One of your middle school students was interested in infinite repeating decimals. He noticed that when he tried to convert N = .999 ... to a rational of form § he got the following:
10 = 9.999
10N - N = 9
9N = 9
N = 1
He asked you how .999 ... could be 1.
b. The fact that .9999 . .. is another name for 1 is an interesting but complex fact. Research this idea and summarize your conclusions.
The concept that .999... is equivalent to 1 is indeed interesting, but it may seem counterintuitive at first. However, this idea can be proven mathematically using different approaches.
One of the most commonly used proofs is similar to the one you have mentioned:
Let N = 0.999...
Multiplying both sides of the equation by 10, we get:
10N = 9.999...
Now, subtracting the initial equation from the above equation, we have:
10N - N = 9.999... - 0.999...
9N = 9
Dividing both sides by 9, we get:
N = 1
This proof demonstrates that N, which represents the infinite repeating decimal .999..., is equivalent to 1.
Another way to show this equivalence is by considering the concept of limits in calculus. In calculus, a limit represents the value a function approaches as the input approaches a certain point. In terms of .999..., we can consider the limit of the function:
lim(n→∞) 0.9 + 0.09 + 0.009 + ... + 0.000...9
As n approaches infinity, the terms on the right side become smaller and smaller, approaching zero. Thus, the sum of these terms, which is .999..., can be considered as the limit of the function. And in this case, the limit is equal to 1.
Furthermore, we can use the concept of geometric series to prove this equivalence. A geometric series is a series in which each term is obtained by multiplying the previous term by a constant ratio. In the case of .999..., we can set up a geometric series with the first term as 0.9 and the common ratio as 0.1:
S = 0.9 + 0.09 + 0.009 + ...
To find the sum of this infinite series, we can use the formula for the sum of a geometric series:
S = a / (1 - r)
where a is the first term and r is the common ratio. Plugging in the values, we have:
S = 0.9 / (1 - 0.1)
S = 0.9 / 0.9
S = 1
Therefore, the sum of the infinite geometric series .999... is 1.
In conclusion, the fact that .999... is equivalent to 1 can be proven through different mathematical approaches, such as algebraic manipulation, limits, and geometric series. While it may initially seem complex and counterintuitive, these proofs demonstrate the validity of this concept.