A guide to numerical literacy.

A Guide to Understanding a Million vs. a Billion vs. a Trillion

A Guide to Understanding a Million vs. a Billion vs. a Trillion

The state of the universe.
Nov. 17 2017 5:10 PM

A Guide to Understanding a Million vs. a Billion vs. a Trillion

It’s really easy to read past these words, and extremely difficult to internalize how dramatically different these amounts are. These thought experiments can help.

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It’s all relative.

Ingram Publishing

This story was originally published on the Conversation and was republished here with permission.

The Conversation

National discussions of crucial importance to ordinary citizens—such as funding for scientific and medical research, bailouts of financial institutions, and the current Republican tax proposals—inevitably involve dollar figures in the millions, billions, and trillions.

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Unfortunately, math anxiety is widespread even among intelligent, highly educated people.

Complicating the issue further, citizens emotionally undeterred by billions and trillions are nonetheless likely to be ill-equipped for meaningful analysis because most people don’t correctly intuit large numbers.

Happily, anyone who can understand tens, hundreds, and thousands can develop habits and skills to accurately navigate millions, billions, and trillions. Stay with me, especially if you’re math-averse: I’ll show you how to use school arithmetic, common knowledge, and a little imagination to train your emotional sense for the large numbers shaping our daily lives.

Unlike Star Trek’s Mr. Spock, scientists and mathematicians are not exacting mental calculators but habitual estimators and analogy-makers. We use “back of the envelope” calculations to orient our intuition.

The bailout of AIG after the mortgage-backed securities crisis cost more than $125 billion. The Panama Papers document upward of $20 trillion hidden in a dark labyrinth of shell companies and other tax shelters over the past 40 years. (The recently published Paradise Papers paint an even more extensive picture.) On the bright side, we recovered $165 million in bonuses from AIG executives. That’s something, right?

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Let’s find out: On a scale where a million dollars is one penny, the AIG bailout cost taxpayers $1,250. The Panama Papers document at least $200,000 missing from the world economy. On the bright side, we recovered $1.65 in executive bonuses.

In an innumerate world, this is what passes for fiscal justice.

Let’s run through that again: If one penny represents a million, then one thousand pennies, or $10, represents a billion. On the same scale, one million pennies, or $10,000, represents a trillion. When assessing a trillion-dollar expenditure, debating a billion dollars is quibbling over $10 on a $10,000 purchase.

Here, we’ve scaled monetary amounts so that “1,000,000” comprises one unit, then equated that unit to a familiar—and paltry—quantity, one penny. Scaling numbers to the realm of the familiar harnesses our intuition toward understanding relative sizes.

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In a sound bite, a savings of $200 million might sound comparable to a $20 trillion cost. Scaling reveals the truth: One is a $2 (200 cent) beverage, the other the $200,000 price of an American home.

Suppose you landed a job paying $1 per second, or $3,600 per hour. (I assume your actual pay, like mine, is a tiny fraction of this. Indulge the fantasy!) For simplicity, assume you’re paid 24/7.

At this rate, it would take 1 million seconds to acquire $1 million. How long is that in familiar terms? In round numbers, 1 million seconds is 17,000 minutes. That’s 280 hours, or 11.6 days. At $1 per second, chances are you can retire comfortably at the end of a month or few.

At the same job, it takes 11,600 days, or about 31.7 years, to accumulate $1 billion: doable, but you’d better start young.

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To acquire $1 trillion takes 31,700 years. This crummy job doesn’t pay enough!

This analogy gives a taste for the absolute size of a billion, and perhaps of a trillion. It also shows the utter impossibility of an ordinary worker earning $1 billion. No job pays a round-the-clock hourly wage of $3,600.

Let’s examine the wealth of actual multibillionaires. Our calculations prove that they acquired more than $1 per second over long intervals. How much more?

Testifying to the Senate Judiciary Committee on July 27, William Browder, an American-born businessman with extensive Russian dealings, estimated that Vladimir Putin controls assets of $200 billion. Let’s assume this figure is substantially correct and that Putin’s meteoric rise began 17 years ago, when he first became president of Russia. What is Putin’s average income?

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Seventeen years is about 540 million seconds; $200 billion divided by this is … wow, $370 per second. That’s $1,340,000 per hour. Yet even at this colossal rate, acquiring $1 trillion takes 85 years.

The Panama Papers document some $20 trillion—the combined fortunes of 100 Vladimir Putins—sequestered in shell companies, untaxed and untraceable. Though the rate of leakage has surely increased over time, for simplicity let’s assume this wealth has bled steadily from the global economy, an annual loss of about $500 billion.

How much is this in familiar terms? To find out, divide $500 billion by 31.6 million seconds. Conservatively speaking, the Panama Papers document an ongoing loss averaging $16,000 per second, around the clock, for 40 years.

American cities are now vying for a $5 billion Amazon headquarters, a windfall to transform the local economy lucky enough to win the contract. At the same time, the world economy hemorrhages that amount into a fiscal black hole every few days. Merely stemming this Niagara (not recovering the money already lost) would amount to 100 new Amazon headquarters per year.

The root cause of our economic plight looms in plain sight when we know the proper scale on which to look. By overcoming math phobia, wielding simple arithmetic, refusing to be muddled by “gazillions,” we become better citizens, avoiding squabbling over pennies when tens of thousands of dollars are missing.

The Conversation
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Andrew D. Hwang is an associate professor of mathematics at the College of the Holy Cross.