  There is currently a great deal of empirical research—and philosophically sophisticated  debate[21]—on the underpinnings of numerical knowledge in adults and children. There is strong evidence for the view that in addition to an exact number system that underlies formal mathematical thinking, adults also have an analog magnitude system for representing approximate number (see Dehaene 1997). For example, if we are very briefly shown 2 bowls of rice, one with 20 grains and one with 50, we can tell immediately which has more, even though we couldn’t say exactly how many grains of rice were in either. Similarly, shown a book with 70 pages and a book with 100 pages, we can see instantly which has more pages—though again without knowing the exact number of pages in either book. Controlled tests show that such judgments are independent of variables that correlate with magnitude, such as the extent of space occupied or the size of the individual stimuli. The ‘signature’ of this analog magnitude system is its ratio dependence: that is, the difficulty in comparing two analog magnitudes decreases as the ratio difference between them grows. Recent studies indicate that the smallest ratio difference needed for adults to successfully discriminate 2 different analog magnitudes is 8:7. If the ratio is smaller, error rates in comparing magnitudes spike up.