Last night I was toying with the idea of a numeration system based on primes.
A number like 24 would mean 3^2*2^4 or 144 in our base 10 system. Notice the position’s are combined multiplicatively instead of additively.
Problem 1: Since there is no regrouping in this system, does 24 mean two 3’s and four 2’s or twenty-four 2’s? We’d need to introduce a grouping symbol like a comma. Instead of using it to divide every position as in the number 3,5,7,13,4 we could simply reserve it for times when the position was filled by a number of two or more digits. So 3,5,7,13,4 could be rewritten 357,13,4. In cases where the leading digit has two or more digits we could put a comma up front. So 14 means one 3 and four 2’s, but , 14 means fourteen 2’s.
Problem 2: Since there is no regrouping, we either need an infinite number of symbols to fill the positions or to use another numeration system (as I have done).
Things that are easy in this system: multiplying and dividing are now as easy as adding and subtracting. Exponentiating now becomes as easy as the usual multiplication.
Things that are hard: adding and subtracting are now very difficult. Also ordering the numbers is hard. For example 24>10000
Problem 3: How do we write one or zero?
Interestingly, we can also expand on the other side of the decimal point.
Here’s the problem. Rewrite 12.75 from our base 10 system using this prime numeration system.
awesome pic by Lillybet Magdalena
Spoiler …
http://primepuzzle.com/tunxis/multiplicative-numeration-system.html
Speaking of Number Systems …
http://primepuzzle.com/tunxis/margaret.html
Please click the above link to see both this text and the images.
I had a 20-minute tutoring session in Number Systems this morning. I was unable to solve the problem my student wanted help on but was later able to figure it out. IMHO, this seemingly “low level,” “elementary” course is among the deepest and most complex Math courses offered at Tunxis and I am not surprised when students have difficulty with it. Another tutor later tutored the same student in the same subject and I could not help hearing some of the questions they discussed. Stuff like “show that 8128 is a perfect number” and “which primes must you use to determine if 599 is prime.” The ideas of “look it up if you have no clue what a word means” and “try all primes less than the square root of the number in question” are pretty deep ideas …
The problem my tutee and I were battling was this:
” … show why the sum of the squares of two consecutive whole numbers is 1 larger than a multiple of 4.”
Play student and see how long it takes you to prove this.
Some very basic properties of numbers and the whole idea of a proof are going on here. A scan of what I came up with follows. I’m embarrassed to tell you how many pieces of paper I scribbled on to eventually produce this. How I came up with the “identity” to be proved is pretty hard to explain.
The word “elementary” does not mean “easy.” It means “fundamental.”
I agree so many good problems can be generated by just asking questions about the naturals and the basic arithmetic ops. I think that is why number theory is one of the most loved areas by math enthusiasts. It is easy to state and understand a problem, but you can work on one for a long, long time.
A picture is worth a thousand words …
http://primepuzzle.com/tunxis/nt.html