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MAGNITUDE |
A Bedtime Story
How the Decibel Got its Name
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MAGNITUDE |
There was once a little boy who wanted to measure sound. But the adults told him: What’s the point of that – you can’t see or touch sound, let alone measure it!
But the little boy was not deterred. He replied: Well, you can’t see or touch heat either, but you feel it all the time. And besides, we even have two measurements for it, Fahrenheit and Centigrade which I had to learn in school.
So, the boy, who was a good reader, read all of the children’s science books in the library and in one of them he found something amazing. This book was trying to entertain young people while teaching them something about science, and so it said that our ears can almost hear molecules of air moving about – so that must be really quiet – and some sounds that are a trillion times stronger! Except that you shouldn’t really try that last part.
The boy was excited – a trillion! How many zeroes was that after the 1 … he thought it was 12, but it’s easy to get lost in counting them.
So, that night he dreamed of millions and billions of sounds of all sizes and shapes, but in the morning he felt confused. How could he measure such a huge range of sounds?
He tried to explain this to his teacher, who thought he was being a bit precocious, but his teacher finally suggested he go to a particular Mathematician at the local University and get things straightened out with her. So that’s what he did one afternoon when he was free.
It turned out the Mathematician wasn’t intimidating at all, and smiled a bit quizzically at him when he arrived in her office. How can I help you, young fellow?
Please ma’m, I want to measure sound, but the numbers are too big.
The Mathematician thought this was an odd problem for a young boy, but instead of laughing, she asked him seriously if he wanted to measure all possible sounds, or just the ones you could hear safely without hurting your ears. The boy thought a moment, and said, just the safe ones we can hear, please. There’s no point in measuring what we can’t hear or are harmful, is there?
Okay, that makes it easier, the Mathematician said, and thought this boy was actually very practical after all. We’ll compare all your sounds to the quietest one you can hear. And she wrote on her whiteboard a ratio:
I / I ref
What’s “ref”, the boy asked? That’s the intensity of the quietest sound you can hear, which is why I wrote it as an I, and we’ll measure everything relative to that. Think of it as a ratio – you know that from school, don’t you?
Of course he knew about ratios, and in fact he always liked how the numbers on top could be related to the ones on the bottom. But then he frowned: But what about the trillion to 1 problem? Wouldn’t that be really top heavy and crush the poor 1 on the bottom? And besides I can’t imagine a ratio like 22 billion, 325 million and …. At this point his voice trailed off as he tried to think about such big numbers.
No problem, said the Mathematician brightly – we’ll just add a “log” to the ratio like this:
log (I / I ref)
– and pausing for effect – that will reduce your trillion-to-one to just 12-to-one! Your biggest sound will be 12 times the size of your quietest one which will be 0, because the log of 1 is zero.
The boy was astounded and his eyes widened as never before. Ma’m that’s brilliant! How did you do that so easily?
Well, we just look at how many times you’d have to multiply 10 by itself to get a trillion – it’s called the exponent, so with the log, we just look at that top part and get rid of the 10.
But the boy wasn’t satisfied. He asked, but what will we call this new unit of measurement, like how we say so many degrees of heat - but it would have to be different from that.
Well, the Mathematician said, scientists like to honour their historic colleagues, so is there anyone you think deserves to be named after your sound measurement?
The boy thought a bit, and said, well I really like talking on the phone, and wasn’t the person who invented it a Canadian who came from Scotland? There’s probably never been a measurement named for a Canadian.
Oh, you mean Alexander Graham Bell, so let’s call it the “Bell”. Your phone even has one …
The boy didn’t get the joke since he’d only seen a cellphone, but since he’d been reading about ships, he said: But that would mean I’d have to say my sound was 10 bells or something like that, and doesn’t it mean how they tell time on a ship? That would be confusing.
The Mathematician thought he had a good point, so she said: well, we could just spell it "Bel", I’m sure Alexander wouldn’t mind. And with that, she turned back to her books, expecting the boy to be satisfied.
But he wasn’t, and that was starting to get annoying to the Mathematician. But the boy looked so pleadingly at her in such an earnest way, she couldn’t send him away just yet.
Then the boy said: it was amazing the way you reduced a trillion to 12 with just that log trick, but now it’s too small and I’ll have to use all sorts of decimal places to be very accurate about my sounds.
Again, the Mathematician wondered if the boy would ever be satisfied, but instead she smiled and said: That’s even easier – we’ll just multiply it by 10, and she wrote the formula on the board:
10 . log (I / I ref)
Now what do we call it, the boy said excitedly? She had an answer for that one too: the decibel!
And that, boys and girls, is how the decibel got its name and why we always capitalize the B.