real analysis: is 0.5=0.4999999999....?

Question

Why (in real analysis texts), it is assumed to be a obvious that 0.5 is equal to 0.4999.... . i find them different. help please to this paradox.

Answer 1

Yes, they are equivalent.

First, you need to know the method of turning a repeating decimal into a fraction.

Let's take 0.83333... as an example.

The fixed portion (0.8) is just 8/10.

The repeating portion would be 3/9, but it is offset by one digit, so it is actually 3/90.

Adding the two:
8/10 + 3/90 = 72/90 + 3/90 = 75/90 = 5/6

We all agree that 0.8333... is equivalent to 5/6, right?

How about 0.16666....

Again 1/10 + 6/90 = 9/90 + 6/90 = 15/90 = 1/6

So 0.16666... is equivalent to 1/6

How about 0.49999....

This is 4/10 + 9/90 = 4/10 + 1/10 = 5/10 = 1/2

Another way to look at this is to subtract 0.4 from both sides:
Does 0.09999... = 0.1?
Multiply both sides by 10:
Does 0.9999.... = 1?

0.1111... = 1/9
0.2222... = 2/9
0.3333... = 3/9 (or 1/3)
0.4444... = 4/9
0.5555... = 5/9
0.6666... = 6/9 (or 2/3)
0.7777... = 7/9
0.8888... = 8/9
0.9999... = 9/9 (or 1)

However you look at it, 0.49999... is the same as 0.5

For some reason it is hard for our brain to see 0.9999... as equivalent to 1.... but it is. We can see that 0.3333... = 1/3 and that 3 x 0.3333... = 0.9999... = 3 x 1/3 = 1, but we seem to have a hard time seeing that 0.9999.... = 1.

Trust me, your text is correct.

Answer 2

the only paradox lies in the ... notations. What does this mean ?

in fact, it hides a limit problem.

defines two sequences c_n and d_n by :

c_0 = 0 c_1 = 5 and for n > 2 c_n = 0
d_0 = 0 d_1 = 4 and for n > 2 d_n = 9


then sum(c_n * 10 ^-n) and sum(d_n * 10 ^-n) are two convergent series (dominated by geometric series of reason 0.1 which is < 1)

their limit is the same. and you can write the limit as 0.5 or as
0.4999.... which is just a way of designing sum(d_n) for n in N

Answer 3

My answer is more or less the same as Steve's, just in words that make sense to me as an engineer.

Let's try to come up with a rule about when two numbers are the same. Practically, two values are the same if the difference between them is so small as to be negligible. Are two numbers the same if the difference between them is 0.01? 0.0001? 0.00000001? It doesn't really matter what you think the small difference value should be, because whatever number you think of, 0.5 - 0.4999999999... will be smaller than that value. So since the numbers are the same according to ANY rule of that sort, they are, in fact, the same.

Answer 4

Hi there Good question with many possible answers. I think it comes down to why do we fight? We should only fight to protect the life's of ourselves and family. Or in older times to hunt for food. Most people fight today because of pure ego, drugs, alcohol and the oldest one of all over a woman. Its not bogus as you as you put it but its just down to maturity! By the time someone has spent over 20 years training there much older and much wiser. Regards idai

Answer 5

If we assume .5 = .4999...

multiplying both sides by 10 yields

5 = 4.999...

Then

Let x = .999...

10x = 9.999...

-x to both sides

10x -x = 9.999... -x
9x = 9.999... - .999...
9x = 9
therefore, x = 1

5 = 4.999... = 4 + .999... = 4 + x = 4 + 1 = 5

Since 5 = 4.999...

Then .5 = .4999...

Answer 6

well since the .9 continues on infinitely theres always gonna be something to fill in the ga f u now wat i mean...
1. 4.9-5 is .1
2. 4.99-5 is 0.01
3. 4.999-5 is 0.001..and so on until the number gets smaller and smaller,,,

Answer 7

let x = .499999, then 10x = 4.9999999, so 10x - x = 4.5, so 9x = 4.5 and solving for x, we have x = 0.5

Steve