Why Big Numbers Are Hard to Grasp: Millions, Billions and the Number Line in Your Head

Million, billion, trillion. The words sound like neighbours, a short step apart, the way Tuesday follows Monday. In fact each is a thousand times the one before: going from a million to a billion is the same thousandfold jump as going from one to a thousand. Most of us know that. Very few of us feel it.

That gap between knowing and feeling is where estimates go wrong, in the news, in budgets and in games. It is why the hardest part of What's Your Guess? for many players is not the reasoning but the size of the answer: knowing that a chain of sensible steps should land in the billions, and then typing a number in the millions because it felt big enough.

The most famous way to see the gap is with time.

• A million seconds is about 11 and a half days. A long holiday.
• A billion seconds is about 31 years and 8 months. A large part of a life.
• A trillion seconds is about 31,700 years, more than twice as long ago as the first farms.

The words differ by a few letters. The spans run from under a fortnight to most of a career to the Stone Age. If those three lines surprise you, even a little, that surprise is the subject of this article.

Some of the clearest evidence about how we picture numbers comes from a simple task. Give someone a line with 0 at one end and 1,000 at the other, and ask them to mark where a number such as 150 goes.

In a 2003 study in Psychological Science, Robert Siegler and John Opfer gave that task to children and adults. Adults and older children placed numbers more or less evenly along the line, as they should. Younger children did not. Their marks followed a logarithmic pattern: they spread the small numbers out generously and crowded the big ones together near the end, as if the distance from 10 to 100 were about as big as the distance from 100 to 1,000.

The same children could be logarithmic on one scale and linear on another: on a range they knew well, such as 0 to 100, many placed numbers evenly, then fell back on the compressed pattern on the line to 1,000. Later work by Siegler and Julie Booth found the same shift a few years earlier on the 0 to 100 line, between kindergarten and second grade. It looks as if we start with a compressed sense of number and learn the even, linear one range by range, as each range becomes familiar.

The compressed pattern is not a childish mistake that schooling simply removes. In a 2008 study in Science, Stanislas Dehaene and colleagues worked with the Mundurucu, an Amazonian people whose language has few number words and whose adults had often had little schooling. Asked to place quantities from 1 to 10 on a line, the adults placed them logarithmically too, with more room between 1 and 2 than between 9 and 10.

And educated adults go back to the old habit as soon as the numbers get big enough to be unfamiliar. In a 2013 study in Cognitive Science, David Landy, Noah Silbert and Aleah Goldin asked adults to place numbers on a line running from one thousand to one billion. One million belongs very close to the left end, at a tenth of one per cent of the way along. A large share of the adults put it near the middle. They were not being careless: they treated thousand, million and billion as three evenly spaced steps, because that is how the words present them.

A compressed scale is not a design fault. For most of what our senses handle, what matters is the ratio, not the difference. One apple against two is a big change; 101 against 102 is not. The rule that the change we notice grows with the size of what is changing goes back to the 19th-century work of Ernst Weber and Gustav Fechner, and the Weber–Fechner law is the name it now goes by.

So a sense of number built on ratios is natural and often useful. It is, in fact, exactly how What's Your Guess? scores a guess: by how many times off it is, so that ten times too big and ten times too small count as the same size of mistake, and the bar on the reveal is drawn on a log scale with the answer in the middle. The trouble is not thinking in ratios. The trouble is losing count of the steps, so that each factor of a thousand between the words feels like one small step.

Our number words help with small numbers and hide the big ones. Thousand, million, billion and trillion each sound like a single unit, and nothing in the words tells you that every step multiplies by a thousand. "Two million" and "two billion" differ by one letter.

The words have not even always meant the same thing. In the older long scale, a billion was a million million, 10¹², and that was the usual meaning in British English until the British government formally adopted the American short-scale billion, a thousand million, in 1974. Much of continental Europe still uses the long scale: in French and German, the word for a thousand million is milliard or Milliarde, and billion or Billion means a million million. A headline translated without care can be off by a factor of a thousand. Wikipedia's article on long and short scales sets out who uses which.

The fix is not more effort but better comparisons. A few that work:

• Turn it into time. Count at one a second and see how long it takes, as above. A thousand is a quarter of an hour; a million is days; a billion is decades.
• Share it out. Divide by the number of people. A billion dollars shared among the world's 8 billion or so people is about 12 cents each; a trillion is about $120 each. Suddenly the difference has a shape.
• Walk it. A million millimetres is a kilometre, a short walk. A billion millimetres is 1,000 km, a bit further than London to Berlin. A trillion is a million kilometres, more than twice the distance to the Moon.
• Climb a ladder of anchors. Hang each word on something you know: a thousand is the pupils in a big secondary school, a million is the people in a big city such as Birmingham, a billion is the people of India or China, and a trillion is the dollars the whole US economy turns over in under a fortnight.
• Count the zeros, or the powers. Write 3 billion as 3 × 10⁹ and 3 million as 3 × 10⁶, and the gap is right there in the exponents: three steps of ten, not one step of anything.

What's Your Guess? is good practice at this, because its answers run from fractions of one to numbers more than twenty digits long, and the game writes every one as a person would say it: "292 million", or "4.1 × 10²¹" once a number is too long for words. As you type a big guess, the readout under the box writes your number out the same way, which is the moment to catch a guess that is a thousand times too small.

Two habits help. Before you type, say aloud which word your answer should end in, thousand, million or billion, and why. And after each reveal, turn the answer into something from the list above: seconds, a share per person, a distance. Do that for a week of runs and the words stop blurring.

To give yourself more rungs on the ladder, keep a pocket list of numbers worth knowing, and practise thinking in powers of ten when you estimate.