From Counting Objects to Positional Notation
Human reasoning starts with objects in the world: stones, apples, days, distances, measurements. Once objects can be compared and counted, symbols can be assigned to quantities. A numeral system is exactly that assignment rule: a finite set of symbols combined with place-value rules that can represent arbitrarily large quantities.
In most modern contexts, the decimal system (base 10) is dominant, using ten symbols from 0 to 9. But mathematically, decimal is only one instance of a general positional system with base \(b\).
Examples of Positional Number Systems
The decimal representation
\[ 234_{10} = 2\cdot 10^2 + 3\cdot 10^1 + 4\cdot 10^0. \]
In hexadecimal (base 16, prefix 0x), the digits extend to \(A=10\), \(B=11\), \(C=12\), \(D=13\), \(E=14\), \(F=15\). For example,
\[ \text{0x}6C8 = 6\cdot 16^2 + 12\cdot 16^1 + 8\cdot 16^0 = 1736_{10}. \]
In octal (base 8, historical prefix 0),
\[ 101_8 = 1\cdot 8^2 + 0\cdot 8^1 + 1\cdot 8^0 = 65_{10}. \]
In binary (base 2),
\[ 101_2 = 1\cdot 2^2 + 0\cdot 2^1 + 1\cdot 2^0 = 5_{10}. \]
Other historically relevant systems include duodecimal (base 12) and sexagesimal (base 60), the latter still visible in time and angle notation.
General \(b\)-Adic Representation
A positional integer in base \(b\ge 2\) has digits \(d_i\in\{0,1,\dots,b-1\}\) and is represented as
\[ n = \sum_{i=0}^{k} d_i b^i = d_k b^k + \cdots + d_1 b + d_0. \]
Example for \(b=10\):
\[ 234 = \underbrace{2}_{d_2}\cdot 10^2 + \underbrace{3}_{d_1}\cdot 10^1 + \underbrace{4}_{d_0}\cdot 10^0. \]
The same idea extends to fractions by allowing negative exponents:
\[ x = \sum_{i=-m}^{k} d_i b^i. \]
For instance, \(10.625_{10}=1\cdot 10^1 + 0\cdot 10^0 + 6\cdot 10^{-1} + 2\cdot 10^{-2} + 5\cdot 10^{-3}\).
Written Addition in Base \(b\)
Written addition works digit-wise from right to left, with carries. In decimal,
789
123
+ ---
912 In binary,
101
111
+ ---
1100 Let \(m=\sum m_i b^i\), \(n=\sum n_i b^i\), and define carry values \(c_i\) by \(c_0=0\). Then for every position \(i\):
\[ t_i = m_i + n_i + c_i, \qquad s_i = t_i \bmod b, \qquad c_{i+1}=\left\lfloor\frac{t_i}{b}\right\rfloor. \]
This formulation is the base-independent core of schoolbook addition and exactly what low-level arithmetic units implement in hardware.