Is 0.999... really equal to 1?

  • Thread starter Dooga Blackrazor
  • Start date
In summary, the conversation was about the equation 0.999... = 1 and whether or not it is true. The teacher believes that it is only approximately equal to 1 because it is being rounded, but the student recalls seeing a formula that mathematically proves it to be equal to 1. Other forum users point out that the teacher is incorrect and provide a demonstration using fractions to show that 0.999... does indeed equal 1. They also mention that this is a common topic of discussion and that 0.999... is defined as an infinite series, making it equal to 1. The conversation ends with questioning the teacher's understanding of basic math concepts.
  • #1
Dooga Blackrazor
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0.999... = 1 (Why?)

Sorry to put such a basic question on here, but it's not for homework so I figured I'd post here.

On these forums, I've saw the issue of .99... = 1 brought up before; however, I recently discovered it in Math class.

My teacher said it equals one because it is being rounded; however, it actually doesn't equal one. I understand what she means; however, for some reason, I recall seeing a formula that mathematically proved .99... = 1 without rounding. Perhaps I am seeing things.
 
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  • #2
That formula (or the closest thing to it, there is no formula) is in those fifty other threads.
 
  • #3
If your teacher said that 0.999... is only approximately 1, then she is wrong.
 
  • #4
Dooga Blackrazor said:
My teacher said it equals one because it is being rounded; however, it actually doesn't equal one.

Your teacher is wrong.

Quickie demonstration:

[tex]\frac{1}{3}=0.\bar{3}[/tex]

[tex]3\left(\frac{1}{3}\right)=3(0.\bar{3})[/tex]

[tex]1=0.\bar{9}[/tex]

And if your teacher still thinks that [itex]0.\bar{9}=1[/itex], then ask him/her to try to find a real number between the two. It can't be done.
 
  • #6
Not only is this not a new topic, it's a regular topic!

My only objection to (1/3)= 0.33333... so 1= 0.999... is that the same people who object to 1= 0.9999... would also object to 3(0.3333...)= 0.999...- and they have a point. Proving one is equivalent to proving the other.

The real point is that, by definition of a "base 10 number system", 0.999... means the infinite series .9+ .09+ .009+... which is a geometric series whose sum is 1.

By the way, what grade is this teacher? And who is his/her principal/college president?!
 

1. What is the mathematical proof behind 0.999... = 1?

The proof behind this statement lies in the concept of infinite series. 0.999... can be expressed as the sum of the infinite series 0.9 + 0.09 + 0.009 + ... which can be simplified using the formula for infinite geometric series to 1. Therefore, 0.999... is equal to 1 mathematically.

2. How can three decimal places equal a whole number?

The concept of decimal places is a human invention to help us represent numbers in a more convenient way. In reality, numbers are not limited to a certain number of decimal places. As we add more and more 9s after the decimal point, the difference between 0.999... and 1 becomes smaller and smaller until it becomes infinitesimally small, making them essentially the same number.

3. Can you give a real-life example of 0.999... = 1?

One example is the infinite staircase paradox. Imagine a staircase where each step is half the size of the one before it. The total height of the staircase is 1, but it can also be expressed as the sum of an infinite series of 0.5 + 0.25 + 0.125 + ... which can be simplified to 1. Therefore, the height of the staircase is equal to 1, just like how 0.999... is equal to 1.

4. Is 0.999... close to 1 or actually equal to 1?

0.999... is not just close to 1, it is actually equal to 1. As mentioned earlier, the difference between 0.999... and 1 becomes infinitesimally small, making them essentially the same number. In fact, in mathematics, we often say that 0.999... is a different way of representing the number 1.

5. Can this concept be applied to other numbers?

Yes, this concept can be applied to other numbers as well. For example, 0.333... is equal to 1/3, and 0.111... is equal to 1/9. This is because these numbers can also be expressed as infinite geometric series, just like how 0.999... can be expressed as an infinite geometric series with a common ratio of 0.1.

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