- #1
Blahness
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Elaborate.
Plus, wouldn't that mean 1.999...8 = 2?
Plus, wouldn't that mean 1.999...8 = 2?
Finding the Is 1.999...8 Really Equal to 2?
- Thread starter Blahness
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In summary, 1.999...8 is not a valid number since the ellipsis indicates an infinite string with no end. However, the limit of the sequence 1.8, 1.98, 1.998, 1.9998, ... is equal to 2.
- #2
hypermorphism
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No. See the Archives.
- #3
Lisa!
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God! Again this question. 
- #4
HallsofIvy
Science Advisor
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By definition of a "base 10 numeration system", 0.999... means the limit of the sequence 0.9, 0.99, 0.999, 0.9999, ...Blahness said:
Those are the partial sums of a geometric series (a+ ar+ ar2+ ...) with a= 0.9 and r= 0.1. It's easy to show that a geometric series converges to a/(1-r) which in this case is 0.9/(1- 0.1)= 0.9/0.9= 1.
For the second question, what do you mean by "1.999...8"? If that ... indicates an infinite string then there is no end to put the 8 on! If you mean the limit of the sequence 1.8, 1.98, 1.998, 1.9998, ... (that's what 1.999... means- without the 8 of course) then it is equal to 2, yes.
FAQ: Finding the Is 1.999...8 Really Equal to 2?
Is 1.999...8 really equal to 2?
Yes, 1.999...8 is equal to 2. This is because in the decimal system, numbers can continue infinitely after the decimal point. Therefore, 1.999...8 is just another way of writing the number 2, with an infinite number of 9s after the decimal point.
How can a number with an infinite amount of 9s be equal to a whole number like 2?
This concept can be difficult to grasp, but it is important to understand that in mathematics, we use the concept of limits to define and understand infinite numbers. In this case, the limit of 1.999...8 is 2, meaning that as we add more and more 9s after the decimal point, the number gets closer and closer to 2. And in the limit, it becomes equal to 2.
Is this concept unique to the number 2?
No, this concept applies to all repeating decimal numbers. For example, 0.333... is equal to 1/3, and 0.999... is equal to 1. It is important to understand that the representation of a number may vary, but the value remains the same.
Why is this concept important in mathematics?
This concept is important because it helps us to understand and work with infinite numbers, which are often encountered in many areas of mathematics such as calculus and number theory. It also allows us to use shortcuts and simplifications in calculations, making problem-solving more efficient.
Can this concept be applied to numbers in other number systems?
Yes, this concept can be applied to other number systems, such as binary or hexadecimal. In these systems, a number with an infinite amount of digits after the decimal point may be represented differently, but the concept of limits still applies. For example, 0.111... in binary is equal to 1 in decimal, and 0.FF... in hexadecimal is equal to 1 in decimal.