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“Great Zero” and the alternative negative number system

May 1, 2012

“Great Zero”

69,720,375,229,712,477,164,533,808,935,312,303,556,800
(41 digits; read as “69 duodecillion, 720 undecillion,… etc.)
This is the lowest common multiple of all numbers up to 100. I always wondered what the number would be like, since it has one property zero has in our everyday multiplication, and that’s all numbers from one to one hundred go into it. The easiest way to get a number like that would of course to be to multiply all numbers from 1 to 100. This is called the factorial of 100. But I wanted the smallest number possible. So the way to do this was to multiply the highest powers of all prime numbers. So it’s 64×81×25×49×11×13…and then all prime numbers up to 97. 100 will go into it, because the 64 contains 4, and then multiplying it by 25 gives you something divisible by 100. And so with so many other numbers. There are many numbers higher than 100 that go into it as well. Also, since all two digit numbers go into it, if you replace the two zeroes at the end with any two digit integer, that number will go into the new number as well.
It was in the 90’s; I don’t have a computer yet, but I got the idea and someone let me use the computer at their office. I don’t even remember what program that was, she just brought up something and I could multiply all the numbers. The calculator in my Windows accessories began doing that “e +40” stuff where it leaves out digits and makes it a huge decimal, after 73. Whatever program the office computer had (I just remember it being done on a blank screen) didn’t do that. I then printed it out.
Just for kicks; I multiplied the whole number by the next prime number, 101 sometimes later. That way, cool numbers like 1111 and its multiples would go into it. 111 already goes into it, because that’s 3×37. 1001 goes into it because that number is 7×11×13; and by extension, multiples like 111,111, which is 111×1001. So the number with 101 multiplied in is:
7,041,757,898,200,960,193,617,914,702,466,542,659,236,800
(7 tredecillion…)

Alternative Negative Number system

These numbers can also help get an idea of an alternate negative number system I had thought of. Instead of the number line to the left of zero being a mirror image of the positive numbers, it would be a backwards extension of the positive numbers. So 0 – 1, which is the same as …0000000000000 – 1, would be “…999999999999”, instead of “-1”. subtract another one, and it would be “…9999999999998”., 0 – 10 would be …9999999990. 0 – 100 would be …9999999900. Think if you had a very large number like “googol”, discussed below [next entry]. That’s a 1 followed by 100 zeroes. Subtract 1 from it, and you have a string of 100 nines. Subtract 10 and you have 99 nines followed by a zero. Subract 100, and you have 98 nines followed by two zeroes. And So on. The only problem, as big as a googol is, many numbers do not go into it. Its prime factors are nothing but 2 and 5. No other prime number (3, 7, 11, etc) or their multiples will go into it. But with these “great zero” numbers, all numbers from 1 to 97 or 101 will go into it, plus many more; so going backwards from it, you get a sense of this new “negative number” system I am talking about.
So the next number divisible by 3 going backwards, instead of “-3”, will be “….97”. The next multiple of 7 will be “…93”. It’s as if I have looped the number line around so that zero is both the beginning and the end (infinity), to a limited extent. Hence in the new “negative” numbers, the infinite number of digits to the left of the ones place (still the rightmost place by default). I had always tried to imagine a hypothetical “end” of the number line. You get one last number consisting all of nines, and then add that final one, and get zeroes. Of course, both the nines and the zeroes would have been infinitely long, to the left. And this is what I have here, but with zero as the infinitely far end. The beginning and end are right before you; it is just a circle with an infinitely long circumference. This is another mathematical concept I mention below.

If you were to hold the zero in the ones place, and realizing that there are an infinite number of zeroes to the left; then subtract 1, but treating it as a very large multiple of a large power of 10, then instead of “-1”, you keep putting 9’s, and then subtracting 1 from the next zero to the left, since you had to take away from the previous zero. You will end up with an infinite series of nines to the left. And so on. One oddity, is that …9999999999 appears to consist of all nines, and any such number is always divisible by both 3 and 9. (9×…1111111111). But this is not the case here, because …99999999999 would be 0 – 1, and zero itself is 3×0 and 9×0. In fact, the …9999999999 number would be prime! Every other number meets at …00000000000, so only 1 would go into …99999999999. It is really -1, remember! …11111111111111 would be the number divided by 9, but that number would be infinitely far away. It would basically be the new -.1111111111…, which is -1/9.

Multiplication and division really don’t work with these numbers. After all, they have infinitely many places to the left of the decimal point. You couldn’t even start multiplying them. (where numbers like 1/9 or 1/3 have infinitely many places to the right of the decimal, you could start multiplying them, and realize that it just continues on left to right forever). Thus, it would not help define the square root of -1 (i.e. “imaginary” numbers) either. So this new negative number system is primarily for addition and subtraction. (The only way to cross over to the negative range with division or multiplication is by directly using negative numbers, and then it works exactly the same as dividing or multiplying positive numbers; only the – or + sign changes). Just thought it would be an interesting alternative. Would make subtraction into the below 0 range easier.

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3 Comments
  1. Rob permalink

    Hi – interesting article, and a very interesting number. Just a quick comment on the name — I think it would actually be 69 duodecillion, 720 undecillion… and multiplying by 101 gives you 7 tredecillion, etc. Not that the name is important but thought I’d comment. (You can check the name at wolframalpha.com)

    • Oh, OK, thanks!
      I guess in counting the groups of three digits and using the Latin names (bi, tri, etc), I must have missed one. (What makes it easy to get thrown off is that the groups of zero don’t match the prefix, because it starts with million at 2×3 “0”s, but then billion is 3, quadrillion is 4, etc).

      Interesting site. I had found number crunching sites before, but not this one. I knew of the wolfram site, but didn’t know they had this.
      (See it doesn’t menion that “googol” is ten duotrigintillion).

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