Million, Billion, Trillion
Million, Billion, Trillion
[A graph with Y-axis labeled “Perceived size of number” and X-axis labeled “Actual size of number (log scale)”.] [The line graph shows points that are labeled with the following numbers from “1 million” to “100 trillion”:]
[ Points on graph: ]
1 million [ (2, 1.5) ] 10 million [ (3, 3.5) ] 100 million [ (4, 4.8) ] 1 billion [ (5, 2.8) ] 10 billion [ (6, 4.4) ] 100 billion [ (7, 5.3) ] 1 trillion [ (8, 4.3) ] 10 trillion [ (9, 5.1) ] 100 trillion [ (10, 5.6) ]
[The perceived size increases between most numbers, but decreases between 100 million and 1 billion, and between 100 billion and 1 trillion. The decreases are shown as dashed lines labeled “?”, in contrast to the solid unlabeled lines between the increases. The increases and decreases in perceived size become smaller as the numbers grow in actual size.] [Caption below the panel:] Talking about large numbers is hard

Much like comic 558 , this comic addresses the difficulty ordinary people have with large numbers. Though most if not all people intuitively understand the difference between one object and two objects, or one object and ten objects, or even one object and a hundred objects, as numbers increase most people's ability to innately conceive of the numbers being discussed decreases remarkably quickly. When numbers reach the millions and the billions, and especially the trillions, most people don't truly process the numbers at all, and instead conceive of them as some version of a drastically-oversimplified concept such as "very big." Where comparing one to ten is simple, comparing "very big" to a different "very big" can prove extremely challenging, and will certainly require non-intuitive, conscious thinking.
The comic represents this challenge by providing a graph which represents Randall 's intuitive conception of the values of various very large numbers, and said conception's misalignment with reality. Though some trends reflect the real value of the numbers on the graph, i.e. 100 million larger than 10 million larger than 1 million and 1 billion larger than 1 million, the curve is far from the linear (exponential on the log-scaled axes) path it should take, with 1 billion being intuitively understood as less than 100 million, based, presumably, on the fact, easily comprehended on an intuitive level, that 100 is larger than 1 , and therefore the presence of 100 in 100 million places it at a higher value than the 1 in 1 billion would place the latter. In reality, of course, 1 billion is ten times larger than 100 million, but the comic deals not with actual reality, but with the perception of reality  of these numbers before conscious thought is applied .
The most interesting parts of the graph, and the parts where the disconnect between intuition and reality becomes clearest, are the dashed sections labeled with question marks, the one between 100 million and 1 billion, the other between 100 billion and 1 trillion. Here two competing intuitive understandings compete for dominance. On the one hand, the intuitive understanding described above, with 100 trumping 1, would see the curve taking a sharp downturn. On the other hand, the path from 100 million to 1 billion is paved with such numbers as 500 million, 700 million, and 900 million, all of which would theoretically be seen intuitively as larger than 100 million, thanks to the fact that 9 is greater than 7, and 7 greater than 5, and so on, bending the curve up rather than down. These two conflicting intuitions leave Randall with no single intuitive path for the two dashed sections, leading to their dashed and questioned state.
The comic's caption and title highlight another problem surrounding the intuitive grasping of large numbers: the flaws in the English words used for them. For instance, nothing about the word "million" suggests smallness relative to the word "billion" on an intuitive scale. This unintuitive language contributes greatly to the "100 trumps 1" intuitive fallacy described above.
The title text references a highly relevant disconnect between the long and short scales of large numbers .
For all English speakers, and for most languages, 1 million constitutes 1,000 thousands, or, less ambiguously, 10^6. However, this is the last of the consensus numbers, and the definition of what should be the "next step" varies depending on how each country's language evolved.
In other words, 1 billion objects in a country using the short-scale would be 1,000 million objects in a country using the long-scale; at the "next step", 1 trillion in the short-scale would be named 1 billion in the long-scale, despite the fact that the number of objects has remained the same. This difference between languages using the short-scale and the long-scale often causes confusion when translating articles with large numbers in them, as translators sometimes fail to change between short-scale and long-scale schemes, wrongly translating large numbers to incorrect values.
The fact that such a staggering difference of terminology was able to exist and be almost completely unknown to many supports Randall's point about the failure of human intuition in the discussion of extremely large numbers.
[A graph with Y-axis labeled “Perceived size of number” and X-axis labeled “Actual size of number (log scale)”.] [The line graph shows points that are labeled with the following numbers from “1 million” to “100 trillion”:]
[ Points on graph: ]
1 million [ (2, 1.5) ] 10 million [ (3, 3.5) ] 100 million [ (4, 4.8) ] 1 billion [ (5, 2.8) ] 10 billion [ (6, 4.4) ] 100 billion [ (7, 5.3) ] 1 trillion [ (8, 4.3) ] 10 trillion [ (9, 5.1) ] 100 trillion [ (10, 5.6) ]
[The perceived size increases between most numbers, but decreases between 100 million and 1 billion, and between 100 billion and 1 trillion. The decreases are shown as dashed lines labeled “?”, in contrast to the solid unlabeled lines between the increases. The increases and decreases in perceived size become smaller as the numbers grow in actual size.] [Caption below the panel:] Talking about large numbers is hard
