Here’s Mathematician KP Hart’s Math Question and Answer for Monday, March 28th!
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You can write every number as a sum of prime numbers so that they can serve as a base for a number system However, the sum is not always unique; e.g., 6=2+2+2 en 6=3+3. Can we create some order in this so that we can also define other operations, like multiplication.
Using only the prime numbers will lead to difficulties, as noted in the explanation, because the expression as sum of primes is far from unique. One can express a preference for certain types of expressions but experience teaches that this usually leads to bad behaviour in relation to addition and multiplication.
RELATED: Read all of KP Hart’s math questions here!
There is a positional system based on prime numbers and their products. Define p0#=1, and define pn# (for n≥1) to be the product of the first n primes. So p1#=2, p2#=6, p3#=30, … Every natural number can be written as d_0p0#+d_1p1#+d_2p2#+…, where always di<pi+1, so d0<2, d1<3, d3<5, etc.
One can work with this system in exactly the same way as with the decimal system; addition and subtraction work with the same rules for `carrying’ and `borrowing’. The main difference is that the number of `digits’ increases with the position: at position 5, that is at p5#(=2310), we use the digits de `cijfers’ 0, 1, 2, 3, …, 11, 12 (all numbers below p6=13).
One could also use the prime factorization to represent numbers but then one needs all natural numbers (including 0) as digits at all positions and though multiplication is quite easy in this system addition and subtraction will be quite cumbersome.
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Read all of KP Hart’s math questions here! If you have any questions for KP Hart, tweet us or leave them in the comment section below!
About Dutch Mathematician KP Hart: In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.
I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.
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