7.1 Introduction

Many people struggle with numeric calculations. However, even people who excel in manipulating numbers tend to overlook their representational properties.

When thinking about the representation of numbers, even simple numbers become surprisingly complicated. This is mostly due to our intimate familarity with particular forms of representation, which are quite arbitrary. When stripping away our implicit assumptions, numbers are quite complex representational constructs.

Main distinctions: Numbers as numeric values vs. as ranks vs. as (categorical) labels.

Example: How tall is someone?

Distinguish between issues of judgment vs. measurement.

When providing a quantitative value, the same measurement can be made (mapped to, represented) on different scales:

  1. size in cm: comparisons between values are scaled (e.g., 20% larger, etc.)

  2. rank within group: comparisons between people are possible, but no longer scaled

  3. categories: can be ordered (small-medium-tall) or unordered (fits vs. does not fit)

Additionally, any height value can be expressed in different ways:

  • units: 180cm in ft?
  • accuracy: rounding
  • number system: “ten” vs. “zehn”, value \(10\) as “1010”" in binary system

Why relevant?

Distinguish between the meaning and representation of objects:

  • Different uses of numbers allow different types of calculations and tests.
  • They should be assigned to different data types.