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Imprenditore5

Member

Registered: 2017-12-27

Posts: 3

Theorization of new numbers

Can you help me in theorization of new numbers like 4,(1)(2)1982(3)8?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Theorization of new numbers

hi Imprenditore5

Welcome to the forum!

4,(1)(2)1982(3)8

I am not understanding what you mean.  Those are the old numbers with a comma (,) and some brackets.  Please explain if you can.  OR tell us how this question arose.  What topic in maths are you studying?

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Imprenditore5

Member

Registered: 2017-12-27

Posts: 3

Re: Theorization of new numbers

I mean numbers with more than one period.
The questione arose by reading "In alternative number systems" at https://en.wikipedia.org/wiki/0.999... (I thought 0,(9) x 10 = 9,(9)0).

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Theorization of new numbers

hi Imprenditore5

Thanks for the reply.

So are you using the brackets to mean recurring digits ?

As people are brought up and educated to count and use the decimal number system some find it hard to accept that there can be anything else.  But numbers are just an abstract concept invented by humans and defined by humans.  There is no such thing as a 'three'.  You cannot hold it or see it; only appreciate its properties.  So 3 apples no problem, but 3 out of context is meaningless.  And not everything in everyday life obeys the rules of arithmetic.  Consider: one pile of sand added to one pile of sand is one pile of sand.

So the decimal number system has to be defined and there are many texts which will tell you a possible set of definitions.  A problem arises with 1.999999 recurring because there is no number that can be sandwiched between it and 1.  I have seen two ways to resolve this: (1) Define 1.9999999 recurring to be equal to 1.  This is my preferred solution and the one described in the Wiki article.  There's lots of maths that works nicely if you use this definition.  (2) The definition used by Shilov in his book 'Elementary Real and Complex Analysis' is to dis-allow 1.9999999 recurring and all decimals ending in an infinite string of 9s completely.  Such 'numbers' are not allowed to exist!

Other number systems you might look at are (to name just two)  (1) complex numbers and (2) quaternions.   

Other recurring digit sequences can by converted to fractions like this:

example: 0.56565656........

Let 0.5656565656... = a/b.  Then 56.56565656..... = 100a/b

So by subtracting the 'infinite decimals' 99a/b = 56.  Therefore a/b = 56/99

The proof that 0.999999.... = 1 follows similar lines.

That's a start anyway.  Post back if you want more.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob