# Is there a way to write 5.000000000 1?

• brum
In summary: It is easy to prove, and it is easy to explain why it must be true.In summary, there is no greatest number less than 5 or least number greater than 5, and the concept of a "greatest number less than" is impossible in an ordered number system. Additionally, 0.499... is numerically equal to 0.5 and there is no number between 4.999... and 5.
brum
ok so we can write greatest number that is < 5
(_
4.9 which means 4.9999999999999...

then.. what's the next highest number > 5?
` _
5.000000000000000...1 or 5.01

can you do that? (put the "repeating" there ^^)edit: didn't realize the spaces would be removed so my repeating signs were at the beginning, not to where i spaced them out to

Last edited:
4.999... = 5
Since there is no smallest number, it is not possible to represent the smallest number greater then something. Dealing with such concepts means you must specifiy intervals. The notation [1,2) implies all numbers greater then and equal to 1 but less then but not equal to 2. 1.999... = 2 so it is not included in the above interval, but any finite number of 9's is. Likewise (1,2] is the interval which contains all numbers greater then 1, but not 1, and less then or equal to 2. This is the concept of an open interval, that means the interval does NOT contain its endpoint.

5.000...1 where the elipsis represents an infinite number of 0s does not represent a real number. In that context it means a finite but unspecified number of 0s because by the definiton of real numbers the 1 MUST occupy a position which corresponds to some integer therefore there must be a finite number of 0s.

i believe this is known as a finite nonstandard number

maybe..

5 + 10^-n

=

5 + 1 / 10^n

(where n is natural number, that in this case gives arbitrary number of digits after the comma)
that, what you meant?

ok so we can write greatest number that is < 5

There does not exist a greatest real number less than 5. ("decimal number" is synonymous with "real number")

There also does not exist a least real number greater than 5.

In fact, in any ordered number system with division and 2, "the greatest number less than x" never exists. Here is a proof by contradiction:

Suppose y is the greatest number less than x. This means that y < z < x can never be true for any z, because that would mean z is greater than y, but z is less than x.

Now, let z = (x + y) / 2

Starting from y < x
add y to both sides yielding y + y < x + y
this is the same as 2y < x + y
dividing by 2 yields y < (x + y) / 2
this is the same as y < z

Starting from y < x
add x to both sides yielding x + y < x + x
this is the same as x + y < 2x
dividing by 2 yields (x + y) / 2 < x
this is the same as z < x

so y < z < x... but this is impossible!

So our assumption that "y is the greatest number less than x" was false. Since there was no restriction placed on y (beyond being the greatest number less than x), there cannot exist a number that is the greatest number less than x.

As integral mentioned, 0.499... is, by definition, numerically equal to 0.5 (so, in particular, it cannot be less than 0.5)

Hurkyl

real numbers axiom

I suppose there should not be any number like that in existence according to an axiom which states that between any two reals there is another real number -> so between 5 and 5+eps (eps small as you want it to be) there is always another number.

Though, another question arises - is then 4.999... same as 5 because 4.999... is larger than any other number smaller than 5?

Yes, As I posted above, 4.999... = 5. This can be demonstrated several different ways and can be rigoursly proven several different ways.

## 1. Can 5.000000000 1 be written as a whole number?

No, 5.000000000 1 cannot be written as a whole number because it contains a decimal point and a fractional value.

## 2. Is there a simpler way to write 5.000000000 1?

Yes, 5.000000000 1 can be written in scientific notation as 5.1 x 10^-9.

## 3. How many significant figures are in 5.000000000 1?

There are 10 significant figures in 5.000000000 1. The zeros after the decimal point are considered significant in this case.

## 4. Can 5.000000000 1 be rounded to a different value?

Yes, since the number has 10 significant figures, it can be rounded to any number of significant figures desired. For example, rounding to 5 significant figures would result in 5.0000.

## 5. What is the difference between 5.000000000 1 and 5.1 x 10^-9?

The difference is in the way the number is written. 5.000000000 1 is written in standard decimal form, while 5.1 x 10^-9 is written in scientific notation. Scientific notation is often used to express very large or very small numbers in a more compact form.

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