Wednesday, January 23, 2008

Wednesday Math, Vol. 12: Logarithmic scales

Remember when we were discussing death statistics? We used the scale of 100,000 population so we could compare different causes of deaths in terms of how common they are in the populations they effect. A low murder rate might be less that 5 per 100,000 population, while a high one in the U.S. might be ten times larger, around 50 per 100,000. The death rate for U.S. soldiers in Iraq last year was between 500 and 600 per 100,000. Infant mortality rates, usually given on the per 1,000 live birth scale, run even higher, with the worst countries on earth seeing between 15,000 to 20,000 babies per 100,000 live births die before they reach their first birthday.

So in death statistics, the ratio between biggest and smallest things we measure is that very big is about 3,000 times bigger than very small. That may seem like a big difference, but what if the ratio of biggest things to smallest things we measure is millions or even billions of times bigger? If we have to write really big numbers (or really small decimal numbers), it's easy to make a mistake and miss a zero when writing, which means we would be off by a factor of ten, which is not good. Is there any alternative method in a situation like this?

The alternative is a logarithmic scale. I put a picture of orange juice, earthquake damage and some speakers together because the pH scale, the Richter scale and the decibel scale are all logarithmic.

On the Richter scale and the decibel scale, some smallest measurable thing is defined as the unit, and we then compare any measurable event as so many times larger than the smallest thing, then take the logarithm base ten of that number and report it.

For example, let's say an earthquake is 3,000,000 stronger than the baseline amount of energy used in the Richter scale. Log(3,000,000) = 6.4771213..., which means 10 raised to the power of 6.477123 is about 3,000,000. This earthquake would be reported as 6.5 (rounding to the nearest tenth), which is definitely noticeable for a Californian such as myself, but the truly destructive quakes are usually at around 7 or higher, which means about three times stronger than this small fry.

If instead it was a sound with 3,000,000 times more energy than the minimum on the decibel system, it would measure 6.477123... bels. Since every time you go up one bel the strength is multiplied by 10, sound engineers decided to split one bel into 10 decibels, so this sound would measure as 65 decibels, or maybe 64.8 decibels if they wanted to round it even closer.

Then we come to the pH scale, where the measurement system is still logarithmic, but a little backwards. pH measures the concentration of hydrogen ions in a chemical compound. What makes it backwards is that the more concentrated the number of ions, the lower the pH number is, as this scale shows.

Not everything listed here is edible, but as we see most of the things that are edible that are listed are somewhere between pH level 2, which is equal to the acidity of the gastric acid mixture in our stomach, and pH level 7, which is distilled water, the midpoint between acids and alkalines. Sea water is 1/10 as concentrated in hydrogen ions as pure water (pH level 8) and the Great Salt Lake is 1/100 as concentrated in hydrogen ions as sea water (pH level 10). As we can see, milk of magnesia is an alkaline at the same level as the Great Salt Lake, and the idea of it is to mix it in with our gastric acids to reduce the pH level in our stomach when it is acting up. I do not recommend using water from the Great Salt Lake in place of milk of magnesia when feeling bad, just as I do not recommend drinking acid rain instead of tomato juice, even though the scale says they are both at the same level of ionization.

I modestly propose that we start thinking about measure money on a logarithmic scale. The amounts that are being thrown around by governments and corporations are huge in comparison to the amounts being spent by you and me. For example, a friend of a friend worked in Sierra Leone as a nurse, and when I asked about the very high infant mortality rate and low life expectancy, she told me that when surgical gloves were removed each day, they were boiled and re-used the next day. Here in the U.S., or any other industrialized nation, the gloves would be discarded, but the cost was just too high in Sierra Leone for such extravagance. The concentrations of money in different parts of the world are so vastly different that these sorts of decisions happen on a daily basis.

Yay, flags of many lands™!
Today, it's Malta. Put an order in for a falcon!

Also, it's not Ireland turned 180 degrees! Hello, it's the Ivory Coast!
Or should I say, Bonjour! C'est Côte d’Ivoire!
They parlay the Fran-say over there, you know.

Now playing: David Bowie - Little Wonder
via FoxyTunes


Splotchy said...

Thank you for yet another informative post!

You are a gentleman and a scholar.

dguzman said...

As I finished that last glass of acid rain, I hit my forehead and said, "Wow, I really COULD'VE had a V-8."

CDP said...

Just leave it to bunny girl to have a wiseguy comeback. Excellent 'splainin'! I always learn something from your math posts.

Matty Boy said...

Thanks for the kind words, y'all.

Tres drole, dg, tres amusant.

FranIAm said...

Fran-say? No kidding! I am so there!

You got mad flags Matty. I have had a big influx lately but I don't track them the same way. I got Turkey, South Africa, I can't remember all of them.

Paul who is a friend o' Padre Mickey has a good flag thing going on too.

As for math... No parlez.