Fundamental flaws in human mathematics?

[Tony Heng]

Hi guys I'm wondering about something, currently in our mathematics our number system goes something like this, 1,2,3,4,5... etc all the way to 9, then the whole cycle is repeated when it reaches 10.

I believe this method of counting seems to stem from the fact that we have 10 fingers and our early ancestors might have acquired this method of counting by the fact that they are only able to count up to 10 with their hands. However this method of counting seems to have some flaw. I shall explain below.

The issue I'm concern about here is the fact that certain answers to an equation seem to go on forever. A typical example here is 1 / 3 = 0.33333333333, personally i feel this implies a fundamental flaw in our number system.

In fact when we use 0.3333333333 to multiply back by 3, we get 0.9999999999 etc, what happen to the remaining 0.0000000001?

The problem here is that the person who divide must agree with the person who multiplied the specific decimal point to round up to, in order for the "1" to be restored.

For me, math is invented by humans to understanding the codes of reality. The answer must utimately be the same everywhere regardless of the consensus of observers.

I feel the reality seems to reset its cycle at 3, instead of 10. Given that we lived in a 3 dimensional world too.

From this new perspective, the number 1-10 should look something like this:


1
2
3
11
12
13
21
22
23
31

The answer to the above equation 1 / 3 will be 0.1 exactly under this new perspective.

By resetting our mathematics fundamentally, seeing the world in "different language", it might allows us to open new insights to our reality and the wonders of mathematics and indeed the coding of reality will reveal itself to us.

 

[micromass]

Hi guys I'm wondering about something, currently in our mathematics our number system goes something like this, 1,2,3,4,5... etc all the way to 9, then the whole cycle is repeated when it reaches 10.

I believe this method of counting seems to stem from the fact that we have 10 fingers and our early ancestors might have acquired this method of counting by the fact that they are only able to count up to 10 with their hands. However this method of counting seems to have some flaw. I shall explain below.

The issue I'm concern about here is the fact that certain answers to an equation seem to go on forever. A typical example here is 1 / 3 = 0.33333333333, personally i feel this implies a fundamental flaw in our number system.

Do you understand what things like 0.333333333... even mean?? Are you familiar with infinite series??

In fact when we use 0.3333333333 to multiply back by 3, we get 0.9999999999 etc, what happen to the remaining 0.0000000001?

The problem here is that the person who divide must agree with the person who multiplied the specific decimal point to round up to, in order for the "1" to be restored.

Please read our FAQ https://www.physicsforums.com/showthread.php?t=507002

For me, math is invented by humans to understanding the codes of reality. The answer must utimately be the same everywhere regardless of the consensus of observers.

It is the same everywhere.

I feel the reality seems to reset its cycle at 3, instead of 10. Given that we lived in a 3 dimensional world too.

From this new perspective, the number 1-10 should look something like this:


1
2
3
11
12
13
21
22
23
31

Did you meant to forget things like 10 and 20? Is 3=10 here?

The answer to the above equation 1 / 3 will be 0.1 exactly under this new perspective.

This won't absolve you from the flaws. For example, here we have

[tex]\frac{1}{2}=0.111111...[/tex]

not any number system you can come up with will absolve you from flaws since the real numbers are uncountable.

By resetting our mathematics fundamentally, seeing the world in "different language", it might allows us to open new insights to our reality and the wonders of mathematics and indeed the coding of reality will reveal itself to us.

Mathematics nowadays relies very little on the decimal representation. So it won't give us anything new. Furthermore, your system of ternary representation is well known: http://en.wikipedia.org/wiki/Ternary_numeral_system

 

[Diffy]

Hi guys I'm wondering about something, currently in our mathematics our number system goes something like this, 1,2,3,4,5... etc all the way to 9, then the whole cycle is repeated when it reaches 10.

I believe this method of counting seems to stem from the fact that we have 10 fingers and our early ancestors might have acquired this method of counting by the fact that they are only able to count up to 10 with their hands. However this method of counting seems to have some flaw. I shall explain below.

The issue I'm concern about here is the fact that certain answers to an equation seem to go on forever. A typical example here is 1 / 3 = 0.33333333333, personally i feel this implies a fundamental flaw in our number system.

In fact when we use 0.3333333333 to multiply back by 3, we get 0.9999999999 etc, what happen to the remaining 0.0000000001?

The problem here is that the person who divide must agree with the person who multiplied the specific decimal point to round up to, in order for the "1" to be restored.

For me, math is invented by humans to understanding the codes of reality. The answer must utimately be the same everywhere regardless of the consensus of observers.

I feel the reality seems to reset its cycle at 3, instead of 10. Given that we lived in a 3 dimensional world too.

From this new perspective, the number 1-10 should look something like this:


1
2
3
11
12
13
21
22
23
31

The answer to the above equation 1 / 3 will be 0.1 exactly under this new perspective.

By resetting our mathematics fundamentally, seeing the world in "different language", it might allows us to open new insights to our reality and the wonders of mathematics and indeed the coding of reality will reveal itself to us.

I was under the impression that time counts as a dimension, so shouldn't 4 feel more natural to you?

And your new number system doesn't have a 0. Shouldn't it? 0, 1, 2, 3, 10, 11, 12, 13, 20, etc...

Fact is that our number system is Base 10 (see http://en.wikipedia.org/wiki/Decimal). You might be interested in other number systems (See http://en.wikipedia.org/wiki/Numeral_system).

You can have a number system base 3 aslo called a ternary number system (See http://en.wikipedia.org/wiki/Ternary_numeral_system)

If you want to have a 0 and the numbers 1, 2, 3 then you have 4 digits, so it is a base four number system or a quaternary system (see http://en.wikipedia.org/wiki/Quaternary_numeral_system)

You can really pick any base number to use, which means you can pick any positive whole number of digits to use, call it x, and your number system will then be base x.

One cool thing that you can learn from the wikipedia links I included is that irrational numbers will always be irrational in base x (where x is a positive whole number > 1). So pi, in no matter the number system, will always be irrational. And irrational number is a number that has a decimal representation that goes on forever and does not have a pattern.

1/3 is rational because there is a pattern to it's decimal representation,
0.3333333...

Notice the ... at the end. That means that it goes on forever and ever. And in this case there is a clear pattern, The next number is always three.

Now if you take 0.333333... and multiply it by 3, you are right, you do get 0.999999...

But 0.99999... is in fact 1. It is just a different representation.

I had trouble seeing that 0.999999... = 1 at first too. But ask yourself, if 0.9999... and 1 are two different numbers, then, what number comes in between them? Our numbers have a neat property that says, if two numbers are different then there must be a number in between them.

In math terms we say if x does not equal z then there exists y such that x < y < z.

I hope I answered a few of your questions, if you have any more regarding the wikipedia articles you can ask them in this thread.

 

[HallsofIvy]

Hi guys I'm wondering about something, currently in our mathematics our number system goes something like this, 1,2,3,4,5... etc all the way to 9, then the whole cycle is repeated when it reaches 10.

No, it doesn't. I think what you are talking about is our numeration system. While that is useful in writing numbers, mathematical statements are about "numbers", not "numerals", and are true or false no matter what numeration system is used.

I believe this method of counting seems to stem from the fact that we have 10 fingers and our early ancestors might have acquired this method of counting by the fact that they are only able to count up to 10 with their hands. However this method of counting seems to have some flaw. I shall explain below.

The issue I'm concern about here is the fact that certain answers to an equation seem to go on forever. A typical example here is 1 / 3 = 0.33333333333, personally i feel this implies a fundamental flaw in our number system.

In fact when we use 0.3333333333 to multiply back by 3, we get 0.9999999999 etc, what happen to the remaining 0.0000000001?

Nothing happened to it. When you "used 0.333333333" you did NOT use 1/3, which, as you say, "goes on forever" (not just "seems to"). Mathematicians have no problem with infinite decimals like that, and if you feel it is a flaw, it is in the base 10 numeration system, not with mathematics.

The problem here is that the person who divide must agree with the person who multiplied the specific decimal point to round up to, in order for the "1" to be restored.

No, it is not necssary to round up to any specific decimal place. Any such number eventually repeats and there are standard ways to indicate the repeating section. For example, anyone who has taken basic algebra should recognize "[itex]0.\overline{3}[/itex]" as meaning that the 3 repeats infinitely and do the arithmetic with that in mind..

For me, math is invented by humans to understanding the codes of reality. The answer must utimately be the same everywhere regardless of the consensus of observers.

I feel the reality seems to reset its cycle at 3, instead of 10. Given that we lived in a 3 dimensional world too.

From this new perspective, the number 1-10 should look something like this:1
2
3
11
12
13
21
22
23
31

The answer to the above equation 1 / 3 will be 0.1 exactly under this new perspective.

By resetting our mathematics fundamentally, seeing the world in "different language", it might allows us to open new insights to our reality and the wonders of mathematics and indeed the coding of reality will reveal itself to us.

Okay, mathematicians would have no problem with "base 3". But now such very useful numbers as 1/2 and 1/4 would be infinite decimals. The fact that our universe has 3 space dimensions doesn't mean there is anything special about the number "3".

(And before you suggest we change to base 3 just because there are 3 space dimensions, think about the fact that physics deals "space-time events" and according to Einstein, as far back as 1905, "We live in a four-dimensional space-time continuum". Now, of course, string theorist want 10 or even 26 dimension, only four of which we can sense. It might be a good idea for you to learn some mathematics and physics before insisting that either one is wrong.)

 

[Millennial]

Ternary number system is useful for numerous computer algorithms though; as it has the lowest radix economy. But personally I think base 60 is better for calculation. 60 has many factors, and those include 2,3 and 4. This means that all of these can be expressed as fractions of 60 when taking their reciprocal. However, 1/7 will still be a number repeating its expansion. Base 2 is more efficient, but not practical (since the numbers are very long.) This isn't a problem for a computer but it is a problem for us.

Returning to the base 60, we would need to come up with some signs representing numbers from 10-59 that are single. For the sake of this reply, I will write these representations in parantheses. In other words, you should consider (20) as the sign corresponding to 20 in the base 60 numerical system. Now, we have the following:

0,(30) = 30/60 = 1/2
0,(20) = 20/60 = 1/3
0,(15) = 15/60 = 1/4
0,(12) = 12/60 = 1/5
0,(10) = 10/60 = 1/6

As you can see, we have nonrepeating expansions of many fractions in the base 60 numerical system. We will never get rid of repeating expansions though: If a base b is not divisable by a prime p, then the multiples of the prime p will have repeating expansions in base b. In our case, it is valid for 7. Also, irrational numbers will always go on forever. If your wish is to decrease these repeating expansions as much as possible, I suggest you use a numerical system such as base 60.