# B How many numbers are there between 0 and 1

#### Brian Preece

The definition of a number on the decimal number line from British Standard 1959 (if my memory serves me right) says that a number occupies a range of values, on the number line, between -50% and +49% of the least significant digit and so each number is separated from its neighbour by 1% of the least significant digit. This means that the number 0 occupies the range -0.5 to +0.49, and the number 1 between 0.5 and 1.49. In this case there are no numbers between 0 and 1.

However if you add another digit (0.0 to 1.0) the answer returns nine. As you add digits the number increases. Some would say that, if you have an infinite number of digits in each number, then you can have an infinite number of numbers on the number line between 0.000... and 1.000... (beware the ellipsis). I refute this condition because, if the number has an infinite number of digits, then the last digit cannot be determined. Hence the number cannot be defined as above and cannot be positioned on the number line.

Another consequence of this definition is that the Quantum Mechanical theories of Heisenberg's uncertainty and Pauli's exclusion principles can be assigned to the number line.

#### fresh_42

Mentor
2018 Award
The definition of a number on the decimal number line from British Standard 1959 (if my memory serves me right) says that a number occupies a range of values, on the number line, between -50% and +49% of the least significant digit and so each number is separated from its neighbour by 1% of the least significant digit. This means that the number 0 occupies the range -0.5 to +0.49, and the number 1 between 0.5 and 1.49. In this case there are no numbers between 0 and 1.
This sounds as a definition for accountants, not mathematicians and other scientists.
However if you add another digit (0.0 to 1.0) the answer returns nine. As you add digits the number increases. Some would say that, if you have an infinite number of digits in each number, then you can have an infinite number of numbers on the number line between 0.000... and 1.000... (beware the ellipsis). I refute this condition because, if the number has an infinite number of digits, then the last digit cannot be determined. Hence the number cannot be defined as above and cannot be positioned on the number line.
Correct. The representation of a number in the decimal system has its limits and is just that: a representation. E.g. $\pi$ cannot be represented exactly in this way. The amount of infinite numbers in this representation exceeds the amount of finite numbers by far!
Another consequence of this definition is that the Quantum Mechanical theories of Heisenberg's uncertainty and Pauli's exclusion principles can be assigned to the number line.
No. You cannot misunderstand the number line and use this flawed concept to press HUP into it. One has nothing in common with the other regardless how you put it.

#### nuuskur

Answer1. There are as many real numbers as there are real numbers between 0 and 1.
Answer2. The cardinality of the interval $(0,1)$ is the cardinality of the continuum.

There are no integers satisfying $0<k<1$.

Quantum mechanical propositions are inapplicable to numbers.

#### Brian Preece

My background is in engineering. In this discipline the numberline is quantised as it refers to quantities (integer), or dimensions, which are quantised by tolerances similar to the above definition (even if we commonly use an approximation of pi). It is also currently the accepted theory that all dimensions are fundamentally quantised (well below the level of any engineering measurement).
As mathematics is the main method of describing nature, then I suggest that the numberline should reflect this and cease to be a continuum.
Furthermore I suggest that if a number has a precise location on the numberline, with an infinite number of digits in any decimal (including hexadecimal, etc) system, then it will be indistinguishable from the next number in the sequence. In other words an infinite number of numbers can occupy a precise position on the numberline. Please forgive my continued use of metaphor, but I contend that numbers are fermions, not bosons.

#### fresh_42

Mentor
2018 Award
It is also currently the accepted theory that all dimensions are fundamentally quantised (well below the level of any engineering measurement).
As mathematics is the main method of describing nature, then I suggest that the numberline should reflect this and cease to be a continuum.
There is a fundamental difference between mathematical objects and real world objects. E.g. there is no such thing as a circle in the real world. Latest under the electron microscope and circle will look like anything but not a circle anymore. Nevertheless, we can happily work with $x^2+y^2=z^2$ and its deduced formulas for sine, Pythagoras etc. Mathematical objects are an idealization. They do not demand a real world correspondence. There isn't even a straight line in the real world, but we all live well with the fact that our I-beams are.
Furthermore I suggest that if a number has a precise location on the numberline, with an infinite number of digits in any decimal (including hexadecimal, etc) system, then it will be indistinguishable from the next number in the sequence.
Your concept of indistinguishable is based on real world limitations. They do not apply to mathematics. From which digit on do you call two numbers indistinguishable? The closest way to such a constructive concept is to define, that two numbers are indistinguishable if a computer program doesn't come to a halt if it tries to figure out a difference. But this allows arbitrary long runs, not limited by human properties. But this leads back to the question of a suitable description of numbers. The constructive approach to mathematics is a stony path, and in my opinion isn't superior to the abstract concept of Plato's heavenly ideals.

#### nuuskur

What is a location of a number on the number line? Do you describe it in terms of..numbers? You would be cheating!

#### Ray Vickson

Homework Helper
My background is in engineering. In this discipline the numberline is quantised as it refers to quantities (integer), or dimensions, which are quantised by tolerances similar to the above definition (even if we commonly use an approximation of pi). It is also currently the accepted theory that all dimensions are fundamentally quantised (well below the level of any engineering measurement).
As mathematics is the main method of describing nature, then I suggest that the numberline should reflect this and cease to be a continuum.
Furthermore I suggest that if a number has a precise location on the numberline, with an infinite number of digits in any decimal (including hexadecimal, etc) system, then it will be indistinguishable from the next number in the sequence. In other words an infinite number of numbers can occupy a precise position on the numberline. Please forgive my continued use of metaphor, but I contend that numbers are fermions, not bosons.
Do you sometimes use calculus? Do you ever need to do integrals, or take derivatives? If you answered YES, you are automatically using the "continuum" of real numbers, not any form of quantization.

#### phinds

Gold Member
My background is in engineering ... I suggest that the numberline should reflect this and cease to be a continuum.
These two statements are in direct contradiction to each other.

#### Drakkith

Staff Emeritus
2018 Award
In this discipline the numberline is quantised
No it isn't.

as it refers to quantities (integer), or dimensions, which are quantised by tolerances similar to the above definition
Again, not true. Even by your original definition you'd get more numbers just by having closer tolerances. Quantities don't even have to be integers. This is a self-defeating argument.

Furthermore I suggest that if a number has a precise location on the numberline, with an infinite number of digits in any decimal (including hexadecimal, etc) system, then it will be indistinguishable from the next number in the sequence. In other words an infinite number of numbers can occupy a precise position on the numberline.
This is also incorrect, as has been said already. I suggest that instead of trying to make your own rules, you learn why we define numbers the way we do.

#### Mark44

Mentor
The definition of a number on the decimal number line from British Standard 1959 (if my memory serves me right) says that a number occupies a range of values
A measurement, to some degree of precision, can be considered to occupy a range on the number line, but every real number (which includes integers) occupies a specific location.
Furthermore I suggest that if a number has a precise location on the numberline, with an infinite number of digits in any decimal (including hexadecimal, etc) system, then it will be indistinguishable from the next number in the sequence. In other words an infinite number of numbers can occupy a precise position on the numberline.
This is flawed reasoning. For any given real number, there is no "next" number in either direction. It's possible to have multiple representations of some real numbers; e.g., the decimal representations 0.5 and 0.499... (the ellipsis has a defined meaning) are both representations of a single number: the fraction $\frac 1 2$.

#### Brian Preece

The problem I have goes back to Newton and calculus. The derivation of dx/dy uses to the statement that delta x and delta y (within the limit) go to zero. They never actually equal zero. It is for this reason that I contend that the number line is not a continuum. In the decimal system 1/3 is an irrational number (1.333 repeating) however in a system to the number fifteen, 1/3 can be written as 0.5. The choice of decimal systems has a bearing on the definition of numbers.
Everyone believes that pi is an irrational number, it cannot be defined precisely as a decimal, but is there proof that it cannot be defined as a finite number in any decimal system?

#### PeroK

Homework Helper
Gold Member
2018 Award
The problem I have goes back to Newton and calculus. The derivation of dx/dy uses to the statement that delta x and delta y (within the limit) go to zero. They never actually equal zero. It is for this reason that I contend that the number line is not a continuum. In the decimal system 1/3 is an irrational number (1.333 repeating) however in a system to the number fifteen, 1/3 can be written as 0.5. The choice of decimal systems has a bearing on the definition of numbers.
Everyone believes that pi is an irrational number, it cannot be defined precisely as a decimal, but is there proof that it cannot be defined as a finite number in any decimal system?
As others have said, you are confusing the representation of a number with its definition and properties. Mathematics is in no way dependent on the decimal notation. The properties of $\pi$, for example, are independent of how you choose to represent it.

In particular, an "irrational" number is a number that cannot be expressed as the quotient of two integers. $\pi$ is irrational. See , for example:

#### Mark44

Mentor
In the decimal system 1/3 is an irrational number (1.333 repeating) however in a system to the number fifteen, 1/3 can be written as 0.5. The choice of decimal systems has a bearing on the definition of numbers.
Any number that is the ratio of two integers is by definition a rational number, excluding of course ratios where the divisor is zero. It's not a coincidence that the root of "rational" is "ratio."

Brian Preece said:
Everyone believes that pi is an irrational number, it cannot be defined precisely as a decimal, but is there proof that it cannot be defined as a finite number in any decimal system?
$\pi$ is very much a finite number, existing as it does between 3 and 4 on the number line.

"How many numbers are there between 0 and 1"

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