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 3×10¹², 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 1,000×1,000², and a trillion is 1,000×1,000³.

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 1,000,000¹, a billion is 1,000,000², and a trillion is 1,000,000³. 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 googolplex (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 10¹⁰¹⁰⁰. 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
101	deka-	da or D	10
102	hecto-	H	100
103	kilo-	K	1,000
106	mega-	M	1,000,000
109	giga-	G	1,000,000,000
1012	tera-	T	1,000,000,000,000
1015	peta-	P	1,000,000,000,000,000
1018	exa-	E	1,000,000,000,000,000,000
1021	zetta-	Z	1,000,000,000,000,000,000,000
1024	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.”