Dec 08 2012

Innumeracy and Magnitude

I know none of my regular readers have a problem with concepts like this because…

Well basically because they’re not ignorant rubes who reject mathematics in favor of magic.

Herr Doktor Professor

Dean Baker catches David Ignatius suggesting that trade liberalization can provide enough economic boost to offset the effects of austerity. As Dean says, the arithmetic is totally off – almost two orders of magnitude off.

Order of magnitude.  What does that mean?

Without overly complicating things (there are 10 types of people, those who know binary and those who don’t) what makes our modern, Arabic system of numeric notation superior for quick calculation to that used by Rome is that we represent numbers positionally using a zero to indicate that no elements occupy a particular multiple of our base.

When you line numbers up in columns this gives you a quick and easy way to see which numbers are about the same size.  Which is bigger- MM or MCMXCIX?



Why MCMXCIX obviously, it has more numbers.

How about this way?



Now that’s actually a bad example because 1999 and 2000 are really close together.  Let’s try something where we are actually displaying some orders of magnitude.




So 10 is 9 more than 1 and 100 is 90 more than 10.  What’s important to notice is not only is 100 more than 10, it’s a lot more than 10.  The difference is bigger between 100 and 10 than it is between 10 and 1, a lot bigger.

Conveniently for us the difference between 100 and 1 is exactly 2 orders of magnitude, just what Herr Doktor ordered.

Let’s look at what Dr. Baker says-

Ignatius’ trade deal will increase growth over the next decade by an average of 0.09 percent a year.

By comparison, the Congressional Budget Office’s projections show that the tax increases and spending cuts associated with the end of year fiscal dispute will reduce GDP by close to 4.0 percent, or roughly 40 times the impact of Ignatius’s trade deal.

40 times?  How does he come up with that number?  If we take out the confusing decimal place it looks like this-



So the real number is something like 45 (44.4) times, but that’s still rather big.

Ok, so what does this have to do with the price of tea in China?  Well, if you don’t know how to deal with orders of magnitude you might end up spending an awful lot more for your Chinese tea.  Let’s look at another example shall we?

Goldman Fined for Failing to Block Trader’s $8.3 Billion Bet

By Silla Brush, Bloomberg News

Dec 7, 2012 4:30 PM ET

Goldman Sachs Group Inc. (GS) will pay $1.5 million to settle U.S. Commodity Futures Trading Commission claims the firm failed to supervise a trader who hid an $8.3 billion position. One CFTC commissioner dissented, saying the penalty is far too small.

Goldman Sachs inadequately policed trades made by Matthew Marshall Taylor on seven days in late 2007, ultimately suffering more than $118 million in losses as his bets were unwound, according to the CFTC. Later, Goldman Sachs didn’t send the regulator “important information” on the incident that was provided to another industry watchdog, the CFTC said.

$1.5 million, that’s a lot of money isn’t it?  Perhaps to you and I, but let’s take a look at those orders of magnitude shall we?

       $8,300,000,000 == Amount of the illegal trade ($8.3 Billion)

         $118,000,000 == Amount the trade lost ($118 Million)

           $1,500,000 == Amount of the fine ($1.5 Million)

So the fine was $116.5 Million dollars less than Matthew Marshall Taylor cost Goldman Sachs just by being stupid and wrong and $8.2985 BILLION less than the amount of money at risk in the trade.

And a billion here and a billion there and pretty soon you’re talking about real money.

The moral of the story is, don’t let verbal shortcuts confuse you.

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