Number System Invariance

 Invariance is an important property in mathematics. A mathematical property is said to be invariant if it does not change under certain transformations or operations. For example the angles of a triangle do not change when the triangle is rotated or dilated. So you can say that the angles of a triangle are invariant under rotation or dilation. However the angles will change if the two axes are scaled differently.

The property of primality of numbers is invariant under number systems. The number 7 is prime whether it is expressed in decimal or binary or any other base. Properties of numbers are invariant under number systems, except those that extend into recreation or superstition. For example, the property that the sum of the digits of a number is a multiple of 9 is not number system invariant.

Now why am I even writing about this. Of course being a mathematician, albeit a lapsed one, I am interested in invariant properties. But are you? However, I can pique your interest by challenging some widely believed superstitions.

The number 9 is considered to be a holy number in Hindu culture. You have your Navagrahas, Navarathri and the Navadhanyas. The Navagrahas are considered to be 9 in number though two of them, the Sun and the Moon,  are not grahas (planets) and two others are only nodal points of intersections of orbits of Sun and Moon. So yes, 9 is considered a very lucky number. So far good. But there is a bit of a problem when this luck of 9 is extended to the belief that any number whose digits add up to a multiple of 9 is also a lucky number. As I have already pointed out this property is not an invariant property over all number systems. For example, 9 in binary is 1001 and 18 is 10010. Clearly these digits in binary don't add up to multiples of 9. 

I have heard that in Italy the number 17 is a feared number. This goes back to the Roman times. The number 17 written in Roman numerals is XVII which is an anagram of VIXI, which itself means "I have lived." The number 17 in roman numerals makes a reference to life in the past tense and is therefore a feared number in Italy. But again this property is dependent on the peculiarity of the Roman numeral system.

One can also talk about linguistic variances particularly that of names. Does it matter how you spell Lakshmi? Is Laxmi luckier than Lakshmi? The two spellings have different implications in numerology. But these properties or implications are not language invariant or alphabet invariant. Though the two names can be considered to be phonetically the same. Ironically numerology itself is not number system invariant or linguistically invariant.

The famous German mathematician Leopold Kronecker is believed to have said "God created the integers, all else is the work of man." But what number system is God using now to express the integers? Has He adopted the digital binary universe like many of us?