Americans vastly overestimate the percentage of fellow residents who are foreign-born, by more than a factor of two, and the percentage who are in the country illegally, by a factor of six or seven. They overestimate spending on foreign aid by a factor of 25, according to a 2010 survey. And more than two-thirds of those who responded to a 2010 Zogby online poll underestimated the part of the federal budget that goes to Social Security or Medicare and Medicaid.

"It's pretty apparent that Americans routinely don't know objective facts about the government," says Joshua Clinton, a political scientist at Vanderbilt University.

Americans' numerical misapprehension can be traced to a range of factors, including where they live, the news they consume, the political rhetoric they hear and even the challenges of numbers themselves. And it isn't even clear how much this matters: Telling people the right numbers often doesn't change their views.