[Guest]

This is only a math question made by a teacher of a friend of mine. It caused a lot of discussion and, although I had my answer, I was not fully satisfied with it. Since I didn't found this question here and I love braindeners discussion, here is the question:

What is the first real number above 1?

Once you think about it, read the spoiler. There the question is complemented, and that's why I'm not satisfied with my answer:

What is the first real number above 1 that isn't mathematically equal to 1?

[Guest]

There isn't one. Which might seem a strange and unsatisfying answer, but there is never any reason to need one. Let's say you needed the next highest real number after 1 for anything. If you assume that number exists then it will be so close to 1 that it would be impossible to define a threshold of error so fine that you wouldn't be able to just use 1 instead.

[Guest]

Let X be a number greater than 1. Then (X+1)/2 will be halfway between X and 1. If there was a number that satisfied "the first real number greater than 1", then there would necessarily be a number halfway between that number and 1, making it not the first real number greater than 1.

[Guest]

1+epsilon

[Guest]

Assume that there exists a positive real number epsilon > 0, such that (1 + epsilon) is the next real number after 1. Then for e = epsilon/2 we have that 1<1+e<1+epsilon since e=epsilon/2<epsilon. Hence we have a contradiction to our original assumption that 1+epsilon is the next real number after 1. Therefore there is not any epsilon positive such that 1+epsilon is the next real number after 1.

q.e.d

[Guest]

lim X

x-»1⁺

[Guest]

2 ha ha

[Guest]

lim X

x-»1⁺

The answer to this limit is 1 not greater than 1.

[Guest]

The answer to this limit is 1 not greater than 1.

Limit should never approach 1.

[Guest]

...because ice cream has no bones.

[Guest]

Some of the others have alluded to this, but I think this puts it in a clearer mathematical expression.

Define a variable r such that r is real and greater than zero. Then the limit of 1+r as r approaches zero is the next real number above 1.

[Guest]

I'll entre my answer here:

{Z, x E R / Z = Lim (x-infinite) 1 + x}

But mathematticaly speaking, this answer is 1, in the same way that, for exemple, 9,999...=10.

Any other idea?

[EventHorizon]

1+epsilon

Anywhere you are representing a real number with a discrete number of bits (calculator, computer, etc), there will be a smallest number greater than 1. This is because there is a value for epsilon which is the smallest difference between two representable numbers.

In math, however, there does not exist one. It has been pointed out earlier, but I'll but it in a formal proof.

Assume X is the smallest real number greater than 1.

Let Y = .5*(1+X).

Since the real numbers are closed under addition and multiplication, Y is also a real number.

By the way Y was created (average of 1 and X) and because X is greater than 1 (by assumption), Y is less than X and greater than 1.

It was shown that Y is a real number greater than 1 and less than X, but it was assumed that X was the smallest real number greater than 1.

Since we've reached a contradiction, the assumption must be false and X is not the smallest number greater than 1. QED

[Guest]

How about 1*10^-infinty

[Guest]

This is a philosophical problem not a mathematical one for it was first postulated in Zeno’s Paradox of the Tortoise and Achilles. The real question is not that we cannot know the number but we cannot define infinity in any real terms. Only God can answer this question and there is an answer but the answer is only known in a philosophical since for philosophy deals with infinite and math finite. God who knows infinite and where finite stops could answer this question but never a man.

[Guest]

While mathematically, this number is difficult to conceptualize and it is near impossible to properly use this in the context of an equation or as a variable, there is one way to represent the number as a decimal. It does use the idea of infinity but is definitely greater than 1.

_

1.01, which reads as one point zero repeating one. Outside of this particular question, this number is quite useless and cannot really be used. But technically, it is a distinct real number, is greater than one, and there is no number smaller than it and greater than one. If there is something I am missing then please

[Guest]

Sorry my bar came out of place. It should be over the zero. However, the way the number is read is still correct.

[Guest]

This is a philosophical problem not a mathematical one for it was first postulated in Zeno’s Paradox of the Tortoise and Achilles. The real question is not that we cannot know the number but we cannot define infinity in any real terms. Only God can answer this question and there is an answer but the answer is only known in a philosophical since for philosophy deals with infinite and math finite. God who knows infinite and where finite stops could answer this question but never a man.

Zeno's Paradox of the Tortoise and Achilles has been solved. While infinity cannot be concretely defined in real terms, it can be worked with in math. Many of Zeno's paradoxes can be resolved with some simple math of infinite geometric series.

[Guest]

firstly the number would have to be irrational, and irrational numbers are uncountably infinite.

which means that there would be no way to identify the first real number after 1.

for example, consider every function that produces irrational numbers.

there all the root functions, logarithms, sine, cosine, tangent, etc.

how would you order them in any meaningful way? in short there isn't a first real number after 1.

yet we can classify numbers as being between two numbers.

so for example, we can say that while there isn't a first real number after 1,

there are numbers that are sequentially larger than 1.

you might want to look at cantors dust, and other such things for a better perspective.

Edited September 23, 2009 by phillip1882

[Guest]

Consider all rational numbers greater than 1. This set is countably infinite, as any textbook on elementary number theory will show. Yet there is no smallest number in this set in this set either.

[Guest]

While mathematically, this number is difficult to conceptualize and it is near impossible to properly use this in the context of an equation or as a variable, there is one way to represent the number as a decimal. It does use the idea of infinity but is definitely greater than 1.

_

1.01, which reads as one point zero repeating one. Outside of this particular question, this number is quite useless and cannot really be used. But technically, it is a distinct real number, is greater than one, and there is no number smaller than it and greater than one. If there is something I am missing then please

it would appear that this number is equal to 2-0.999999999999(repeating). However, I disagree that this number is a distinct real number greater than 1.

1/9 can be represented as 0.11111111 (repeating)

2/9 can be represented as 0.22222222 (repeating)

3/9 can be represented as 0.33333333 (repeating)

4/9 can be represented as 0.44444444 (repeating)

5/9 can be represented as 0.55555555 (repeating)

6/9 can be represented as 0.66666666 (repeating)

7/9 can be represented as 0.77777777 (repeating)

8/9 can be represented as 0.88888888 (repeating)

9/9 can be represented as 0.99999999 (repeating)

9/9 can also be represented as 1. 0.9999 (repeating) and 1 are the same. Therefore, I believe 2-0.9999 (repeating) is exactly 1.

[Guest]

1+epsilon where epsilon is 1/infinity. This problem deals with infinity, so that's why normal number rules don't apply. In essence, it is 1.00000... with an INFINITE number of zeroes, then a one. Which is why the density property of number isn't applicable.

[superprismatic]

Nonstandard analysis has your answer:

http://en.wikipedia.org/wiki/Nonstandard_analysis

[Guest]

In my classes, this value would be denoted as 1⁺.

It would be defined as

limit as x goes to 0 of 1+x for x>0

[Izzy]

This is a philosophical problem not a mathematical one for it was first postulated in Zeno’s Paradox of the Tortoise and Achilles. The real question is not that we cannot know the number but we cannot define infinity in any real terms. Only God can answer this question and there is an answer but the answer is only known in a philosophical since for philosophy deals with infinite and math finite. God who knows infinite and where finite stops could answer this question but never a man.

Zeno's Paradox of the Tortoise and Achilles has been solved. While infinity cannot be concretely defined in real terms, it can be worked with in math. Many of Zeno's paradoxes can be resolved with some simple math of infinite geometric series.

(My emphasis) Bow down to mathematics as your god, then. I recommend reciting pages 406 and 407 of your math textbook tonight in prayer.