Is there a limit to the decimal expression .999?

Thread starter ram1024
Start date Apr 27, 2004

In summary, the argument revolves around the nature of numbers and their decimal representation. Some argue that .999 contains a limit and is therefore equal to 1, while others argue that it has no limit and will never be equal to 1. The misunderstanding may lie in confusing infinite digits with infinite magnitude. The decimal representation of a number implies an infinite series, but this is different from a sequence of numbers approaching a limit.

Apr 28, 2004

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

Anyways, seeing how everyone participating here other than ram1024 is in agreement on the subject, and ram1024 has rejected all arguments in advance, I see no further point for this thread.

<h2>1. Is .99~ really equal to 1?</h2><p>Yes, .99~ is equal to 1. This is because the tilde symbol (~) indicates that the number is repeating infinitely, so the number is actually 0.999999... which is equal to 1.</p><h2>2. How can a number that is less than 1 be equal to 1?</h2><p>While .99~ may appear to be less than 1, it is actually infinitely close to 1. In mathematics, the concept of limits allows us to understand that as the number of decimal places increases, the difference between .99~ and 1 becomes infinitely small, making them essentially equal.</p><h2>3. Can you prove that .99~ is equal to 1?</h2><p>Yes, there are several mathematical proofs that demonstrate the equality of .99~ and 1. One of the most common proofs involves using the geometric series formula to show that 0.999... is equal to 1.</p><h2>4. Is this concept unique to .99~ and 1, or does it apply to other numbers as well?</h2><p>This concept, known as the concept of infinitesimals, applies to any repeating decimal that has an infinite number of digits. So, any number that can be expressed as 0.999... is equal to 1.</p><h2>5. Why is it important to understand that .99~ is equal to 1?</h2><p>Understanding that .99~ is equal to 1 is crucial in mathematics and science, as it allows us to accurately represent and manipulate numbers. It also helps us to better understand the concept of infinity and the role of limits in mathematics.</p>

1. Is .99~ really equal to 1?

Yes, .99~ is equal to 1. This is because the tilde symbol (~) indicates that the number is repeating infinitely, so the number is actually 0.999999... which is equal to 1.

2. How can a number that is less than 1 be equal to 1?

While .99~ may appear to be less than 1, it is actually infinitely close to 1. In mathematics, the concept of limits allows us to understand that as the number of decimal places increases, the difference between .99~ and 1 becomes infinitely small, making them essentially equal.

3. Can you prove that .99~ is equal to 1?

Yes, there are several mathematical proofs that demonstrate the equality of .99~ and 1. One of the most common proofs involves using the geometric series formula to show that 0.999... is equal to 1.

4. Is this concept unique to .99~ and 1, or does it apply to other numbers as well?

This concept, known as the concept of infinitesimals, applies to any repeating decimal that has an infinite number of digits. So, any number that can be expressed as 0.999... is equal to 1.

5. Why is it important to understand that .99~ is equal to 1?

Understanding that .99~ is equal to 1 is crucial in mathematics and science, as it allows us to accurately represent and manipulate numbers. It also helps us to better understand the concept of infinity and the role of limits in mathematics.