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# The Planet of Base 4

February 9, 2017

Somewhere in the universe there is a planet inhabited by beings called the Quaternary Counters or QC for short.  This friendly race is quite similar to us humans. However the QCs only have two fingers on each hand. Four fingers in total.

When they count from zero onwards, it goes as follows:

0, X, Y, Z, X0, XX, XY, XZ, Y0, YX, YY, YZ, Z0, ZX, ZY, ZZ, X00, X0X, X0Y, X0Z, XX0, XXX, XXY, XXZ, XY0, XYX, XYY, XYZ, XZ0, XZX, XZY, XZZ, Y00, Y0X,....

Like us contemporary humans, the number of digit symbols that they have is the same as the number of fingers. They just live with 4 symbols because they have 4 fingers. Their digit symbols are 0, X, Y and Z. They manage with this base 4 number system just fine.

They denote their 0 like our 0. Then our 1 is their X, our 2 is their Y and our 3 is their Z. But after Z they run out of digits, so their next number is X0. They reuse their digits at higher place values, in the same way that we do in our base 10 number system.

One day a QC traveled through the universe and met a human. It was the first such experience for her. As she observed the human, she mumbled to herself, “Look at that creature, he has Y eyes and X0 limbs. This is just like me". But then she looked closer and shrieked "He has YY fingers! Creepy!” She wasn't used to seeing so many fingers on a creature. So she ran off.

But her curiosity overcame her fear and she eventually returned to greet the human, “Hello”.

Hi” he replied waving the XX fingers of his right hand in the air. The QC smiled back waving her Y fingered right hand. The connection between the two began. The human told the QC about our planet Earth and she told him about her home. Soon enough, they got to speak about numbers...

How old are you?” he asked. “X000” she replied.

The human was puzzled because he didn't understand this number.

Aiming for some clues he asked, “How old were you last year?

ZZZ”, she replied.

And next year?”, he asked. “X00X,” she replied, continuing, “next year I will retire from work and have much time to spend with all of my grandchildren, I have X00 of them.

He still didn’t understand the meaning of all those (base 4) numbers. So he asked, “how many children do you have?” She then smiled, and raised both hands, showing her X0 fingers and said “I have X0 children”. He felt accomplished determining that X0 = 4.

The human paused to think, perhaps you can do the same now. How come X0 is 4? What is the place value of the second digit in this base 4 number system? To compare, he thought about the place value of the second digit in our number system.

The human continued, “So how many children does each of your children have?” She smiled and said, “each has exactly X0”. She then continued, “hence with X0 children, and each having X0 children of their own, I have X0 X0 = X00 grandchildren. I am blessed with so many grandchildren”.

The human then understood that X00 is actually 16. "How many fingers in total, do all of your grandchildren have?”, he asked. She thought it was a silly question, but being the courteous QC that she was, she went along with it and answered “Well, X00 grandchildren times X0 fingers for each grandchild, that makes X000." Realizing the coincidence she continued, "exactly like my age! Remember, I am X000 years old." He finally understood that she is 64 years old. Last year she was ZZZ so that must be 63.

He stopped to think more and realized that the place values in base 4 are "1", "4", "16", "64", "256", and so on... so for example ZZZ is,

Z*16+Z*4+Z*1=3*16+3*4+3*1=63.

If you are reading this to your children, I wonder if they understand base 4 now? You may have to re-read again and explore a bit yourself if not.

Notice that the choice of digits {0XY, Z} is arbitrary.  You will often find people using the digits {0, 1, 2, 3} for base 4. This is fine but requires caution when explaining this concept. For example, when looking at the number 13 you need to also indicate that it is in base 4 (it actually equals 7 in base 10). So I find that using {0, X, Y, Z} causes less confusion for an initial exploration. Also, using these symbols is better than say {0, A, B, C} because it causes less confusion if you also think about the famous base 16 (hexadecimal). In that base, the most common digit symbols are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 , A, B, C, D, E, F}.

The chat between the QC and the human continued. “How old are you?” She asked. “Me, I’m 43 he said”. She looked completely puzzled and didn't understand what 43 means.

So she just asked, “Do you have any grandchildren?”…. “No, just my children so far, they are young. I have 3 kids – I mean Z kids”.

He continued, “but sometimes it feels like 10 kids – they are really intense.” Again she shrugged, not understanding what 10 means. So he thought…. X0 is 4, XX is 5, XY is 6, XZ is 7, then Y0 is 8, YX is 9, “Yes, YY kids! Like the total number of fingers – but just a feeling, not for real.” She understood.

The QC was still puzzled about his age. She didn't understand 43. She asked him to clarify and translate this to her number system.

The human did his best...

X00 is 16 he thought. So Y00 must be twice that, 32. And Z00 is three times that, 48. So what is 43? It is somewhere between Y00 and Z00.

We could start to count up from Y00. But let’s try YX0, what is that?

YX0=2*16+1*4+0*1=36.

YY0=2*16+2*4+0*1=40.

Well, so YY0 is again not 43? What is then 43? Try YYZ

YYZ=2*16+2*4+3*1=43.

He did it! He converted his age to base 4 and yelled out, "I am YYZ years old!".

She immediately understood, smiled and replied, “Incredible! Your age is prime!

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