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Positional Numeral System
The common numeral system we use for calculation is called decimal system, which means it is based on powers of 10 (english: ten, latin: decem, ancient greek: deka). When we write $652$ we mean $2\cdot 10^0 + 5\cdot 10^1 + 6\cdot 10^2$. When we write $256$ we mean $6\cdot 10^0 + 5\cdot 10^1 + 2\cdot 10^2$.
Ten symbols are necessary to define ten multipliers, e.g. the symbol (digit) '5' means 'multiply by five'. The ten elementary symbols used in the decimal system are $\{0,1,2,3,4,5,6,7,8,9\}$.
The most striking achievement in modern numeral systems is that the position of a symbol determines the power of 10 it has to be multiplied with. In the example $652$ the symbol $6$ at position 3 from the right has to be multiplied by $10^2$. The position is also known as digit.
Decimal System, Base $10$
Elementary symbols, multipliers: $\{0,1,2,3,4,5,6,7,8,9\}$
| Position $p$ | … | $5$ | $4$ | $3$ | $2$ | $1$ | $0$ | . | $-1$ | $-2$ | … |
| Decimal weight $w=10^{p}$ | … | $10^5$ | $10^4$ | $10^3$ | $10^2$ | $10^1$ | $10^0$ | . | $10^{-1}$ | $10^{-2}$ | … |
| before decimal point | . | after decimal point | |||||||||
| Digit | … | $a_5$ | $a_4$ | $a_3$ | $a_2$ | $a_1$ | $a_0$ | . | $a_{-1}$ | $a_{-2}$ | … |
Examples
$652_{10} = 2\cdot 10^0 + 5\cdot 10^1 + 6\cdot 10^2 = 2+50+600$
$53.71_{10} = 1 \cdot 10^{-2} + 7 \cdot 10^{-1} + 3 \cdot 10^{0} + 5 \cdot 10^{5} = 0.01+0.7+3+50$
General System, Base $b$
Elementary symbols, multipliers: set of $b$ elements.
| Position $p$ | … | $5$ | $4$ | $3$ | $2$ | $1$ | $0$ | . | $-1$ | $-2$ | … |
| Decimal weight $w=b^{p}$ | … | $b^5$ | $b^4$ | $b^3$ | $b^2$ | $b^1$ | $b^0$ | . | $b^{-1}$ | $b^{-2}$ | … |
| before decimal point | . | after decimal point | |||||||||
| Digit | … | $a_5$ | $a_4$ | $a_3$ | $a_2$ | $a_1$ | $a_0$ | . | $a_{-1}$ | $a_{-2}$ | … |
$\mathrm{Value} = \sum_k a_k b^k$
Binary System, Base $2$
Elementary symbols, multipliers: $\{0,1\}$
| Position $p$ | … | $5$ | $4$ | $3$ | $2$ | $1$ | $0$ | . | $-1$ | $-2$ | … |
| Decimal weight $w=2^{p}$ | … | $2^5$ | $2^4$ | $2^3$ | $2^2$ | $2^1$ | $2^0$ | . | $b^{-1}$ | $b^{-2}$ | … |
| before decimal point | . | after decimal point | |||||||||
| Digit | … | $a_5$ | $a_4$ | $a_3$ | $a_2$ | $a_1$ | $a_0$ | . | $a_{-1}$ | $a_{-2}$ | … |
Examples
$1010_2 = 0\cdot 2^0 + 1 \cdot 2^1 + 0 \cdot 2^2 + 1 \cdot 2^3 = 0\cdot 1 + 1 \cdot 2 + 0 \cdot 4 + 1\cdot 8 = 9_{10}$
$10.11_2 = 2 + 0 + 1/2 + 1/4 = 2 + 0.5 + 0.25 = 2.75_{10}$
Hexadecimal System, Base $16$
Elementary symbols, multipliers: $\{0,1,2,3,4,5,6,7,8,9,A(10),B(11),C(12),D(13),E(14),F(15)\}$
| Position $p$ | … | $5$ | $4$ | $3$ | $2$ | $1$ | $0$ | . | $-1$ | $-2$ | … |
| Decimal weight $w=16^{p}$ | … | $16^5$ | $16^4$ | $16^3$ | $16^2$ | $16^1$ | $16^0$ | . | $16^{-1}$ | $16^{-2}$ | … |
| before decimal point | . | after decimal point | |||||||||
| Digit | … | $a_5$ | $a_4$ | $a_3$ | $a_2$ | $a_1$ | $a_0$ | . | $a_{-1}$ | $a_{-2}$ | … |
Examples
$12_{16} = 2\cdot 16^0 + 1 \cdot 16^1 = 17_{10}$
$\mathrm{E}_{16} = 14_{10}$
$\mathrm{FF_{16}} = 15\cdot 16 + 15 = 255_{10}$
$\mathrm{FFFF_{16}} = 15 \cdot 16^0 + 15 \cdot 16^1 + 15 \cdot 16^2 + 15 \cdot 16^3 = 65535_{10}$
$\mathrm{FFFF_{16}} = 10000_{16} - 1 = 16^4 - 1 = 65536 -1 = 65535_{10}$
Exercises
Hint:
The website https://en.wikiversity.org/wiki/Numeral_systems shows algorithms to convert numeral systems, e.g. Dec to Hex or Dec to Bin.
Convert Hex, Dec, Bin
| Hex | Dec | Bin |
|---|---|---|
| $\mathrm{F}$ | ||
| $\mathrm{AA}$ | ||
| $\mathrm{55}$ | ||
| $123$ | ||
| $1111$ | ||
| $1111$ | ||
| $1111$ | ||
| $00111100$ | ||
| $11000011$ |