Who’s Afraid of Big Numbers?

Pretty much everyone. But it doesn’t have to be that way, two mathematicians contend.,

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“Billions” and “trillions” seem to be an inescapable part of our conversations these days, whether the subject is Jeff Bezos’s net worth or President Biden’s proposed budget. Yet nearly everyone has trouble making sense of such big numbers. Is there any way to get a feel for them? As it turns out, there is. If we can relate big numbers to something familiar, they start to feel much more tangible, almost palpable.

For example, consider Senator Bernie Sanders’s signature reference to “millionaires and billionaires.” Politics aside, are these levels of wealth really comparable? Intellectually, we all know that billionaires have a lot more money than millionaires do, but intuitively it’s hard to feel the difference, because most of us haven’t experienced what it’s like to have that much money.

In contrast, everyone knows what the passage of time feels like. So consider how long it would take for a million seconds to tick by. Do the math, and you’ll find that a million seconds is about 12 days. And a billion seconds? That’s about 32 years. Suddenly the vastness of the gulf between a million and a billion becomes obvious. A million seconds is a brief vacation; a billion seconds is a major fraction of a lifetime.

Comparisons to ordinary distances provide another way to make sense of big numbers. Here in Ithaca, we have a scale model of the solar system known as the Sagan Walk, in which all the planets and the gaps between them are reduced by a factor of five billion. At that scale, the sun becomes the size of a serving plate, Earth is a small pea and Jupiter is a brussels sprout. To walk from Earth to the sun takes just a few dozen footsteps, whereas Pluto is a 15-minute hike across town. Strolling through the solar system, you gain a visceral understanding of astronomical distances that you don’t get from looking at a book or visiting a planetarium. Your body grasps it even if your mind cannot.

Likewise, vast sums of money become more comprehensible if they are reframed in terms of more familiar amounts. In a 2009 blog post, the mathematician Terry Tao rescaled the entire United States federal budget to the annual household spending for a hypothetical family of four. In Dr. Tao’s rescaling, a $100 million line item in the budget became equivalent to a $3 expenditure for the family.

Research in psychology and science education supports Dr. Tao’s strategy. In 2017, cognitive scientists found that students could grasp extremely long time periods — say, between the extinction of dinosaurs and emergence of humans — more readily if they created a personal timeline of the most significant events in their lives and rescaled it to progressively longer time spans: all of American history, all of recorded history and so on. These students were also better than controls at estimating numbers in the billions, an ability that is vital to understanding geological time, astronomical distances or the bewildering sums in the federal budget.

To that end, we thought it could be instructive to update Dr. Tao’s exercise, this time using the numbers in Mr. Biden’s proposed 2022 budget. For simplicity, the total money entering the federal budget — call it “income” — has been scaled to be $100,000. Meanwhile, as the graphic shows, this hypothetical nation-family spends about $144,000 a year, exceeding the budget by about $44,000. Most of the expenditure goes to four big-ticket items: about $29,000 to pay for Social Security, $18,000 for Medicare, the same for Defense and around $14,000 for Medicaid.

Scaling the Budget

The federal budget includes vast sums that are hard to comprehend, but rescaling the amounts to mimic a household budget can help. Below, total revenue of $4.174 trillion has been scaled to a hypothetical family income of $100,000.

Total Revenue

Individual income tax revenue

Social insurance, retirement taxes

Corporate income tax revenue

Other receipts

$100,000

48,850

35,026

8,888

7,259

Total Mandatory Spending

Social Security

Medicare

Medicaid

Other mandatory programs

$96,263

28,654

18,352

13,680

35,601

Discretionary Defense Spending

Department of Defense

$18,112

17,130

Non-Defense Discretionary Spending

Agriculture

Commerce

Education

Energy

Health and Human Services

Homeland Security

Housing and Urban Development

Interior

Justice

Labor

State, other international programs

Transportation

Treasury

Veterans Affairs

Major Agencies:

. Corps of Engineers

. Environmental Protection Agency

. NASA

. National Science Foundation

. Small Business Administration

. Social Security Administration

. General Services, other agencies

$22,329

668

276

2,463

1,107

3,203

1,251

1,394

417

846

340

1,524

616

359

2,710

163

268

594

244

22

235

647

Total Deficit

Primary deficit

Net interest

$44,011

36,703

7,307

Gross Federal Debt

$776,785

Total Revenue

Individual income tax revenue

Social insurance, retirement taxes

Corporate income tax revenue

Other receipts

$100,000

48,850

35,026

8,888

7,259

Total Mandatory Spending

Social Security

Medicare

Medicaid

Other mandatory programs

$96,263

28,654

18,352

13,680

35,601

Discretionary Defense Spending

Department of Defense

$18,112

17,130

Non-Defense Discretionary Spending

Agriculture

Commerce

Education

Energy

Health and Human Services

Homeland Security

Housing and Urban Development

Interior

Justice

Labor

State, other international programs

Transportation

Treasury

Veterans Affairs

Major Agencies:

. Corps of Engineers

. Environmental Protection Agency

. NASA

. National Science Foundation

. Small Business Administration

. Social Security Administration

. General Services, other agencies

$22,329

668

276

2,463

1,107

3,203

1,251

1,394

417

846

340

1,524

616

359

2,710

163

268

594

244

22

235

647

Total Deficit

Primary deficit

Net interest

$44,011

36,703

7,307

Gross Federal Debt

$776,785

By The New York Times | Note: List is simplified; totals may not add due to rounding and omitted line items.

Taken together, these four items add up to almost $80,000 in expenses for our nation-family. In addition, we must still pay off the interest on the national debt, for another $7,000, plus $36,000 on other assorted mandatory programs. So exceeding the budget by as much as Mr. Biden is proposing leaves only about $22,000 to spend on the other things we care about, the so-called nondefense discretionary spending.

When the numbers are reframed this way, the trade-offs become clearer. Want to increase funding to historically Black colleges and universities? Mr. Biden does, and he is asking the nation-family to chip in 36 cents (in these rescaled terms) to that end. What about former President Donald J. Trump’s border wall? Our nation-family spent about $388 on it in 2021. In comparison, Mr. Biden is proposing to spend $255 next year to ensure clean, safe drinking water in all communities and $5 to expand school meal programs. These choices are political ones, but at least now we can wrap our minds around how much money we’re talking about.

Why not employ a more typical diagraming strategy, like a bar chart? Well, a bar chart would reduce most items to barely visible slivers. Sometimes such large numbers are recast as percentages of the whole, but that approach suffers from the same drawback, generating confusingly small figures, like 0.01 percent. As Dr. Tao recognized, $100,000 trades on a scale with which most people are intimately familiar. Few among us, alas, will ever be a billionaire, much less a trillionaire. But we can all reasonably budget like one.

Aiyana Green is an undergraduate majoring in policy analysis and management in the College of Human Ecology at Cornell University. Steven Strogatz is a professor of mathematics at Cornell University and the author, most recently, of “Infinite Powers: How Calculus Reveals the Secrets of the Universe.”

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