Yota'mik 2017-06-28

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.

This article deals with these particularities and especially highlights the differences to the human numeral system.

Annotation

The 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.

Numbers converter

Decimal:
Nazra'yot:

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The hexal system

In 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.

Examples

Decimal Nazra'yo 1 n1 2 n2 5 n5 6 n10 7 n11 11 n15 12 n20 18 n30

Decimal Nazra'yo 100 n244 102 n250 108 n300 129 n333 200 n532 1'000 n4'344 1'080 n5'000 1'865 n12'345

Simple fractions

Fractions are always represented in the Nazra'yo as [http] 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]

In this way, there are no infinite decimal places in the usual decimal fractions, as the following table shows:

Examples

Fraction Decimal Nazra'yo 1/2 0,5 n0.2 1/3 0,333333... n0.3 1/4 0,25 n0.4 1/5 0,2 n0.5 1/6 0,166666... n0.10 1/7 0,142857... n0.11

Fraction Decimal Nazra'yo 1/8 0,125 n0.12 1/9 0,111111... n0.13 1/10 0,1 n0.14 1/11 0,090909... n0.15 1/12 0,083333... n0.20 1/13 0,076923... n0.21

Complex fractures

At 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 (n3.11.1.2 - rest 0,25)
• Calculate 1/0,25 results in 4
• Remove 4 and memorize (n3.11.1.2.4 - kein Rest)
• Result is n3.11.1.2.4

Examples

Decimal Nazra'yo 0,1234 n0.12.13.1.1.1.3.1 0,69777... n0.1.2.3.4.5 2,41666... n2.2.2.2 0,444444 n0.2.3.1.332'333.15'300'442.1.2 0.444... n0.2.3.1 Sqr(2) n1.2.2.2... Sqr(3) n1.1.2.1.2.1.2... Φ (Phi, golden number) n1.1.1.1... π (Pi, circle constant) n3.11.24 (approximation to 7 digits)

Recalculate fractions

The 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 n5.3.32.2 [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

Examples in Nazra'yo