There is no Such a Thing as a Next Number

[Farid7]

There is no such a thing as a next number. What do I mean by that? Let's use 1 as an example. 1 does not have a number that is next to it. You might think 2 is the next number, but 1.5 is actually in between them. You might think 1.5 is the next number, but 1.25 is between them. This goes on infinitely for every number.

 

[Warden_of_the_Storm]

And yet, there is a sequence created that goes 1 2 3 4 5...

So what if there's actually an infinite number of half numbers between each whole number? There is still a sequence of sequential numbers created by Arabic scholars and mathematicians in the 10th century AD. Why? Because it's easy and simple.

 

[PloverWing]

There is no such a thing as a next number.

It's important to define "number", as there are different sets of numbers.

The integers have the property that each number has a unique successor. The natural numbers {0, 1, 2, 3, ...} also have this property.

The rational numbers and the real numbers do not have the unique-successor property. As you point out, between any two rational numbers, there is another rational number, and between any two real numbers, there is another real number.

 

[Farid7]

And yet, there is a sequence created that goes 1 2 3 4 5...

So what if there's actually an infinite number of half numbers between each whole number? There is still a sequence of sequential numbers created by Arabic scholars and mathematicians in the 10th century AD. Why? Because it's easy and simple.

I was just thinking about how time seems to increment by going to the next number, but there is no such a thing as the next number.

It's important to define "number", as there are different sets of numbers.

The integers have the property that each number has a unique successor. The natural numbers {0, 1, 2, 3, ...} also have this property.

The rational numbers and the real numbers do not have the unique-successor property. As you point out, between any two rational numbers, there is another rational number, and between any two real numbers, there is another real number.

I am talking about all numbers.

 

[Warden_of_the_Storm]

I was just thinking about how time seems to increment by going to the next number, but there is no such a thing as the next number.

I am talking about all numbers.

Yes, that might be true, but it's also to say that a system of sequential numbers, or real numbers as was pointed out before, allows to go from one number to the next in sequence.

 

[Pommer]

There is no such a thing as a next number. What do I mean by that? Let's use 1 as an example. 1 does not have a number that is next to it. You might think 2 is the next number, but 1.5 is actually in between them. You might think 1.5 is the next number, but 1.25 is between them. This goes on infinitely for every number.

Zeno’s Paradox with numbers?

 

[Bradskii]

There is no such a thing as a next number. What do I mean by that? Let's use 1 as an example. 1 does not have a number that is next to it. You might think 2 is the next number, but 1.5 is actually in between them. You might think 1.5 is the next number, but 1.25 is between them. This goes on infinitely for every number.

You need to investigate the difference between natural numbers and real numbers.

 

[AV1611VET]

There is no such a thing as a next number. What do I mean by that? Let's use 1 as an example. 1 does not have a number that is next to it. You might think 2 is the next number, but 1.5 is actually in between them. You might think 1.5 is the next number, but 1.25 is between them. This goes on infinitely for every number.

Farid, you say there's no such thing as a next number, then you proceed to give more next numbers that only infinity can hold.

What gives here?

 

[sjastro]

There is no such a thing as a next number. What do I mean by that? Let's use 1 as an example. 1 does not have a number that is next to it. You might think 2 is the next number, but 1.5 is actually in between them. You might think 1.5 is the next number, but 1.25 is between them. This goes on infinitely for every number.

Mathematicians are very fastidious when it comes to terminology, in this case “next”.

The correct term is "nearness” as will become evident.
In mathematical lingo there is the expression.

Translating into English means there is a point “a” near the set X if there exists an element x in the set X where the distance between x and a is less than some distance r which can be infinitesimally small.

In this case r is a positive real number which is classified as follows.

In this example point A is near the set denoted by the coloured region.

In this example the point B is not near the set.

The concept of nearness can be extended to continuous functions.
For example the function y = x² is a continuous function as each successive value for the real numbers x is near each other.

The function y = 1/x on the other hand is discontinuous at the point x = 0 as there is no positive or negative real number for x which is near x =0

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The bottom line is numbers can be near or next to each other.

 

[Bob Crowley]

I'm always bemused by the term Pi,which is an irrational number yet we use it all the time for circles and trigonometry.

We can draw a circle and yet the diameter which is 2 * pi * r could not be exactly pinned down as to it's length as pi is irrational.

Pardon the obtuse Pi symbol but I didn't know how to insert the symbol in the same size as the text.

 

[Bradskii]

I'm always bemused by the term Pi,which is an irrational number yet we use it all the time for circles and trigonometry.

We can draw a circle and yet the diameter which is 2 * pi * r could not be exactly pinned down as to it's length as pi is irrational.

Pardon the obtuse Pi symbol but I didn't know how to insert the symbol in the same size as the text.

In the book Contact by Carl Sagan, our hero (heroine in the film, which has a different ending), is given a message by some alien intelligence to 'search pi'.

Spoiler alert: He does this by running a very powerful computer 24/7 to look for anything unusual in the sequence of numbers. Which is known not to repeat. After quite some time (it might have been years I think) he is deeper into the number than anyone has ever been. And then the computer signals him. It has found a sequence that repeats.

He examines it and discovers a run of 8,100 numbers that repeat endlessly. He eventually recognises the number as the square of 90. So he plots the numbers out on a grid 90 x 90. There seems to be nothing there. Yet there are a lot more uneven numbers than even. So he assigns a zero to the even numbers and a 1 to the odd. When he shows that on the screen, it shows a shape. Yeah, a circle. Hidden in the maths of a circle is the shape of the circle itself. Somebody had designed circles.

I remember reading that line and the hairs stood up on the back of my neck.

 

[Bob Crowley]

He examines it and discovers a run of 8,100 numbers that repeat endlessly....

Does Pi include a sequence of "8,100 numbers that repeat endlessly?" or did Carl Sagan undertake some artistic licence in the story?

Newcastle University had this to say about it.

Comment: Pi might look random but it's full of hidden patterns

Writing for The Conversation, Steve Humble discusses the diversity of patterns that can be seen in the digits that make up Pi.

www.ncl.ac.uk

There were some interesting tidbits -

For example, at position 768 in the pi digits there are six 9s in succession.

Apparently they call that the Feynman point after the famous physicist.

At position 17,387,594,880 you find the sequence 0123456789

But I presume you might have known that after checking every digit to position 17,387,594,880 when you had a bit of spare time?

 

[sjastro]

Ah the mysteries of π which can turn up in mysterious places such as in this infinite sum.

 

[Bradskii]

Does Pi include a sequence of "8,100 numbers that repeat endlessly?" or did Carl Sagan undertake some artistic licence in the story?

Sagan made it up. In fact, I reckon he might have thought of the idea that then built a story around it.

 

[Bob Crowley]

I might borrow the book from the library and read it while I eat a Meat Pi.

 

[Pommer]

Pi is like my ex…irrational and goes on and on…

 

[durangodawood]

In this example point A is near the set denoted by the coloured region.

View attachment 336494​

In this example the point B is not near the set.

View attachment 336495​

Do you set r at whatever value you want to set your own terms for "nearness"?

 

[sjastro]

Do you set r at whatever value you want to set your own terms for "nearness"?

The criteria for nearness require the distance between a unique point x in the set X and a point “a“ outside the set such that the distance |x-a| < r.
This means all other points xₖ in the set meets the requirement |xₖ-a| ≥ r.
As long as these requirements are met r can be any positive value.

The example using the graph y = 1/x doesn’t meet the nearness requirement as the line x = 0 is a limit.
The graph gets closer and closer to x = 0 as | y | approaches infinity which means there is no unique value for x that is nearer to x = 0 than any other value.

 

[Bob Crowley]

You seem to have a strong mathematical base. Do you have qualifications in mathematics, teach mathematics or work in some mathematical field?

 

[sjastro]

You seem to have a strong mathematical base. Do you have qualifications in mathematics, teach mathematics or work in some mathematical field?

Academically my field is in applied mathematics but employed as a scientist in private industry requiring very little maths but lots of physics and engineering.

The maths in this thread required pure mathematics which I did as an undergraduate.