Thursday, June 28, 2007

Unnatural Numbers

What is the next number on the sequence: 180, 90, 45, ... ? Since each number appears to be obtained by halving the previous number in the sequence, it should be 22.5. This shows that even if you start randomly with a large natural number (which is not a power of 2) and you keep halving the successive values, you will reach an odd natural number and when you halve that, you end up with a fraction (a real number). Here's a little Python code that demonstrates the point:
import random

r = random.randrange(100,1000,2)
print r

while True:
r /= 2
print r 
if r % 2 != 0: break

Run it several times and you will see that the sequence, although generally not long, contains an unpredictable number of iterates. This is easily understood by noting that division by 2 is equivalent to a bitwise right-shift (i.e., >> operator in C, Python and Perl), so each successive division simply marches the bit-pattern to the right until a '1' appears in the unit column. That defines an odd number and another right-shift produces '1' as a remainder, which I can either ignore as with integer division or carry as a fraction. It also means that you can predict when you will hit an odd number by first representing the starting number as binary digits, then counting the number of zeros from the rightmost position to the first occurrence of a 1-digit.

In the manufacture of microprocessors and memory chips, the increase in speed follows Moore's law, which is also a halving sequence like the above. Therefore, we see natural numbers like 90 nanometer (nm), 45 nm, and so on. For reference, the diameter of the HIV virus (a molecular machine) is about 100 nm. As I've discussed elsewhere, 45 nm is the next-generation technology that IBM, Intel and AMD will be using to fabricate their microprocessors. But the current smallest technology being used by AMD and Intel is not 90 nm, it's 65 nm. And the next generation after 45 nm will be 32 nm, not 20-something. So, where do these 'unnatural' numbers come from?

Each progression in shrinking the silicon-based photolithographic process is identified by a waypoint or node on a roadmap defined by the Semiconductor Industry Association or SIA. The current SIA roadmap looks like this:
Year  2004    2007    2010    2013    2016    2020 
nm     90      65      45      32      22      14

A technology node is defined primarily by the minimum metal pitch used on any product. Here, pitch refers to the spacing between the metal "wires" or interconnects. In a microprocessor, for example, the minimum metal pitch is the half-pitch of the first layer of metal used to connect the actual terminals of one transistor to another. Notice that IBM and Intel technology is moving faster than SIA expectations, so the roadmap already needs to be updated again.

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