+2446 I'm pretty sure that you mean what does \(0.\overline{9999}\) equal? I'll use some algebra to show the real value here. This method is well-known, but here it goes anyway:
| \(0.\overline{9999}=x\) | I'm going to set this answer equal to a variable. I'll multiply 10 on both sides |
| \(9.\overline{9999}=10x\) | This is probably the trickiest step to understand. Subtract \(0.\overline{9999}\) on the left hand side and \(x\) on the right. I can do this because of the first statement I made |
| \(9=9x\) | Divide by 9 on both sides |
| \(x=1\) | |
Therefore, \(0.\overline{9999}=1=x\).
Now, I have a challenge for you.
\(...9999999=x\)
Using the same algebra I utilized, what does this equal? You should get a bizarre answer
+2446 I think I have an explanation that is simple to understand! In advance, I have not referenced any of Hecticlar's sources before writing this explanation, so mine could be similar or different. To start, I am going to make a table of powers that we can calculate:
| \(2^n\) | Written-out | Result |
| \(2^5\) | 2*2*2*2*2 | 32 |
| \(2^4\) | 2*2*2*2 | 16 |
| \(2^3\) | 2*2*2 | 8 |
| \(2^2\) | 2*2 | 4 |
| \(2^1\) | 2 | 2 |
| \(2^0\) | ? | ? |
| \(2^{-1}\) | 1/(2) | 1/2 |
| \(2^{-2}\) | 1/(2*2) | 1/4 |
| \(2^{-3}\) | 1/(2*2*2) | 1/8 |
| \(2^{-4}\) | 1/(2*2*2*2) | 1/16 |
| \(2^{-5}\) | 1/(2*2*2*2*2) | 1/32 |
Do you notice a pattern? I do. As you go down the list, you can divide by 2 to get the next number in the sequence! FIrst, I'll generalize this statement:
\(\frac{2^n}{2}=2^{n-1}\)
What I have done here is manipulate the powers so that I can circumvent raising to the power of 0. If I make n=1, I will raise to the power of zero and get a result of what that answer should be. Let's try it!
| \(\frac{2^1}{2}=2^{1-1}\) | Let's simplify the right hand side first by doing 1-1 |
| \(\frac{2^1}{2}=2^0\) | Woah! Evaluate the left hand side to figure out what 2^0 truly equals. |
| \(\frac{2}{2}=2^0\) | |
| \(1=2^0\) | |
We can generalize this further to say that any number raised to the power of zero is 1 using some algebra:
| \(1=\frac{x^n}{x^n}\hspace{1mm},x\neq0\) | This statement is true because any number divided by itself is one! I'll use an exponent rule that says that \(\frac{x^n}{x^n}=x^{n-n}\) |
| \(1=x^{n-n}\) | n-n=0, so let's simplify that |
| \(1=x^0\hspace{1mm},x\neq0\) | This is saying that any number to the power of zero is one. |
