Arithmetic and our place value system

[mark2142]

TL;DR

Does arithmetic in certain base of powers lead to number system in different bases ?

A power has two parts. Base and Exponent.

A number 423 in base 10 can be written in other bases as well:
1. 4* 10^2 + 2*10^1 + 3*10^0 = 423
2. 1*7^3 + 1*7^2 + 4*7^1 + 3*7^0 = 1143
3. 7*60^1 + 3*60^0 = 73

All three expressions are equal in quantity. But I have written the multiplier of powers to form numbers in different bases. Is this what place value system is in essence ?

 

[jedishrfu]

I would say yes, but it's good to get more opinions here. @fresh_42 ? @Mark44 ?

https://en.wikipedia.org/wiki/Positional_notation

In the Wikipedia page, it basically says the same thing.

 

[Mark44]

TL;DR Summary: Does arithmetic in certain base of powers lead to number system in different bases ?

A power has two parts. Base and Exponent.

A number 423 in base 10 can be written in other bases as well:
1. 4* 10^2 + 2*10^1 + 3*10^0 = 423
2. 1*7^3 + 1*7^2 + 4*7^1 + 3*7^0 = 1143
3. 7*60^1 + 3*60^0 = 73

All three expressions are equal in quantity.

I would call them equal in value.

But I have written the multiplier of powers to form numbers in different bases. Is this what place value system is in essence ?

For the most part, yes. However, the term "power" is synonymous with exponent, so for example, $3^4$ isn't a "power" -- it's an expression where 3 is raised to the fourth power; i.e. with an exponent of 4.

Your examples show that $423_{10} = 1143_{7} = 73_{60}$, with subscripts shown to indicate the various bases.

Speaking of number-system bases, 7 is not commonly used, but there is some history with base-60, as it is the base of our system of time (60 minutes in an hour and 60 seconds in a minute) as well as in angle measures, with the same subdivisions of a degree as of an hour. My understanding is that the Babylonians were the first to work in base-60.

In computer science two of the most commonly used number bases are binary (base-2) and hexadecimal (base-16). Some lesser-used number systems are octal (base-8), which was used in early computers, and base-64, which is used in some internet protocols to transmit data in a compact form.

 

[fresh_42]

TL;DR Summary: Does arithmetic in certain base of powers lead to number system in different bases ?

A power has two parts. Base and Exponent.

A number 423 in base 10 can be written in other bases as well:
1. 4* 10^2 + 2*10^1 + 3*10^0 = 423
2. 1*7^3 + 1*7^2 + 4*7^1 + 3*7^0 = 1143
3. 7*60^1 + 3*60^0 = 73

All three expressions are equal in quantity. But I have written the multiplier of powers to form numbers in different bases. Is this what place value system is in essence ?

Yes. It is called $b$-adic notation (not to be confused with $p$-adic numbers). The $b$ is the basis, in your examples $b=10, 7, 60.$ It's likely our ten fingers that led to the success of decimal ($10$) numbers.

It is a choice. The numbers are the same, only the notation differs. Computers work with powers of two as bases. It has an interesting consequence: You can do numerical analysis based on a deliberately chosen basis, which is all about computational algorithms, but you cannot do number theory based on a deliberately chosen basis. That's why number theory is mainly about divisibility as its central point of interest.

The logical foundation of arithmetic isn't so simple as it may look at first glance. How do we define our numbers without referring to a deliberate choice? It leads you directly into set theory.

 

[mark2142]

It is called b-adic notation

Can you give me a link where I can read about it? I cannot find on google.

 

[fresh_42]

The best I could find is probably:
https://www.gutenberg.org/files/17920/17920-pdf.pdf
I like that it includes historical considerations: the entire second part.

Here is a book:
https://www.amazon.com/Numbers-Grad...er-Ebbinghaus/dp/0387974970?tag=pfamazon01-20
I have found it online, but I assume it is copyright-protected, so I won't link it here even though it was on a British website.

The subject itself is so basic that it is hard to find, as it is more of a sidenote in regular calculus books rather than an isolated topic. Searching for it brought me to pages on which their digitalization is discussed, or other specific properties.

Here is "An Historical Survey of Number Systems":
https://www.math.chalmers.se/Math/Grundutb/GU/MAN250/S04/Number_Systems.pdf

I also found things like
https://www.sathyabama.ac.in/sites/default/files/course-material/2020-10/unit1_7.pdf
but I'm not so sure what it is,
or a fancy slideshow
https://www.cs.princeton.edu/courses/archive/spr15/cos217/lectures/03_NumberSystems.pdf

 

[fresh_42]

Can you give me a link where I can read about it? I cannot find on google.

My book called it so: b-adic quotients $\displaystyle{\pm \sum_{n=-k}^\infty a_nb^{-n}}.$ I'm not quite sure whether this is a general term for it, which I thought it was, since I learned it from that book, or just a personal notation by that particular author.