I publish this article with a little reservation. I am not sure if all my readers will find this interesting and I wondered if I should share this post here. But then Of This And That is supposed to be ramblings about nothing or anything and all things in between and I decided to chance it. I beg your indulgence but it is possible that I will lose some of you half way through (I am of course making the assumption that that has never happened before).
First, a puzzle to warm up. I came up with this a few years ago and shared it with a few friends when we were all exchanging different puzzles. This blog post itself is the outcome of my further reflection on the final poser.
10 + 5 = 15
5 + 13 + 31 = 144
Then: 7 + 11 = ?
When you have solved that, try 1 + 1 = ? using the same logic.
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Numbers have always fascinated me. I have written about numbers before pointing out how the interpretation of various things represented by a numeric measure affects us (see: Life by Numbers). Provided there is an agreed basis, we can compare different things by measuring them against that basis. This does not have to depend on language. If I count the number of bananas in a basket, I am going to come up with a fact, not an opinion. But if you ask me if they are large or small, then there is room for debate.
The way we represent numbers can differ though the world has standardized on the decimal representation and place values. When we write a number, it is understood by default that the base is 10. The decimal system uses the symbols 1 through 9 and 0. It is possible to use a different base - the Babylonians used 60 as the base. This sexagesimal system needed symbols for the numbers 1 through 59 but it did not have 0. The Romans used a completely different way of writing numbers but without the place value concept. It resulted in a very cumbersome system not suited for performing calculations easily. You can read more about my rant on it in A Roman Affair.
The place value concept along with zero makes it possible to have an elegant and compact way to write numbers, and to do advanced mathematics. You can have any number as the base - you will need separate symbols for all the numbers below the base and 0 to be able to make it work. In a base-8 system (octal) for example, which will use the symbols 1 through 7 and 0, the positional weights will be 1, 8, 64, 512, etc. just as in the decimal system we have 1, 10, 100,1000, etc. In the digital world, the binary system rules. Everything is 0 and 1. The positional weights go 1, 2, 4, 8, and so on. Note that 10 in any system will represent the base e.g. 10 will be the base value 8 in the octal system.
Now something strange happens when we try to use 1 as the base. I can certainly imagine such a system but how will I represent numbers using it? In any number system as seen above, the actual base does not have a separate symbol which means that '1' cannot be used. This presents a problem for the 'unary' system. Based on the examples above, I will need symbols to represent all numbers smaller than the base of 1 but no number is smaller. If I use 0, I cannot write any number other than zero because the place value will always be zero. Does this mean we cannot construct a 'unary' number system?
Imagine that the universe is the set of natural numbers {1,2,3...}. Using 1 as the base, there is no way to express the numbers in writing. It is a kind of singularity if I may use the term where all the numbers can only exist in potential form. Now consider the binary system where all we are doing is introducing the zero which really represents nothing but makes every number visible. All of a sudden we have all these numbers becoming manifest. A big bang of numbers, one might say, which immediately made me think of the origin of our universe.
Our universe is said to have begun with the Big Bang. What was there before that point of singularity? Science does not talk about it and perhaps cannot talk about it. On the other hand, the Chandogya Upanishad declares, 'All this was Existence or 'sat' (Being) in the beginning, one only, without a second'. Note that Existence does not come into being. It just was, is and will always be. This 'sat' (or Brahman) is the essence of everything in the manifest universe. Before the universe came into being, there was just pure undifferentiated Existence that is Brahman. How did the universe come about? The universe of names and forms arose because of the principle of Maya, the mysterious power of Brahman.
In the natural number universe, can we say that there is just 'number-ness' in the beginning, and all the numbers become manifest by using the principle of zero? That 'number-ness' is the 'sat' or the essence of all numbers and zero is the differentiating principle or Maya? Maya literally means 'that which is not but seems to be'. Zero though not strictly a number, yet seems to be one. It may be a loose analogy but I found the connection interesting.
One final thought on this. I sort of resolved the issue with the unary representation by dropping the zero and using just 1. I know this breaks the rules but now every place value is 1 and every number is represented by 1s. In this set, there are 1s everywhere you look. All numbers are just bundles of 1s - {1, 11, 111, 1111...}. Knowing '1' is knowing all. As for our universe, the Upanishads say, 'All this is Brahman'. And by knowing the essence ('sat'), we know everything.
Ah, how the mind can go from zero to infinity in no time at all! I do hope this ramble about 'being' and 'number-ness' has not led to some kind of numbness but has proved stimulating. You can decide if it is profound or silly or profoundly silly.
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[For those still trying to solve the puzzle, all the numbers on the left side of the equations use the decimal system. On the right hand side, the answer is expressed using different base. The base is the first number on the left side of that line. For example, 3 + 8 = 11 (decimal). Converted to base 3, this is 102. On the last line, 7 + 11 is 18 (base 10) or converted to base 7, it is 24 (base 7) so the answer is 24. Now applying the rule to 1 + 1, the answer is 2 (decimal) and we should convert this to base 1, but how do we express any number with 1 as the base?]
