Calculus, as I remember it, is the study of limits.

Y= 0.99999... and y= 1 are parallel on an x,y plane, never to intersect. They have no points in common.

Can you prove this rigorously?

Originally Posted by Wayne Bruinekool

It most certainly is. It is not, however, exactly measurable. It can only be approximated as the integral approaches infinity. Calculus, as I remember it, is the study of limits.

Again, you confuse process with product. The approximate area under a curve, , is given by a discrete sum known as a Riemann sum:

where is the number of intervals between lower limit and upper limit and is the width of a finite interval between and . The area may be calculated exactly from the definite integral of the function . The definite integral is the Rieman sum in the limit of :

where is an infinitesimal interval between and .

If the function is given by the derivative of another function :

,
then is the antiderivative of . In this case, the Fundamental Theorem of Integral Calculus gives

.
This result is not the process of taking the limit because the limit has already been taken. In the case of is a closed-form function, then the area under the curve can be calculated in closed-form. It is unclear what exactly you mean by:

Originally Posted by Wayne Bruinekool

... It is not ... exactly measurable. ...

In the case of the integral representing a measurable quantity, then that quantity can be measured as exactly as anything else.

Originally Posted by Wayne Bruinekool

... The difference between 1 and 1 is 0. The difference between 1 and 0.99999... is infinitely small, yet greater than zero.

No. You have been shown time and time again that the difference between and is exactly .

Originally Posted by MisterMe

Nothing that you said about in your previous post is true. In business and finance, negative numbers have very definite meanings. Science and engineering, negative numbers also have very definite meanings. To be certain, there are physical quantities that cannot be negative. However, there are many, many others that can be.

No everything I said is true depending on your point of view. I have no argument with the math. My argument is with application to physical reality!
The problem is I accept what the math tells us but I don't accept that it is completely true to reality. You however are of the school of thought where once you learn something then you just accept it as total fact without any thought to the actual "real" application. I find this to be narrow minded. sorry.

I see the argument for 0.9999.. to = 1 I see how this is true using the language of math. My argument is that in physical application infinity has no value of any kind since it has no definitive conclusion. You cannot define a infinite value in physical reality. This is why I argue that 0.999... has no meaning its nonsensical.

Minus values are just scalar, again they have no value in physical reality. I can set a scale and assign what ever value I please from - infinity through to + infinity. So what? Its physical application has no value. In finance, business and the like its just about juggling the figures. The universe is the prime example of this! Conservation always balances the books, its never on the whole a minus value! its either zero or positive.

Originally Posted by David M W

No everything I said is true depending on your point of view. I have no argument with the math. My argument is with application to physical reality!
The problem is I accept what the math tells us but I don't accept that it is completely true to reality. You however are of the school of thought where once you learn something then you just accept it as total fact without any thought to the actual "real" application. I find this to be narrow minded. sorry.

I see the argument for 0.9999.. to = 1 I see how this is true using the language of math. My argument is that in physical application infinity has no value of any kind since it has no definitive conclusion. You cannot define a infinite value in physical reality. This is why I argue that 0.999... has no meaning its nonsensical.

Minus values are just scalar, again they have no value in physical reality. I can set a scale and assign what ever value I please from - infinity through to + infinity. So what? Its physical application has no value. In finance, business and the like its just about juggling the figures. The universe is the prime example of this! Conservation always balances the books, its never on the whole a minus value! its either zero or positive.

The OP asked the question:

Is 0.9999... equal to 1?

The answer to the OP's question is "Yes." This answer has no racial, religious, cultural, handicapped, sexual, sexual-orientation, or age bias. Any person with with the requisite mathematics education will give the same answer. You are arguing a very different point. You are arguing about practicality of using in your calculations. This is strictly an issue of value judgment.

Giving you that value judgment argument, some of your comments are teeth-grindingly off-base. The repeating decimal is just one decimal with an infinite number of digits. There is an infinite number of other numbers with an infinite number of digits following their decimal point. Among the more common ones are , the square root of , and , the ratio of the circumference of a circle to its diameter.

Are all such numbers nonsensical? Is it only that is nonsensical? Explain your reasoning.

Originally Posted by MisterMe

The OP asked the question:
The answer to the OP's question is "Yes." This answer has no racial, religious, cultural, handicapped, sexual, sexual-orientation, or age bias. Any person with with the requisite mathematics education will give the same answer. You are arguing a very different point. You are arguing about practicality of using in your calculations. This is strictly an issue of value judgement.

Giving you that value judgement argument, some of your comments are teeth-grindingly off-base. The repeating decimal is just one decimal with an infinite number of digits. There is an infinite number of other numbers with an infinite number of digits following their decimal point. Among the more common ones are , the square root of , and , the ratio of the circumference of a circle to its diameter.

Are all such numbers nonsensical? Is it only that is nonsensical? Explain your reasoning.

Yeah whats the point of the number? what is its purpose if its = 1, why complicate matters?

So its shows that its impossible to "exactly" calculate the circumference of a circle from its diameter. so pi is a useful approximation.

I'm not arguing the math I'm arguing that math is a tool, a language discovered by us to make accurate predictions about what we observe. This does not mean that its the absolute. For me these examples show that there is some slight flaw in the absolute logic and that sometimes math is not always "exactly" accurate to physical application.

Originally Posted by David M W

Yeah whats the point of the number? what is its purpose if its = 1, why complicate matters?

You have it backwards. The OP asked:

Is 0.999... equal to one?

That is the title of this thread. The answer is his question is "Yes." Stated as an undirected question: Given the infinite repeating decimal , what is its value? The point of the number in this context is that it is the given. Contrary to your assertion that assuring the OP that is equal to complicates things, the opposite is true. Assuring the OP that dramatically simplifies things. It reduces an infinite string of digits to a single digit. You can't get any simpler than that.

Originally Posted by David M W

So its shows that its impossible to "exactly" calculate the circumference of a circle from its diameter. so pi is a useful approximation.

The purpose of numerical values to calculate and quantify, not to "show." The value of is a transcendental number, not a "useful approximation." There are approximations of . However, claims that they are "useful" are dubious because it is not 1970 anymore. It is 2011. We have calculators now. Actually, more and more of us use calculator apps on our cell phones. My own favorite is the HP 48GX emulator on my iPhone. You may believe that is a useful approximation. Pressing four keys on your calculator for 3-digit accuracy? Yeah, right! With the press of a single key, the []-key, the calculator gives correct to 12-14 digits.

Originally Posted by David M W

I'm not arguing the math I'm arguing that math is a tool, a language discovered by us to make accurate predictions about what we observe. This does not mean that its the absolute. For me these examples show that there is some slight flaw in the absolute logic and that sometimes math is not always "exactly" accurate to physical application.

Mathematics is much more than a tool. As for the rest of this statement, you are stating the obvious. Engineers, physical scientists, life scientists, and social scientists devote an enormous amount of their resources studying the limits of their knowledge and developing tests to ensure the reliability of their findings.

Originally Posted by MisterMe

You have it backwards. The OP asked: That is the title of this thread. The answer is his question is "Yes." Stated as an undirected question: Given the infinite repeating decimal , what is its value? The point of the number in this context is that it is the given. Contrary to your assertion that assuring the OP that is equal to complicates things, the opposite is true. Assuring the OP that dramatically simplifies things. It reduces an infinite string of digits to a single digit. You can't get any simpler than that.

Fair enough.

Originally Posted by MisterMe

The purpose of numerical values to calculate and quantify, not to "show." The value of is a transcendental number, not a "useful approximation." There are approximations of . However, claims that they are "useful" are dubious because it is not 1970 anymore. It is 2011. We have calculators now. Actually, more and more of us use calculator apps on our cell phones. My own favourite is the HP 48GX emulator on my iPhone. You may believe that is a useful approximation. Pressing four keys on your calculator for 3-digit accuracy? Yeah, right! With the press of a single key, the []-key, the calculator gives correct to 12-14 digits.

Agreed, but its still an approximation which becomes more accurate the more digits after the decimal in you include in your calculation. I am very familiar with this since I use it regularly in my job.

Originally Posted by MisterMe

Mathematics is much more than a tool. As for the rest of this statement, you are stating the obvious. Engineers, physical scientists, life scientists, and social scientists devote an enormous amount of their resources studying the limits of their knowledge and developing tests to ensure the reliability of their findings.

Yes its a language also, used by scientists, engineers and such.. to describe what they observe, measure and predict. Its not a commandment set in absolute truth. Again I'm not arguing the importance of math, (I know because I rely on it everyday for my work) I'm only arguing that its not infallible.

I feel Its come to a point in this thread where its just going round in circles, which I am one of the main instigators of. We find ourselves discussing the semantics.

The OP has been answered.

Originally Posted by David M W

Fair enough.

Agreed, but its still an approximation which becomes more accurate the more digits after the decimal in you include in your calculation. I am very familiar with this since I use it regularly in my job.

It's not an approximation. It is exactly the ratio of the circumference of a circle to its diameter. It has an exact value, and that value does not change. Now, 3.14 is an approximation of pi, but pi itself is defined as the ratio, not an approximation of the ratio. This is an important concept to understand, as it is the real topic of debate here: the difference between approximate value and exact value, and why and when each is applied.

Originally Posted by David M W

Yes its a language also, used by scientists, engineers and such.. to describe what they observe, measure and predict. Its not a commandment set in absolute truth. Again I'm not arguing the importance of math, (I know because I rely on it everyday for my work) I'm only arguing that its not infallible.

The power of mathematics is that it is logically consistent, and for all intents and purposes (those encountered by the layman) it is infallible and is the very definition of absolute truth, i.e. it is impossible to derive two inconsistent theorems from the same axioms. Now, the absoluteness of the axioms can come into question, but no correctly derived theorem can possibly be false within the system.

Originally Posted by David M W

I feel Its come to a point in this thread where its just going round in circles, which I am one of the main instigators of. We find ourselves discussing the semantics.

The OP has been answered.

I guess you could say we've found ourselves in a loop

Originally Posted by ADH

It's not an approximation. It is exactly the ratio of the circumference of a circle to its diameter. It has an exact value, and that value does not change. Now, 3.14 is an approximation of pi, but pi itself is defined as the ratio, not an approximation of the ratio. This is an important concept to understand, as it is the real topic of debate here: the difference between approximate value and exact value, and why and when each is applied.

Ok agreed, pi is the exact ratio between the circumference and the diameter. But what is the exact value of pi?

Originally Posted by ADH

The power of mathematics is that it is logically consistent, and for all intents and purposes (those encountered by the layman) it is infallible and is the very definition of absolute truth, i.e. it is impossible to derive two inconsistent theorems from the same axioms. Now, the absoluteness of the axioms can come into question, but no correctly derived theorem can possibly be false within the system..

Agreed.