Thread: Does 0.9 repeated = 1?

Originally Posted by Xei

There's no reason you can't extend the decimal notation to state that.

It would of course represent 0 + 0/10 + 0/10^2 + ... + 1/10^infinity which is equal to 0.

But naturally it is completely redundant and inefficient, just like writing .999~.

In other news, how a more knowledgeable person explain about how I solve kx = e^x? What about functions like kx = sinx?

I was messing about with a grapher and I've found that there's one root in the first case when k = e. I also plotted k = e^2 and I found a root at pi??

You're getting to deep. Turn back now before it's too late!

But kx=e^x is an equation with only 1 variable. How can you plot that on a 2D graph?

Edit: Never mind, I just realized you were making two equations (y=kx and y=e^x) equal to each other. My bad.

Originally Posted by Xei

There's no reason you can't extend the decimal notation to state that.

It would of course represent 0 + 0/10 + 0/10^2 + ... + 1/10^infinity which is equal to 0.

But naturally it is completely redundant and inefficient, just like writing .999~.

In other news, how a more knowledgeable person explain about how I solve kx = e^x? What about functions like kx = sinx?

I was messing about with a grapher and I've found that there's one root in the first case when k = e. I also plotted k = e^2 and I found a root at pi??

Some equations can't be solved exactly. You're just going to have to accept that, I'm afraid.

There are ways to do them, but the roots are transcendental I think. Series of fractions such as those used to define pi and e.

Anyway, this was a question for a Cambridge interview; solve kx = e^x for different values of k.

At the very least, how do I find out the number of roots? Why is there one at k=e?

Iteration works but that's not good enough...

What you said doesn't make sense. What do you mean by a root at k=e? x is the variable, not k.

kx = e^x

Consider the two intersecting lines; the number of roots will depend on k. I should probably have been a bit clearer, by 'at k=e' I mean that the equation will have a root when k=e (at some x).

I've used a graphical calculator to find that there's one root at ex = e^x (and hence two roots for k>e and zero for k<e). I don't know why though.

When k=e and x=1, then:

kx = e(1) = e
e^x = e^1 = e

But that seems too obvious. I don't think I understand what you don't get.

The problem is you had to solve that by trial and error. So for k=e,

ex = e^x

If you want, log both sides.

lnx = x - 1

But this is no more helpful than the first. Sure, with a little bit of guessing you find that it works out when x = 1. But how do you solve it properly?

Try do what you did for k = 3.

Originally Posted by Xei

The problem is you had to solve that by trial and error. So for k=e,

ex = e^x

If you want, log both sides.

lnx = x - 1

But this is no more helpful than the first. Sure, with a little bit of guessing you find that it works out when x = 1. But how do you solve it properly?

Try do what you did for k = 3.

How many times to I have to say this? You can't solve it. Just accept that. The best you can hope for is Newton's method or bisection or whatever.

Can somebody explain how 0.000...1 is a number? That is a decimal followed by an infinite number of 0's and then a 1. It represents infinite smallness, and it is the difference between 1 and 0.999... How can something come after infinity?

Originally Posted by Universal Mind

Can somebody explain how 0.000...1 is a number? That is a decimal followed by an infinite number of 0's and then a 1. It represents infinite smallness, and it is the difference between 1 and 0.999... How can something come after infinity?

Whole thread. See above.

Originally Posted by ClouD

Whole thread. See above.

I have not seen a direct answer to how something can come after infinity. Can you give one? A 1 after an infinite number of 0's after a decimal seems like something on the other side of an infinite universe. I do not see how it is possible. However, I do believe that infinite smallness exists. I just cannot answer the questions I posed. Can anybody?

Originally Posted by Universal Mind

I have not seen a direct answer to how something can come after infinity. Can you give one? A 1 after an infinite number of 0's after a decimal seems like something on the other side of an infinite universe. I do not see how it is possible. However, I do believe that infinite smallness exists. I just cannot answer the questions I posed. Can anybody?

No such number exists. The notation is as nonsensical as 0.1.5 or something.

Originally Posted by drewmandan

How many times to I have to say this? You can't solve it. Just accept that. The best you can hope for is Newton's method or bisection or whatever.

Well once so far. And actually I managed to derive by a proper method that the one root occurs at k=e; you do it by observing that at this point, de^x/dx = dkx/dx and that the tangent crosses the origin.

My Cambridge interview went ok. Completely went into meltdown and failed at a basic question about inequalities though... luckily got the other 5 questions right, although I felt like I wasn't pushed enough which is concerning me.

Originally Posted by drewmandan

No such number exists. The notation is as nonsensical as 0.1.5 or something.

Huh? Then what is 1 - 0.99999(repeating forever)?

Do the math...

1.00000000000000000000000000000000000000000000...( forever repeating)
- 0.99999999999999999999999999999999999999999999...( forever repeating)

Like I said, that really means 1/10^0 + 0/10^1 + 0/10^2 + 0/10^3 + ... + 1/10^x which tends to 1 as x tends to infinity.

Originally Posted by Universal Mind

Huh? Then what is 1 - 0.99999(repeating forever)?

Do the math...

1.00000000000000000000000000000000000000000000...( forever repeating)
- 0.99999999999999999999999999999999999999999999...( forever repeating)

Originally Posted by Xei

Like I said, that really means 1/10^0 + 0/10^1 + 0/10^2 + 0/10^3 + ... + 1/10^x which tends to 1 as x tends to infinity.

What he said. Regular old elementary school subtraction is a special case of true subtraction in the reals, which uses convergent series.

Originally Posted by Xei

Like I said, that really means 1/10^0 + 0/10^1 + 0/10^2 + 0/10^3 + ... + 1/10^x which tends to 1 as x tends to infinity.

That is 1 + a bunch of 0's, which equals 1. So, you say the answer is 1?

Also, I am looking for something more exact than "tends to". When such limits are involved, the boundaries are never reached. They are only forever approached. The gap never gets closed.

Originally Posted by drewmandan

What he said. Regular old elementary school subtraction is a special case of true subtraction in the reals, which uses convergent series.

So you are saying the answer is not a real number? Is it an imaginary number? What is your official answer?

That is 1 + a bunch of 0's, which equals 1. So, you say the answer is 1?

Also, I am looking for something more exact than "tends to". When such limits are involved, the boundaries are never reached. They are only forever approached. The gap never gets closed.

Yes, it's 1.

Tends to is the proper mathematical term. Formally that's how you describe the thing you just wrote out. In layman's terms means that that term equals 0.

So you are saying the answer is not a real number? Is it an imaginary number? What is your official answer?

1 is a real number. 1.000...1 = 1. So it is the same real number.

Originally Posted by Xei

Yes, it's 1.

Tends to is the proper mathematical term. Formally that's how you describe the thing you just wrote out. In layman's terms means that that term equals 0.

"Tends to" means it never actually reaches it. It is what happens with limits in calculus and the approaching of a hyperbola toward an asymptote. Infinite approach is not the same as officially reaching, in theory and on paper, but they strangely seem to be the same in reality. It is so bizarre.

It seems that infinite smallness and 0 are the same and that 0.999... and 1 are the same, but it leaves open the question of how the gap is ever fully closed. How does any 9 after a decimal ever bring the number to the next level? It seems to be a paradox with a lot of the same elements involved in Zeno's Paradox. It is one of those philosophical mindfucks that does not have a solid answer humans know, a lot like the question of what is at the root of consciousness or why there is existence instead of nothing.

The last two I'll agree with, but this isn't something that troubles me. Decimals don't 'exist' in any case.

Originally Posted by Xei

Decimals don't 'exist' in any case.

But what they represent does.

Originally Posted by Universal Mind

So you are saying the answer is not a real number? Is it an imaginary number? What is your official answer?

I'm saying the crude method of subtraction you're using doesn't work in general. It's a shortcut taught to school children, not to be taken as something rigorous.

Originally Posted by Universal Mind

But what they represent does.

This quote is out of context, but I think you know what I'm referring to with the following.

The reals are defined on a structure of limits. The details are quite complicated, but in the reals, limits are how equality is defined, in this case. The reals are dense, so you can't use something as crude as whole number equality.

But what they represent does.

I'm not so sure. Decimals represent numbers. Do numbers exist? Essentially numbers are adjectives which have consistent meaning in our reality. Do adjectives exist?

Adjectives manifest themselves as properties of nouns; concrete and abstract.