More Big Numbers

Millions and !!Billion!!s

There's been a bit of comment in recent months about the use of million and billion in the media, particularly with all the bank and big business bailouts that have occurred, and the apparently insatiable appetites of their fat cat executives for telephone number bonuses. And yet, just as truth is the first casualty in any conflict, so perspective is rarely allowed to stand in the way of popular journalism.

As is often the case, Randall Munroe's XKCD nailed it best (click through for the full strip and punchline):

But I'm particularly fond of the way newsreaders and politicians, in mitigation of the above, invariably explode the word "billion" at us, almost as if it begins and ends with double exclamation marks. There seems always to be a multiply exaggerated pause in the middle of the discourse, while they build up the head of steam they judge necessary to shock us out of our reverie, wherein we would otherwise have been certain to have misheard them say merely "million". Then all of a sudden, there's that precisely timed unleashing of - what can only be described as - a bomb of high pressure air, spit, and thunder: !!Billion!!

4,294,967,296

Speaking of billions, I was shocked - shocked I tell you! - on attending my first local [programming language name redacted] users' group those years ago. For some well forgotten reason, the question arose of assigning a unique IP address to each and every one of the "four point something billion" people who happened to be on the planet that day. Did anyone have any idea if IPv4 would suffice? Rhubarb ran around the room.

What's shocking about that? Well, trouble is, the industry was just then at the midpoint of its 32-bit processor "plateau", which had been heralded by the Intel 80386 processor, and so everything was encoded into 32 bits. Now if there's one arithmetical thing more important than multiplication tables to a software engineer, it's her powers of two. Each of the first sixteen is an old friend, the next sixteen are at least familiars, and the last and most important of these, two to the power of thirty-two, is famously around four billion and something. I think it may be a wisdom that's passed on through mother's milk. Remember that old lullaby?

♫ Forty-two, ninety-four, ninety-six; ♫
♫ Seventy-two, ninety-six. ♫

Anyway, I sat there in complete and utter disbelief, chin on floor. In a room full of the very people - software developers of the 32-bit era - most likely to know the answer to this one very specific question, "Can a 32-bit number represent four billion?" To be fair, I don't know how many others were similarly appalled, but I'd certainly like to think it was most of us.

Update, related: Five billionth device about to plug into Internet.

Miles And Miles From Watford

Back again to editorial perspective, and a Telegraph article on the latest Hubble eye candy, NASA's "stunning" new image of a spiral galaxy, places it "trillions of miles from earth". This attempt at an indication fails abysmally, by some nine orders of magnitude, or in other words, by a factor of a billion. The object in question is in fact almost two sextillion miles afar.

It's a significant and most unfortunate failure, in the entire history of the concise and accurate conveyancing of scientific facts. And yet all of the information that would have been needed to improve upon it, why that's already present in the body of the article.

One light year is about 6 trillion miles, and our sun's nearest stellar neighbour, Proxima Centauri, is more than 4 light years distant. So anything at all outside of the solar system is quite literally "trillions of miles from earth". Including, of course, every other single, double and multiple star in our Milky Way galaxy.

But most objects are not in our galaxy, nor anywhere near it. The island universe in question is the very beautiful (and incorrectly reported as "edge-on") spiral NGC-4911 in the Coma cluster. Its distance from ""earth", and indeed from everything else that's in our galaxy and visible in the night sky, is about 320 million light years. That's what I call almost two sextillion miles.

Perhaps the Telegraph article could have read "billions of trillions"? As it stands, it might almost as accurately and meaningfully have said "hundreds of miles from Britain." Either way, it looks like the writer knew what he was talking about, and his editor made him look foolish. Oh well, never mind, thanks for the NASA picture anyway.

Yes, it is a beautiful galaxy. I wonder what its inhabitants think of ours?

Update: here is a pretty good picture (actually more of a photo essay) of a quadrillion.