Alex’s Adventures in Numberland Read Online Free PDF

turned into the modern 1, 2, 3 and 4.)
    In fact, there is some debate about whether the limit of the number of lines we can grasp instantly is three or four. The Romans actually had the alternatives IIII and IV for four. The IV is much more instantly recognizable, but clock faces – perhaps for aesthetic reasons – tended to use the IIII. Certainly, the number of lines, dots, or sabre-toothed tigers that we can enumerate rapidly, confidently and accurately is no more than four. While we have an exact sense of 1, 2 and 3, beyond 4 our exact sense wanes and our judgements about numbers become approximate . For example, try to guess quickly how many dots are at the top of the page opposite.

     
    It’s impossible. (Unless you are an autistic savant, like the character played by Dustin Hoffman in Rain Man , who would be able to grunt in a split second ‘Seventy-five’.) Our only strategy is to estimate, and we’d probably be far off the mark.
    Researchers have tested the extent of our intuition of amounts by showing volunteers images of different numbers of dots and asking which set is larger, and our proficiency at discriminating dots, it turns out, follows regular patterns. It is easier, for example, to tell the difference between a group of 80 dots and a group of 100 dots than it is between two groups of 81 and 82 dots. Similarly, it is easier to discriminate between 20 and 40 dots than it is between 80 and 100 dots. In both A and B below, the left set of dots is larger than the right set of dots, yet the length of time it takes us to process the information is noticeably longer in case B.

     
    Scientists have been surprised by how strictly our powers of comparison follow mathematical laws, such as the multiplicative principlsti his book The Number Sense , the French cognitive scientist Stanislas Dehaene gives the example of a person who can discriminate 10 dots from 13 dots with an accuracy of 90 percent. If the first set is doubled to 20 dots, how many dots does the second set need to include so that this person still has 90 percent accuracy in discrimination? The answer is 26, exactly double the original number of the second set.
    Animals are also able to compare sets of dots. While they do not score as highly as we do, the same mathematical laws also seem to govern their skills. This is pretty remarkable. Humans are unique in having a wonderfully elaborate system of counting. Our life is filled with numbers. Yet, for all our mathematical talent, when it comes to perceiving and estimating large numbers our brains function just like those of our feathered and furry friends.
     
     
    Human intuitions about quantities led, over millions of years, to the creation of numbers. It is impossible to know exactly how this happened, but it is reasonable to speculate that it arose from our desire to track things – such as moons, mountains, predators or drum beats. At first we may have used visual symbols, such as our fingers, or notches on wood, in a one-to-one correspondence with the object we were tracking – two notches or two fingers means two mammoths, three notches or three fingers means three, and so on. Later on we will have come up with words to express the concepts of ‘two notches’ or ‘three fingers’.
    As more and more objects were tracked, our vocabulary and symbology of numbers expanded and – accelerating to the present day – we now have a fully developed system of exact numbers with which we can count as high as we like. Our ability to express numbers exactly, such as being able to say that there are precisely 75 dots in the top picture on the previous page, sits cheek-by-jowl with our more fundamental ability to understand such quantities approximately. We choose which approach to use depending on circumstance: in the supermarket, for example, we use our understanding of exact numbers when we look at prices of products. But when we decide to join the shortest checkout queue we are using our