Discrete vs Continuous Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
Shoe sizes in the UK come in steps of 12\frac{1}{2} (e.g., 6,6.5,7,7.5,…6, 6.5, 7, 7.5, \ldots). Is shoe size discrete or continuous? What about actual foot length in centimetres?

Solution

  1. 1
    UK shoe size: takes values from a fixed set with gaps of 12\frac{1}{2}. You cannot have size 6.36.3. This is discrete (even though the steps are fractional).
  2. 2
    Actual foot length: can in principle be any real number of centimetres (e.g., 24.724.7 cm, 24.7324.73 cm, …\ldots). This is continuous.
  3. 3
    The shoe size system is a discretisation of continuous foot length, rounding to the nearest half-size.

Answer

Shoe size is discrete (finite set of allowed values); foot length is continuous.
Discreteness does not require whole numbers — any quantity with isolated allowed values is discrete. Shoe sizes are a practical discretisation: feet grow continuously, but the product is manufactured in discrete sizes for efficiency.

About Discrete vs Continuous

The distinction between quantities that take separate, distinct values (discrete, like number of students) and quantities that can take any value in a range (continuous, like height or temperature).

Learn more about Discrete vs Continuous →

More Discrete vs Continuous Examples

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Example 3 easy

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