Yota'mik is the name of the nazran numeral system resp. the nazran mathematics. In contrast to the decimal system, it contains only 6 different digits (hexal system) and represents rational numbers as continued fractions.
Mik (numbers) from 0 to 5
This article deals with these particularities and especially highlights the differences to the human numeral system.
AnnotationThe nazran symbols for the numbers are shown top-right. However, for the sake of simplicity, they aren't used in this article. Instead, arabic digits with a preceding "n" for the purpose of differentiation, are used.
Examples: ( 2501 ) is represented as n2501.
The hexal systemIn the decimal system, it's counted from 0 to 9, followed by 0, with 1 being attached ahead - thus 10 is formed. Instead, only 0 to 5 is counted in the hexal system, then again 0 follows and 1 is being attached ahead. So instead of 6 number 10 follows (wherein 1 is a six). From there, it's counted again up to 15, followed by 20, etc. It's helpful not to pronounce these hexal numbers as "ten" and "twenty" but "one-zero" and "two-zero". Because fewer digits are used, hexadecimal numbers seem to be larger in comparison - for example, the decimal 500 corresponds to a nazran n2152.
Simple fractionsFractions are always represented in the Nazra'yo as [Wikipedia] continued fraction, which is best explained by some examples:
- The number "a second" is displayed as n0.2 in the Nazra'yo.
- The number "a third" is displayed as n0.3 in the Nazra'yo.
- The number "a quarter" is displayed as n0.4 in the Nazra'yo.
- The number "a sixth" is displayed as n0.10 in the Nazra'yo. [n10 is 6]
Complex fracturesAt first glance, the simple system can also be quite complicated. For example, there's no integer equivalent in the Nazra'yo for the fraction "three quarters" (that's 0.75). One calculates 1 / 0.75 and receives 1.333 ... and thus a fraction again. The problem is handled as follows: The part before the comma is transferred into the result and the remaining fraction is divided further, until (if) it's solved.
In the example ¾, this looks like this:
- Output number is ¾, also 0,75
- Remove 0 and memorize (n0 - Rest 0,75)
- Calculate 1 / 0,75 results in 1,333...
- Remove 1 and memorize (n0.1 - rest 0,333...)
- Caculate 1 / 0.333... results in 3
- Remove 3 and memorize (n0.1.3 - no rest)
- Result is n0.1.3
A larger example:
- Output number is 3,13
- Remove 3 and memorize (n3 - rest 0,13)
- Calculate 1/0,13 results in 7.6923...
- Remove 7 and memorize (n3.11 - rest 0,6923...) [n11 is decimal 7]
- Calculate 1/0,6923... results in 1,444...
- Remove 1 and memorize (n3.11.1 - rest 0,444...)
- Calculate 1/0,444... results in 2,25
- Remove 2 and memorize (n22.214.171.124 - rest 0,25)
- Calculate 1/0,25 results in 4
- Remove 4 and memorize (n126.96.36.199.4 - kein Rest)
- Result is n188.8.131.52.4
Recalculate fractionsThe recalculation of a continued fraction is simple and linear: One begins with the last floating point number, divides 1 with this number, adds the result to the next floating point number and repeats everything until one has passed all decimal points.
Here's a detailed example:
- Output number is n184.108.40.206 [n32 is decimal 20]
- Calculate 1/2 results in 0,5
- Add 0,5 to the next number (n32) (20 + 0,5 = 20,5)
- Calculate 1/20,5 results in 0,0487...
- Add 0,0487... to the next number (n3) (3 + 0,0487... = 3,0487...)
- Calculate 1/3,0487... results in 0,328
- Add 0,328 to the last number (n5) (5 + 0,328 = 5,328)
- Result is 5,328