# Hexadecimal Notation

## 4-Bit Substring Pattern Replacement

- 4-bit substrings of 0's and 1's are replaced with single characters
- the character used for any specific 4-bit substring is based on the value of the substring
if treated as an unsigned binary encoded number
- values of 0 through 9 use their respective digits as hexadecimal shorthand symbols
- values of 10 through 15 use the alphabetic symbols A through F respectively (10=A;
11=B; 12=C; 13=D; 14=E; 15=F)

### Dividing Bit Strings with Non-Multiple of 4 Lengths

- Start division of the string from the right end
- Pad with leading zeros (on the left)

## Arithmetic in Hexadecimal

**Decimal - Hexadecimal Conversions**
__Hexadecimal as a Positional Notation__
- Positional Weights for Hexadecimal Notation

(etc.) |
(16*16*16) = 4096 |
(16*16) = 256 |
16 |
1 |

- Example, 5050 (decimal) expressed in hexadecimal
- = 1(4096) + 954
- = 1(4096) + 3(256) + 186
- = 1(4096) + 3(256) + 11(16) + 10
- = 13BA (hexadecimal) ...remembering that B is the symbol for 11 and A is
the symbol for 10

**Addition and Subtraction in Hexadecimal**
- in decimal addition and subtraction, carries to or borrows from the next column
are always worth 10 of the current column
- in hexadecimal addition and subtraction, carries to or borrows from the next
column are always worth 16 of the current column
- Sample (Partial) Addition

.. |
3 |
7 |
9 |

**+** |
5 |
E (=14) |
C (=12) |

**(+ Carry In)** |
.. |
(+1) |
(-16) |

**Result** |
.. |
.. |
5 |

- Sample (Partial) Subtraction

.. |
B (=11) |
F (=15) |
3 |

-.. |
6 |
9 |
D (=13) |

**(Borrow from)** |
.. |
(-1) |
(+16) |

**Result** |
.. |
.. |
6 |

**2's Complement Encoding in Hexadecimal**
- The "subtraction from 0 method" is the simplest form for converting a
hexadecimal encoded value into its negative form
- For example, -(04E7) in 2's Complement

.. |
.. |
0 |
0 |
0 |
0 |

**-**.. |
.. |
0 |
4 |
E (=14) |
7 |

**Borrow
from** |
(-1) ignored |
(-1)(+16) |
(-1)(+16) |
(-1)(+16) |
(=16) |

**Result** |
.. |
F (=15) |
B (=11) |
1 |
9 |