Thread: Who can explain why x= 1 = 0.99999 ....

1. Who can explain why x= 1 = 0.99999 ....

Lets say

10x = 9.99999999 ... , then x= 0.999999999 ....

Thus, 9x = 10x - x = 9.9999999... - 0.9999999.... = 9
thus, since 9x = 9 , then x=1

Wait a second, I thought x=0.99999999 ....

0.999999... does not equal 1

2. Re: Who can explain why x= 1 = 0.99999 ....

Decimal representations are not unique. Every number that terminates with an infinite string of zeros can be represented by an expression that terminates with an infinite string of 9's. To understand this further, you would need to learn some topology and/or analysis. The equality comes from something called the $\varepsilon$-principle. Given two numbers $x,y$, if for all $\varepsilon>0$, $|x-y|<\varepsilon$, then $x=y$. Here, $|x-y|$ is the absolute value function, so I am referring to real numbers. If you are using a different metric $d$, then if $d(x,y)<\varepsilon$ for all $\varepsilon>0$, you have $x=y$.

3. Re: Who can explain why x= 1 = 0.99999 ....

Yes, 0.999999... is equal to 1. And 0.5999... is equal to 0.6 etc.

4. Re: Who can explain why x= 1 = 0.99999 ....

Originally Posted by SelfTaughtMaths
Lets say

10x = 9.99999999 ... , then x= 0.999999999 ....

Thus, 9x = 10x - x = 9.9999999... - 0.9999999.... = 9
thus, since 9x = 9 , then x=1

Wait a second, I thought x=0.99999999 ....

0.999999... does not equal 1
Try this as well:
$\frac{1}{3} = 0.33333333...$ if you do the division.

But then:
$3 \cdot \frac{1}{3} = \frac{3}{3} = 1$

and
$3 \cdot 0.333333333.... = 0.99999999,,,,,$

-Dan

5. Re: Who can explain why x= 1 = 0.99999 ....

Originally Posted by SelfTaughtMaths
Lets say

10x = 9.99999999 ... , then x= 0.999999999 ....

Thus, 9x = 10x - x = 9.9999999... - 0.9999999.... = 9
thus, since 9x = 9 , then x=1

Wait a second, I thought x=0.99999999 ....

0.999999... does not equal 1
You have to understand the basic rule of mathematics that if a = b and b = c, then a = c. In other words, if two things are equal to the same thing, then they are equal to each other.

Since you have shown that x is equal to both 0.9999... and 1, then that means 0.9999... = 1.

6. Re: Who can explain why x= 1 = 0.99999 ....

Think of it this way. If two number are different, then there are an infinite amount of numbers between them. The problem with .99999... and 1 is that there are NO numbers in between. This means that they are the same number.