How Big is a Billion?

In my favorite New Yorker cartoon, by Bob Mankoff, a businessman sits at his desk in a well-appointed office with high-rise views and a framed portrait of a distinguished predecessor. He’s speaking, or perhaps shouting, into the phone, “A billion is a thousand million? Why wasn’t I informed of this?”

Mankoff’s businessman has reason to be dismayed. On the scale of personal finance, a billion dollars is a cartoonishly large amount of money. To borrow an analogy from a popular tweet from late 2019, one way to become a billionaire is to start making prudent financial decisions and set aside 5000 dollars each day. If you manage this, congratulations! You’ll only need to keep it up for about 550 years. To become a millionaire using this same method, you’d only have to set aside 50 dollars per day for 55 years: a challenge, no doubt, but one that doesn’t require bargaining with demons to extend your lifespan.

Needless to say, this is not how vast fortunes are actually amassed, and analogies like this have their limitations: they ignore all distinctions between different kinds of assets and don’t account for things like compound interest or inflation. Even so, this example effectively communicates just how large these large numbers are and how different they are from each other. To use an analogy that avoids messy financial details, a million seconds is about 10 days, while a billion seconds is roughly 30 years. Yet we frequently refer to these very different numbers in the same breath, implicitly collapsing the distinction between them.

I’ve been reminded of Mankoff’s cartoon often this year. Extremely large numbers appearing regularly in the news is nothing new, but as in so many other respects, 2020 has upped the ante: more than a million COVID-19 cases every week in the US alone, a record $14 billion spent on the 2020 elections, trillion-dollar stimulus packages, and even the preposterous claim by economist Charles Cicchetti that Joe Biden had only a one-in-a-quadrillion chance of winning the electoral votes of several key swing states given Donald Trump’s initial lead in tabulated ballots. Read enough of these figures in quick succession and they start to blur together. Perhaps you will experience, as I did while writing this, the sneaking suspicion that quadrillion is too silly to be the name of an actual number.

Our inability to intuitively grasp the largeness of large numbers is the subject of “Number Numbness,” a wonderful 1982 column in Scientific American by the cognitive scientist Douglas Hofstadter. In a largely playful tone, Hofstadter offers plenty of colorful examples to help readers overcome what he terms “innumeracy,” by analogy to illiteracy.

But Hofstadter never loses sight of the high stakes of innumeracy in the real world. He wants readers to understand what a trillion dollars in American military spending really means. Numbers like this should not leave us numb.

The example of America’s bloated military budget remains as relevant today as when Hofstadter was writing, and the COVID-19 pandemic has made widespread numeracy more important than ever. Of course, understanding how to interpret the overwhelming deluge of COVID-19 data is not as simple as knowing the difference between thousands and millions. As of this writing, COVID-19 has killed only half as many Americans as die each year of heart disease, but nearly nine times as many as an average flu season; the virus is likely to be the third largest cause of death in the US in 2020 after heart disease and cancer. These examples illustrate that the perceived largeness of any number depends on what we choose to compare it to.

Such comparisons, like all statements about COVID-19 data, exist in a social and political context. The Washington Post reported in July that Trump administration officials were hoping Americans would “grow numb to the escalating death toll and learn to accept tens of thousands of new cases a day.” This didn’t pan out as an electoral strategy, but public desire for a return to normalcy in the face of rising case rates remains a major impediment to keeping the pandemic in check.  

Unfamiliarity with large numbers can also be manipulated to provoke opposition rather than complacency. In Colorado, where I live, a 1992 amendment to the state constitution known as TABOR (the Taxpayer Bill of Rights) prohibits state and local governments from raising taxes without voter approval. TABOR specifically requires that any ballot measure proposing a tax increase begin with the total revenue to be collected, invariably an intimidatingly large number in the absence of context; a representative recent example is “SHALL STATE TAXES BE INCREASED BY SEVEN HUNDRED FIFTEEN MILLION ONE HUNDRED THOUSAND DOLLARS ANNUALLY.” It’s perhaps unsurprising that requests for statewide education and infrastructure funding phrased this way have been repeatedly voted down in recent years.

This brings us back to the value of large sums of money. A billion dollars is an outrageous amount of money for an individual, but comes out to only three dollars per person if equally divided among the US population of 330 million people. Whether the $8.3 billion annual budget of the National Science Foundation, the $14 billion spent last year feeding 30 million children through the National School Lunch Program, or the $110 billion projected cost of 200 B-21 Stealth Bombers counts as a lot of money is not an objective statement about numbers; rather, it inherently involves a value judgment. To make informed decisions about such questions we must consider how the costs are borne, who they benefit, and what would be lost by spending the money elsewhere.

Yet while numeracy is only a small part of responsible citizenship, it is an indispensable and underappreciated part. Mankoff’s businessman remains pompous, but having learned the costs of innumeracy the hard way, he now wants to be informed. That’s a small step in the right direction.

For a fun read about the truly mind-boggling largeness of much much much much larger numbers, check out this post on Kevin Davenport’s sadly defunct blog GravitonBomb.