Where Can I Find More Information on Patterns in Mathematics?

[Square1]

consider the statement:

"every positive integer can be written in a unique way as the sum of powers of 2"


I like to "play" and look for patterns like these. What are of math is occupied with these sorts of patterns. I am looking for a book that would talk more about these kinds of patterns. Can anyone point me in the right direction?

Thanks

 

[micromass]

I guess it's just elementary number theory.
What you state is the existence and uniqueness of binary expansions. A proof can be found here: http://planetmath.org/encyclopedia/ExistenceAndUniquenessOfDecimalExpansion.html

 

[Square1]

mmm number theory. That's what I thought too. Is there a particular area of it that I should look into maybe or keywords to maybe google? Otherwise, thanks for the quick reply!

 

[DeadOriginal]

mmm number theory. That's what I thought too. Is there a particular area of it that I should look into maybe or keywords to maybe google? Otherwise, thanks for the quick reply!

I like to use Wikipedia to explore the different areas of mathematics. Of course I take everything I see on there with a grain of salt.

 

[SteveL27]

consider the statement:

"every positive integer can be written in a unique way as the sum of powers of 2"

4 = 2² = 2¹ + 2¹ = 2⁰ + 2⁰ + 2⁰ + 2⁰

so the statement is false as written.

What's true is that every positive integer can be written in a unique way as the sum of distinct powers of 2. Picky picky picky! :-)

 

[mathwonk]

good point. think about why a positive integer has a unique expression as a decimal, i.e. a sum of multiples of powers of ten, where each multiplier is less than 10.

 

[Number Nine]

Much of it would be covered under elementary number theory, though bear in mind that quite a few simple seeming statement like the one you've written require the use of complex analysis (giving us analytic number theory).

 

[Anti-Crackpot]

A simple way to think about this question:

What integer can you not express in base 2?

A few random examples...
101 = 5
110 = 6
1101 = 13
11111 = 31
1111111111111 = 8191

- AC

 

[Akshay_Anti]

in computers, we have to convert one system of expansion into another, what is called as change of base. i think you should search for base change in number theory. try reading computer programming related books. you will find all necessary info.

 

[chiro]

Following on from what Akshay_Anti said, the algorithm to convert number into a given base is called the DIV/MOD algorithm.

 

[awkward]

consider the statement:

"every positive integer can be written in a unique way as the sum of powers of 2"


I like to "play" and look for patterns like these. What are of math is occupied with these sorts of patterns. I am looking for a book that would talk more about these kinds of patterns. Can anyone point me in the right direction?

Thanks

In addition to books on number theory, you might also find books on discrete mathematics interesting, as they also discuss these kind of problems. (There is considerable overlap between number theory and discrete mathematics.) One such book is "Concrete Mathematics" by Graham et. al.

 

[luis20]

good point. think about why a positive integer has a unique expression as a decimal, i.e. a sum of multiples of powers of ten, where each multiplier is less than 10.

I believe that is somehow the definition of our decimal system, no?

 

[20Tauri]

I don't know how much math you've taken, so these may or may not be appropriate, but I used two different books for elementary number theory. The first was Number Theory by Pommersheim, Marks, and Flapan. We were expected to have had through integral calc, but I honestly don't think we needed that more than a couple times. It starts at the beginning and teaches you proof techniques as you go. I imagine even an advanced high school student would be fine with it. It's pretty clear, but it's also very silly (lots of puns, etc.), so if you don't like to mix your math and humor you might not like it. It also doesn't have any answers in the back.

The other book I used (for a follow-up independent study the next trimester) was Elementary Number Theory in Nine Chapters, by Tattersall. I think it covers a lot of the same material, but I started near the end because I'd already done a lot of that sort of stuff. It's more serious and more difficult to read, and it also doesn't have answers in the back. My professor said it was a good resource, but I found it hard to read and it had a lot of typos in it.