 There are several uses for which we want a number system with no gaps, and thus we use the real numbers. If you try to measure the amount of water in a large vessel by counting out a specific number of small cups, there’s no guarantee the number of cups will be an integer. If you try to measure a long distance by counting out the length of your foot, there’s no guarantee the number of feet will be an integer. We might know that something is more than 4 of the units and less than 5. By moving to fractions of these units, we can get more precise—4 cups and 5 to 6 ounces, or 4 feet and 5 to 6 inches—but there’s still no guarantee that a specific rational number will give the precise amount. But we can generate a sequence of approximations that get closer and closer by using smaller and smaller fractions of these units. So in measuring ‘how much’ of something there is, or giving coordinates to describe the location of a point in geometric space, we use the real numbers, to guarantee, as we show in the next section, that there is some precise value the sequence of approximations converges to.