# Blog Archives

## Billions and Billions: A math lesson for NBC

An article by Patrick Rizzo appeared on NBC’s website today entitled, “World’s Billionaires Grow Even Richer, Led by Bill Gates.”, he wrote:

“In total, Forbes found that there were 1,645 billionaires around the world, with an aggregate net worth of $6.4 trillion (yes, that’s trillion, with a T), up from $5.4 trillion last year.”

Despite all of the political and socioeconomic ramifications of that sentence, what struck me was the parenthetical. I couldn’t decide whether (a) Rizzo was actually surprised that the aggregate net worth of over a thousand billionaires is over a trillion dollars, or (b) Rizzo thought his readers don’t know what “billion” and “trillion” mean. Either one of those possibilities makes me sad.

I briefly pondered whether he was trying to address an international audience, but dismissed the thought immediately, as his entire article would mean something different to most of Europe (and South America, and the rest of North America…) and the parenthetical in that case would be quite wrong.

In the U.S., a billion is 1,000,000,000 and a trillion is 1,000,000,000,000 (a thousand billion). It would be quite impossible to have over a thousand billionaires and not have their total net worth be over a trillion (yes, a trillion, with a T).

The good news in all of this, however, is that it gives me an excuse to dust off an article I wrote a long time ago and breathe some new life into it:

## Big numbers

Start with a three. Put 12 zeros after it. At a glance, tell someone what that number is. Our convention of grouping digits in threes helps. 3,000,000,000,000 is easier to parse mentally than 3000000000000. Still, there must be a better way, and there is.

People have been naming numbers for as long as we’ve had language. Almost nobody would refer to that number in the prior paragraph as “three followed by twelve zeros.” An American television newscaster would call it three trillion. A British newscaster before 1974 would have called it three billion (there’s a *reason* George Bernard Shaw said that “England and America are two countries separated by the same language”!). A scientist would write 3×10^{12}, or say it out loud as “three times ten to the twelfth power.” A computer programmer would write 3E12, which is a different way of saying the same thing. In fact, these are all different ways of saying the same thing.

Wait! What’s this “before 1974” stuff?

The United States has always been a bit of a rebel. Where most of the rest of the world changed number names every six digits, we silly Americans changed every three digits. Where the British would say sixty thousand million, we’d say sixty billion. This, as you can imagine, caused problems, especially for countries like poor Canada, which had to work with both the U.S. and the U.K., so in 1974, the U.K. officially changed to the U.S. system.

When I was learning number names in school, we called the “count by powers of three” system the “American” system and the “count by powers of six” system the “British” system. Once the British adopted the American system, that had to change, and they became known as the short scale and long scale, respectively.

## Names of Large Numbers

Both systems make logical sense. In the short scale (nee “American system”), our number names change with every three digits we add. Americans count the groups of three digits starting after 1,000. If you add two groups of three digits, it is called a billion (“bi” being the root for “two”). Add three groups of three digits to a thousand, and you have a trillion (“tri” for three). In exponential notation, a billion is 1,000×1,000^{2}, and a trillion is 1,000×1,000^{3}.

The British, on the other hand, changed the names with every six digits, or powers of one million. A billion was a million million. A trillion was a million billion, and so forth. The exponential notation for the long scale is easy: a million is 1,000,000^{1}, a billion is 1,000,000^{2}, and a trillion is 1,000,000^{3}. Most of Europe still uses this style of numbering, as does Mexico and most of South America (Canada uses both systems). Here are the commonly-accepted names in both systems:

# of Zeros |
Short Scale (formerly “American”) Name |
Long Scale (formerly “British”) Name |

3 | thousand | thousand |

6 | million | million |

9 | billion | milliard |

12 | trillion | billion |

15 | quadrillion | thousand billion, or billiard |

18 | quintillion | trillion |

21 | sextillion | thousand trillion, or trilliard |

24 | septillion | quadrillion |

27 | octillion | thousand quadrillion |

30 | nonillion | quintillion |

33 | decillion | thousand quintillion |

36 | undecillion | sextillion |

39 | duodecillion | thousand sextillion |

42 | tredecillion | septillion |

45 | quattuordecillion | thousand septillion |

48 | quindecillion | octillion |

51 | sexdecillion | thousand octillion |

54 | septendecillion | nonillion |

57 | octodecillion | thousand nonillion |

60 | novemdecillion | decillion |

63 | vigintillion | thousand decillion |

… | … | … |

100 | googol | googol |

… | … | … |

303 | centillion | quinquagintilliard |

… | … | … |

600 | cennovemnonagintillion | centillion |

1 googol | googolplex | googolplex |

Some of these numbers are so unimaginably huge that they have no meaning in the “real world.” A sexdecillion is larger than the number of hydrogen atoms you’d have to put in a line to span the entire universe. Yet, strangely, that still isn’t quite enough.

The googol is a huge number, bigger than the total number of hydrogen atoms in the visible universe. That massive business called Google got its name as a play on the number (albeit spelled differently), because Google’s founders were trying to build a search engine that could index a seemingly infinite amount of information.

In the never-ending quest to name a bigger number, the contest so far has been won by the googolplex. As you can see in the chart above, a googol is a one followed by one hundred zeros. A googol*plex* (no, I am *not* making this up) is a one followed by one googol zeros. This is a number so large that you couldn’t even write down the number given your entire lifetime, an endless supply of paper, and all the #2 pencils you wanted. The only way to reasonably express it is to write 10^{10100}. It’s a number only a mathematician could love.

## Prefixes of the Metric System

Just for fun, as long as we’re talking about large numbers, let’s take a moment to look at the metric system.

The allure of the metric system is that is uses a system of consistent prefixes that allow you to easily name larger and smaller units. Even though the United States isn’t using the metric system widely today, it has infiltrated our speech. We all know what a megabuck is, and can cope with a gigabyte hard disk. Electronic components are all measured in metric units, like microfarads, megawatts, and milliohms. Here, for your reading pleasure, are the common metric prefixes:

Exponent |
Prefix |
Symbol |
Number |

10^{-18} |
atto- | a | 0.000 000 000 000 000 001 |

10^{-15} |
femto- | f | 0.000 000 000 000 001 |

10^{-12} |
pico- | p | 0.000 000 000 001 |

10^{-9} |
nano- | n | 0.000 000 001 |

10^{-6} |
micro- | µ (mu) | 0.000 001 |

10^{-3} |
milli- | m | 0.001 |

10^{-2} |
centi- | c | 0.01 |

10^{-1} |
deci- | d | 0.1 |

10^{1} |
deka- | da or D | 10 |

10^{2} |
hecto- | H | 100 |

10^{3} |
kilo- | K | 1,000 |

10^{6} |
mega- | M | 1,000,000 |

10^{9} |
giga- | G | 1,000,000,000 |

10^{12} |
tera- | T | 1,000,000,000,000 |

10^{15} |
peta- | P | 1,000,000,000,000,000 |

10^{18} |
exa- | E | 1,000,000,000,000,000,000 |

10^{21} |
zetta- | Z | 1,000,000,000,000,000,000,000 |

10^{24} |
yotta- | Y | 1,000,000,000,000,000,000,000,000 |

To use these, take any unit of measurement, and put the prefix in front of the unit (no hyphens). If you’re abbreviating the unit, preceed it with the symbol instead of the full prefix. So 3,000 meters is 3 kilometers (3 km). One hundredth of a liter is a centiliter (1 cl). Two million watts is two megawatts (2 MW). Five millionths of a gram is five micrograms (5 µg). Note that the symbols greater than one are uppercase and the symbols less than one are lowercase. They’re all English letters except for the Greek letter µ (mu), which is used for “micro.”

## Online Medical Information for Patients: Where Quality Beats Quantity

*I have been pretty lucky as a freelance writer. There have been very few times that I haven’t been paid for my work. In one of those cases, I was approached by a medical magazine in Canada — it’s always nice when they call you — about doing a guest column for medical professionals about online medical information. The editor and I discussed all of the details: pay, deadlines, bylines, residual rights, and so forth. One thing I forgot to bring up was a kill fee.*

*You guessed it. When I finished the column and submitted it, nothing happened. After a few months, I was informed that the magazine had been sold and the new editor wasn’t interested in my column. Since it was a relatively small amount of money, the magazine was in another country, and I had no kill fee in the contract, I never collected a dime from the column. It’s been sitting on my hard drive for the last nine years or so (I save everything), so I decided to post it on my blog.*

*I hope you enjoy the article. If you want another story about my lymphoma experience, take a look at Rodeos, beer, and cancer.*

I found an odd lump on my chest. Neither my family doctor nor the dermatologist he called in had ever seen anything like it. They took a biopsy, and called a few days later to tell me I had an appointment with an oncologist. It was a frightening time, but I assumed it would be a melanoma, and they would simply cut it out and send me on my way. I was wrong.

The oncologist informed me that I had a dermal large b-cell lymphoma—it had manifested in my skin rather than my lymph system. He took a bone marrow biopsy, and recommended immediate monoclonal antibody and chemotherapy treatments. I left his office reeling from the news. I walked out with a little informational pamphlet about lymphoma, a fistful of prescriptions, about a half-liter of barium sulfate for the next morning’s C-T scan, and a bloodstream full of Demerol.

I’m not afraid of anything I understand, but I knew nothing about lymphoma, Rituximab, or CHOP. I hadn’t checked into a hospital since I was seven years old. I was downright terrified. My immediate mission: find out everything there is to know about this cancer in my system and its treatment methods.

I knew I’d lose the hair on my head, but what about body hair? Would I be able to keep food down? Would I lose a third of my body weight, as a friend did when he went through chemo in the 70s? Were there new, more-effective anti-nausea drugs or would I have to use medicinal marijuana, as he did? With my ravaged immune system, how could I protect myself against infection when I worked on my ranch?

The pamphlet was only mildly useful. It was just too generalized. So I turned to my usual resource for first-line research: the Internet. I found a whole lot of highly technical information that was probably wonderful for an oncologist, but useless to me. I couldn’t understand it. I also found thousands of Web pages written by patients and their friends and families. Nice, but of questionable accuracy. There just wasn’t much to be found that met my criteria: it had to be authoritative, understandable, and comprehensive.

Had I been suffering from a common disease with a well-understood and long-unchanged course of treatment, things might have been different. Dermal lymphomas, however, are rare, and monoclonal antibodies are still becoming established as mainstream treatments. There is a lot of conflicting information online, and it’s very difficult for a layman to determine which information is truly authoritative.

Medical patients simply can’t get all of their information from friends, phone-in radio shows, and newspaper columns. We can’t rely on what some unknown author keyed in to their personal Web site—even if there is an “MD” after their name. Our *lives* are on the line here.

Medical journals are written for medical professionals, and that is as it should be. The information is carefully vetted and peer-reviewed. There is a crying need for trained medical professionals to produce a body of peer-reviewed medical information written for laymen, with a meaningful stamp of approval on the individual articles.

It would not be necessary for all of this information to reside on a central Web server, nor for a single administrator to oversee it. All that would be required is a central directory or search engine for information carrying the aforementioned stamp of approval.

I ended up getting the information I wanted. I used up a lot of my chemo nurse’s time (she’s not only an RN, she’s a cancer survivor), and asked my oncologist many questions. I read articles online and in magazines. I got data from cancer support networks. Eventually—that’s the key word—I learned about my disease and its treatment. But that was no substitute for what this scared and lonely patient needed the afternoon of my diagnosis: fast, clear, information I knew I could count on.