Today, October 23rd, is the National Mole Day, created not for the tiny burrowing mammal, but to celebrate the basic unit in chemistry. One mole, also known as Avogadro's Number, is 6.02x10^23 of whatever you are measuring.
So a mole of moles would be 602,200,000,000,000,000,000,000 animals.
Now, can you imagine how many moles that is? Probably not. The way our brains are set up, truly understanding that vast a number is pretty much impossible.
"Our cognitive systems are very much tied to our perceptions," said Daniel Ansari, a researcher at the Numerical Cognition Laboratory at Western University in Canada. "The main obstacle is that we're dealing with numbers that are too large for us to have experienced perceptually."
By contrast, we constantly experience small numbers. "Smaller numbers are more frequent in our daily vocabulary," Ansari said. "When you lay the table you ask your child, how many knives do we need? It's never going to be 10,000 unless you have a very big dinner party."
And in line with our normal life experiences, our brains have become capable of representing small numbers, but helpless at accurately reflecting very large ones. "What you don't have is an intuitive sense of what this number means," said Brian Butterworth, a cognitive neuroscientist at University College London and author of The Mathematical Brain. "The reason for this is that our sense of number is based upon two innate systems which essentially deal with small numbers accurately or large numbers only approximately."
Using the first system, we can visualize, say, five balloons but can't imagine 500 of them. Using the second system, we can tell 500 balloons are significantly more than 50 balloons. Butterworth and his team have found evidence that other animals have these two systems as well. They've seen that fish, which benefit from the protection offered by joining a larger shoal, are pretty good at estimating which of two groups of fish is larger. They can also accurately distinguish between small numbers of fish. "What this tells us is that our innate system has deep evolutionary roots that go back at least to the common ancestor for small fish and humans," Butterworth said.
But unlike us, fish don't have symbols to represent large numbers. "So they can't get up to large number exactly the way that we can, because we've got counting words and the digit system," said Butterworth.
This ability stems from a connection between the two innate number system and other parts of the brain. In humans, brain systems dedicated to processing numbers seem to reside in the parietal lobe, in two key areas. One is the intraparietal sulcus—studies in monkeys have found that some groups of neurons in the intraparietal sulcus code numerosity linearly: the more objects the animal sees, the more they fire. In contrast, neurons in another part of the intraparietal sulcus fire preferentially for a specific number— one fires for 3, another for 4, and so on up to 5, the highest number tested in this way, Butterworth said. Similarly, studies of number processing in humans have pointed to the intraparietal sulcus, and have found that certain structural and functional abnormalities in this region in children are linked with dyscalculia, an impairment of mathematical ability. The other brain area involved in number processing is the temporal-parietal junction, which seems to deal with small number comparisons.
"In the course of development…the innate systems for representing numbers link up the language system where verbal counting is represented and the visual system where the familiar digits are represented," Butterworth said.
Still, many people struggle to work with large quantities. To get a better grasp on these numbers, it helps to round up or down, said Butterworth; 1,000 miles is easier to imagine than 1,235.
The larger a number grows, the harder it becomes to deal with. But sometimes, extremely large numbers lurking in the levels of billions and trillions and more, actually are relevant to the lives of everyday people. Take the national debt and government deficit for example. In order to understand such numbers, it's important to have an understanding of the context that number falls into.
"How do you understand a deficit of 28 billion pounds—is it a lot, is it a little?" he said. "That's a question not only of understanding what 28 billion means but also what it means in terms of the things that it's describing, namely the economic activities of the country."
We may not be able to change our brain systems for grasping big numbers, but we can get better at dealing with them, with practice. In his own research, Butterworth has seen the difference that regularly working with large numbers can make. Some years ago, he and his colleagues came up with a math test to assess neurological damage by finding out whether patients could still perform calculations. The team made the math problems on the test increasingly more difficult by throwing in larger numbers.
"We had patients who could deal with quite small numbers but when you got up to the thousands they broke down," Butterworth said. But when they compared results with their collaborators in Italy, they saw something that surprised them.
The Italians were better at understanding and comparing big numbers. "It turned out that Italian patients were very good with thousands and millions, whereas English patients where not," he said.
Did the Italians have some kind of a special math gene? Not really. Their skills were traced back to lira, the Italian currency at the time. "One thousand lire was worth about a dollar, ten thousand lire was worth about 10 dollars, and if you wanted to buy a car or a motorbike or a house, you had to deal in millions or even billions of lire," said Butterworth. "So people got quite used to that, and they were pretty good at it. Of course now, they won't be so good because the currency is the euro."
For the rest of us, too, regular practice with symbols and units could make it easier to work with large numbers (perhaps even curb math anxiety?), even though we likely won't ever have an instinctive sense of how many moles make up a mole.