The Current Base Mathematics System
Currently our mathematics base system is described as working as such: a number of a specific base ‘X’ will have ‘X’ designated characters (I.e. base 10 = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and an unlimited possible amount of places. The first place is the base system raised to the power of “0” (I.e. base 10 = 10^0) as any number raised to the power of 0 = 1, this is always known as the “ones place”. The place to the LEFT of the ones place has the value of base ‘X’ raised to the power of 1 (I.e. base 10 = 10^1). This means every designated character (number) in the second place would be equal to the base value itself multiplied by the character’s value (as X^1 = X), and the next place to the left would be ‘X’ raised to the power of 2. Each additional place to the left of the ‘ones place’ simply increases the power ‘X’ is raised by according to the number of places it is to the left of the ‘ones place’.
Examples
Base: 3 8 10
Trinary Octal Decimal
102 13 11
2011 37 31
Problems with the Current Base Mathematics System
Now while the current base mathematical system appears to work at first glance, it does have some MAJOR flaws. The most immediately apparent flaw is when trying to use a “negative base value”. The first flaw becomes apparent as more and more numerical positions are placed in the number as shown below:
Yes, you saw that right… If you force a negative base using the current base system the value would oscillate between positive and negative as the number gets more numerical positions. What’s worse is how the value of the whole number increases… using just the first two digits of a number from a negative base will show this problem clearly. For this example let’s simply count (by count I mean increase the ones place by 1, reverting it to 0 and increasing the tens place by one when appropriate):
Base: -10
9 = 9 10= -10 11= -10+1= -9 12= -10+2= -8 13= -10+3= -7 14= -10+4= -6 15= -10+5= -5
16= -10+6= -4 17= -10+7= -3 18= -10+8= -2 19= -10+9= -1 20= -20 21= -20+1 = -19
You see, all this is technically mathematically correct using the current rules, just a massive headache with varying levels of illogicality. Which brings us to the biggest problem forcing a negative base in the current base system causes. A positive and a negative cancel each other out. Now I’m not just saying +10 and -10 cancel each other… because x8 and /8 also cancel ( / is the negative equivalent of * ) in fact the simple list below shows common methods of negation…
Note that multiplication (nor division) is not negated by it’s negative? This is because their negatives are used for inversion (changing positive to negative and vice-versa). This list is in no way a complete list (even considering just using the base mathematical operations, the method of negating a rooted number is not listed, the answer of which would naturally be an exponential raising of the power). However as can be seen, the inverse function as well as a negative value with the original function (given above exceptions) would each mathematically return the value of ‘n’ to it’s original value.
Due to this realization, I endeavored to find a way to legitimately make a negative base system that corrects ALL these minor flaws, (the oscillation, value jumps, rules shifting, and of course negation of inverse). The solution I found was quite simple, but first let’s look at the process that led to my solution, as well as other outcomes discovered while seeking the answer… If, however you want to jump straight to the solution you can check out Article 4: Correcting How our Base System works.
(upcoming articles)
Article 2: Seeking answers in other numerical systems
Article 3: Correcting How our Base System works
Comments