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 US federal budget to the annual household spending for a hypothetical family of four. In Dr. Tao’s rescaling, a $100 mn 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. 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.
The writers are journalists with NYT©2021
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