8.3 Numbers

We typically think we know numbers. However, we typically deal with numbers that are represented in a specific numeral system. Our familiarity with specific numeral systems obscures our dependency on arbitrary conventions.

8.3.1 Types of numbers

Different types of numbers (integers vs. doubles):

  • integers
  • positive vs. negative
  • decimals, fractions
  • real numbers

Some oddities:

  • integers
  • errors due to floating point (im-)precision

Rules for rounding numbers.

8.3.2 Representing numbers

When reading a number (like \(123\)), we tend to overlook that this number is represented in a particular notational system. Technically, we need to distinguish between the numeric value \(123\) and the character string of numerical symbols (aka. digits or numerals) “123.”

Generally, numbers are represented according to notational conventions: Strings of dedicated symbols (e.g., Arabic numeral digits 0–9), to be interpreted according to rules (e.g., positional systems require expansion of polynomials).

Overall, our decimal system is a compromise (between more divisible and more unique bases) and a matter of definition. Importantly, our way of representing numbers is subject to two arbitrary conventions:

  1. a positional system with a base value of 10,

  2. representing unit values by the numeral symbols/digits 0–9.

Only changing 2. (by replacing familiar digits with arbitrary symbols) renders simple calculations much more difficult (see letter arithmetic problems).

In the following, we will preserve 2. (the digits and their meaning), but change the base value. This illustrates the difference between numeric value and their symbolic representation (as a string of numeric symbols/digits). (We only cover natural numbers, as they are complicated enough.)

Example

The digits “123” only denote the value of \(123\) in the Hindu-Arabic positional system with a decimal base (i.e., base 10 and numeric symbols 0–9).

Positional notation: Given \(n\) digits \(d_i\) and a base \(b\), a number’s numeric value \(v\) is given by expanding a polynomial sum:

\[v = \sum_{i=1}^{n} {d_i \cdot b^{i-1}}\]

with \(i\) representing each digit’s position (from right to left). Thus, the number \(123\) is a representational shortcut for

\[v = (3 \cdot 10^0) + (2 \cdot 10^1) + (1 \cdot 10^2) = 3 + 20 + 100\]

Representing numbers as base \(b\) positional system

Just like “ten” and “zehn” are two different ways for denoting the same value (\(10\)), we can write a given value in different notations. A simple way of showing this is to use positional number systems with different base values \(b\). As long as \(b \leq 10\), we do not need any new digit symbols (but note that the value of a digit must never exceed the base value).

Note two consequences:

  1. The same symbol string represents different numeric values in different notations:
    The digit string “11” happens to represent a value of \(11\) in base-10 notation. However, the same digit string “11” represents a value of \(6\) in base-5 notation, and a value of \(3\) in base-2 notation.

  2. The same numeric value is represented differently in different notations:
    A given numeric value of \(11\) is written as “11” in decimal notation, but can alternatively be written as “1011” in base-2, “102” in base-3, and “12” in base-9 notation.

Examples of alternative number systems

Alternatives to the base-10 positional system are not just an academic exercise. See

Examples in R

Viewing numbers as symbol strings (of digits): Formatting numbers (e.g., in text or tables)

  • num_as_char() from the ds4psy package:
ds4psy::num_as_char(1:10, n_pre_dec = 2, n_dec = 0)
#>  [1] "01" "02" "03" "04" "05" "06" "07" "08" "09" "10"
  • base2dec() from the i2ds package:
i2ds::base2dec("11")
#> [1] 3
i2ds::base2dec("111")
#> [1] 7
i2ds::base2dec("100", base = 5)
#> [1] 25