𝜐 - New way to express the remain

[Valerii Kryvchyk]

Hello everyone,

For several days I have been thinking about the fractions like 10/3 and 100/16. Calculations show that when you do these problems, usually you would think about the recurring numbers at the end. However if we do the reverse action, the answer won't really equal to the first number we took the result times from the Second number. This is due to remain from the recurring numbers being taken away from the first number. Let's take for example this problem:

10 / 3 (to 1 significant figure)

From this question we see that this is going to be a recurring number as any number that isn't a multiple of 3 - will definitely have recurring end. However this problem says to solve to 1 significant figure. So literally the answer will be 3. However, where does the remaining 1 go? It is likely that this question either has no proper answer or has a remainder hidden from answer like 3 and 1 remaining. or we will just have 3 as a proper answer like a calculator would answer this question. The real answer to the problem from these two would be the first one, with 1 remaining. However, it is too tiring to write 1 remaining and if a calculator would be asked that question; he would never answer with such a long answer. So there must be a simple and understandable answer. Let's have another problem:

10 / 3 (to 3 significant figures)

From this question we could get the answer of 3.33. This would be quite correct and it would be both useful in calculators and in writing out. But nobody said that this question can't have a long answer like and 0.01 remaining. This answer would be less popular in use but would also be correct. And it would be the only correct answer as the first one doesn't show the remainder. Anyways, we still have to find the simple way to correctly and simply express the remain.

Actually, you can put as many significant figures as you want to make yourself happy. However, if you express your answer the short way, you would get an incorrect result when checking the answer. You can still put as many significant figures you like but when using the short way - You would get an incorrect outcome. Here is the reason:

10 / 3 (to 7 significant figures)

My answer is expressed in short way as 3.333333

Let's check if we got this right:

3.333333*3=9.999999

Wow! Our outcome is not equal to our first number. But why? Because our answer was expressed incorrectly! Now let's correct our question and do it the long way:

10 / 3 (to 7 significant figures)

My answer is now corrected and expressed in a long way as 3.333333 and 0.000001 remaining.

Let's check if we got it right this time or not:

3.333333*3+0.000001=10

This is a correct answer! But as we can see it, the numbers are way too long and we have to express it somehow shortly. Here is why we have recurring symbols:

3.(3 recurring) * 3 = 9.(9 recurring)

9.(9 recurring) is not equal to 10

10 - 9.(9 recurring) = 0.(0 recurring)1

So at some point at this type of problems we always going to have the rest and the rest is expressed much better over here. But I have made a better and maybe the best way to express the rest in the problem. I introduce you to 𝜐! 𝜐 is a new simple way to express the rest bit of an improper fraction! From now on, you can express the improper fraction result with Greek letter upsilon = 𝜐. Now, the answer may look something like this:

10 / 3 (to 3 significant figures):

10 / 3 = 3.3 + 𝜐

Now check if it correct:

3 * 3.3 + 𝜐 = 10

So actually, this symbol works as it should! Now, let's try another fraction:

100 / 16 (to 1 significant figure)

100 / 16 = 6 and 4 remaining

4 remaining = 4𝜐

 

100 / 16 (to 3 significant figures)

100 / 16 = 6.66 + 4𝜐

Now, we can see that 4 remaining is equal to 4𝜐 just like x + x + x + x = 4x.

Also this theory can be used on a circle diagram where if we divide it by 3 it will have 2 equal side and 1 tiny bit bigger side. To express the circle remain:

100 / 3 = 33.(3 recurring) + 33.(3 recurring) + [33.(3 recurring) + 𝜐]

I believe that introducing and using upsilon (𝜐) in maths will help understanding the fractions and converting into decimals. Providing a fresh and simpler perspective into the related topic.

If you have any questions on my thought I am free to discuss the benefits or problems of upsilon. Feel free to Ask me anything about 𝜐!

 

Kind Regards,

Valerii Kryvchyk.

[pzkpfw]

9.999... does equal 10.

0.0...1 is meaningless (except maybe in the esoteric infinitesimals) as you cannot stick a 1 after infinite zeroes.

(I'm using "..." as the more common way to represent your "(n recurring)".)

Edited 21 hours ago by pzkpfw

[Valerii Kryvchyk]

1 minute ago, pzkpfw said:

9.999... does equal 10.

Honestly, I think it doesn't as it is rounding of an infinite amount of 9s into a closest number. Like if we had 3.333 + 3.333 + 3.333 we would have  a 9.999 or if we take to 7 significant figures it would be 3.333333+3.333333+3.333333 and it would be 9.999999. This just shows that you can put as many numbers as you want and add them up so in fact it would equal to 9.999999... as 3 + 3 + 3 is 9 and the same thing will continue to infinity. I kind of understand what do you mean but in fact, my expression with 𝜐 would just be perfect without any rounding. But i understand what do you mean and I just think more about the perfect expression. If you have any questions please reply. Thanks!

[pzkpfw]

You're not the first to miss what an infinity of decimal places is doing here.

Search for discussions on 0.999... = 1

Math says 'yes', a few individuals argue 'no'.

I'm not going to get into it further here, just to note that reinventing math itself is not a good way to introduce a new symbol.

[Valerii Kryvchyk]

13 minutes ago, pzkpfw said:

9.999... does equal 10.

0.0...1 is meaningless (except maybe in the esoteric infinitesimals) as you cannot stick a 1 after infinite zeroes.

(I'm using "..." as the more common way to represent your "(n recurring)".)

I mean yes, It is true that you can't put another non-recurring number after the recurring one but in this case, I use it as an infinitestimal to show that there is still a remainder from the answer but that there is atom-size number that still exists but is not shown as 0 as 0 is nothing. My point is that there is still a remainder from 10/3 but it is so tiny we cant write it out fully but it still exists. 

[studiot]

Just now, Valerii Kryvchyk said:

Honestly, I think it doesn't as it is rounding of an infinite amount of 9s into a closest number. Like if we had 3.333 + 3.333 + 3.333 we would have  a 9.999 or if we take to 7 significant figures it would be 3.333333+3.333333+3.333333 and it would be 9.999999. This just shows that you can put as many numbers as you want and add them up so in fact it would equal to 9.999999... as 3 + 3 + 3 is 9 and the same thing will continue to infinity. I kind of understand what do you mean but in fact, my expression with 𝜐 would just be perfect without any rounding. But i understand what do you mean and I just think more about the perfect expression. If you have any questions please reply. Thanks!

I don't know what your background in mathematic is but have you seen the proof that 0.9 recurring and 1.0 refer to the same decimal number ?

This proof is part of a first year university 'foundations of mathematics' course.

Edited 20 hours ago by studiot

[pzkpfw]

On 𝜐 itself:

1. It seems the value depends on the number of significant figures you wish to work with. That would mean both of these would "work":

10 / 3 = 3.3 + 𝜐

10 / 3 = 3.33333 + 𝜐

So 𝜐 has inconsistent values, and couldn't be used in self consistent math. How's that help? (Even worse, what if you we're mixing fractions with different dividends in one formula?)

 

2. You are inconsistent, showing both:

10 / 3 = 3.3 + 𝜐

100 / 16 = 6.66 + 4𝜐

Why not 3𝜐 in the first?

 

3. This is weird:

100 / 3 = 33.(3 recurring) + 33.(3 recurring) + [33.(3 recurring) + 𝜐]

You divide by 3, and don't get three things the same size, "if we divide it by 3 it will have 2 equal side and 1 tiny bit bigger". What problem does that solve? And wasn't 𝜐 so you didn't have to write "recurring"?

 

4. Have you thought about other kinds of repeating?

22/7 = 3.142857

Edited 15 hours ago by pzkpfw

[studiot]

11 hours ago, Valerii Kryvchyk said:

I believe that introducing and using upsilon (𝜐) in maths will help understanding the fractions and converting into decimals. Providing a fresh and simpler perspective into the related topic.

If you have any questions on my thought I am free to discuss the benefits or problems of upsilon. Feel free to Ask me anything about 𝜐!

 

There is already a standard symbol for this purpose  - another greek letter epsilon.

 

Since the same epsilon concept is also used for other purposes in Analysis (eg the epsilon - delta process) what benefits does uisng upsilon bring ?

 

11 hours ago, Valerii Kryvchyk said:

I use it as an infinitestimal to show that there is still a remainder from the answer

You upsilon is not an infinitesimal in the strict sense of the word and again there is standard symbolisn for infinitesimals, which have the advantage that they can distinguish what quantity or quantities is being referred to when the algebra of infinitesimals is being used.

 

For the purposes of decimal expansions of real number you need to refer to Archimedes Condition which states that

Given any positive real number x there exists an integer n such that

1/10ⁿ <  x

(Note that none of these are infinitesimals.)

This condition is used twice to create a decimal expansion of any real number.

 

More detailed analysis of that expansion of say 0.9 recurring and 1.0 leads to the proff I referred to earlier.