Decibels made ultra-simple: Part 2
The dreaded arithmetic: multiplying by 10
Deci is the prefix meaning one-tenth, so a decibel is one-tenth of a bel. But to remove even that minor complication, let's just think about bels for a moment. To show how easy it can be to work in bels, here are some examples. Remember that decibels (and of course, bels) are used to compare power, so we will compare two powers:
If one is 10 times larger than the other, the level difference is one bel.
If one is 100 times larger than the other, the level difference is two bels.
If one is 1000 times larger than the other, the level difference is what? ... (clue: count the number of zeroes) ...
You've got it! Three bels.
Bels are almost never used in practice, they are just not a very handy amount. So we should convert the bels into decibels. There are 10 decibels in a bel, so, looking at power ratios:
10:1 corresponds to a level difference of 10 decibels
100:1 corresponds to a level difference of 20 decibels
1000:1 corresponds to ... (clue: multiply the number of bels by 10) ...
You've got it! 30 decibels.
Why not the decibell?
As we're going to refer to decibels a lot, it's time to abbreviate. The bel is named after Alexander Graham Bell, but they had to drop an 'l' for fairly obvious reasons. Units are ordinary words, not names, so it's spelt with a lower case 'b'. But units that are named after people do get to keep their capital letter in the abbreviation, so it's B for bel.
The abbreviation for deci is d, so the abbreviation for decibel is dB. It is NOT db, nor DB nor Db, whatever you may read elsewhere. I'm sorry to say that the Oxford English Dictionary gets it wrong, but now you know better.
Let the calculator do the work
So far so good. We can deal with any ratio that is an exact power of 10 such as 100, 1000, and so on, just by counting the zeroes. But suppose we have a power ratio of 750:1? How can we turn that into decibels?
Well, this is where the scientific calculator comes in. Use any calculator with a "log" key. If you're using Microsoft Windows, there is a built-in calculator: to put it into scientific mode use the "view" menu.
Or you can use this Pop-up calculator (courtesy of www.calculator.org).
(On the pop-up calculator, if necessary use AC to clear the entry.)
First, enter the number 10.
Next, find and press (or, on a computer, click) the log key.
(Note: on some calculators, you press the log key first, and the number second.)
If you've done it right, the answer will be 1. Now enter 100, and press the log key (or vice versa, if your calculator works that way round). The answer should be 2. And so on. Do those numbers look familiar? (Clue: count the zeroes.)
"Log" is short for logarithm. If you don't know what logarithms are, don't worry. You've just found the log of 10, 100 and so on without a calculator, by counting the zeroes! And bels are nothing more than logarithms. So, you can work them out on any calculator that has a log key. Then it's easy enough to turn them into decibels - just multiply by 10.
What's so wonderful about logarithms is that they don't work just for the numbers like 1, 10, 100, and 1000, they also work for all the numbers in between. And indeed, any positive number, as large or small as you like.
Have a play with the calculator, working out the log of various numbers (or "taking the log"). You'll find that you can take the log of any positive number. But, not zero: if one side of a ratio is zero, then it will be infinitely smaller than the other side. Calculators can't generally deal with infinity so if you try to take the log of zero, Microsoft's calculator and most pocket calculators will give an error message. The pop-up calculator does slightly better: it says "minus infinity" which is technically correct. But it also says "NaN" which means that it is "not a number" that you can do a calculation with!
You can't have a log of a negative number either. That makes sense: how could a ratio of two powers be a negative number? Power isn't positive or negative, it just is.
Logs themselves will be negative, though, if the ratio is less than unity. As you know what the log of 10 is by now, can you guess what the log of 1/10 (or 0.1) is? Try it and see!
Two and a half easy steps
So, if the power of a sound is 750 times that of another, we could work out the level difference in decibels in two and a half easy steps with the aid of the calculator:
Take the log of 750. The result should be 2.875 ... ending with a string of figures. That gives you the answer in bels.
Multiply by ten to convert to decibels: 28.75 ...
That's two steps. The half step (you really can't call it more than that) is to reduce the long string of numbers you get, to something more convenient. There's no point in having more than one digit after the decimal point, as most things that we use decibels for, can't be measured to that precision. In fact for many purposes, whole numbers are plenty good enough, so round the answer to the nearest whole decibel.
That gives the answer: 29 decibels. A few paragraphs back, we found that a ratio of 1000:1 corresponds to a level difference of 30 dB. So you would expect the answer for a ratio of 750 to be a bit less than that.
And so it is! Now read on! (Part 3)