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This representation is correct, but it nonetheless has its limits: how can we represent—and interpret—a fraction such as 11/6? Saying that it means taking 11 out of six equal slices presents some fairly immediate practical difficulties…
A fraction whose numerator is greater than its denominator is called improper, as opposed to a proper fraction, an old-fashioned term for one whose numerator is dutifully smaller than its denominator.
Unlimited slices
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To escape the impasse created by our first pie chart, we can interpret 11 / 6 as 11 times the fraction 1 / 6. Assuming that we have as many sixths of a pie as we like, we need only take 11 of them to obtain a representation of the fraction.
Of course, that is not very illuminating… A more appealing approach is to arrange the sixths into one complete pie with six slices, and place the five remaining slices beside it; they do not quite make a second pie.
This new arrangement makes clear that 11 / 6 is one whole (that is, 6 / 6) plus 5 / 6, which we can write as 11 / 6 = 6 / 6 + 5 / 6, or alternatively as 11 / 6 = 1 + 5 / 6. Even if we are not necessarily aware of it, this is how we usually think of improper fractions. For example, we are more likely to ask a baker for one and a half baguettes than for three half-baguettes!
Notation in other countries
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In some countries, improper fractions are often written as an integer followed by a proper fraction. What we write as 11 / 6 thus takes the form 1 + 5 / 6 (known in English as a mixed number), with the proper fraction generally set in smaller type to reduce the risk of confusion. By convention, the + sign is omitted, which can be disconcerting at first.
