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Section 5.4 Hexadecimal Numbers

And finally, a brief excursion into hexadecimal numbers:

Aside: Video Description.

Activity 5.6.

Please convert the decimal number 20,000 into hexadecimal.
Answer.
0x4E20
Binary is the native number system of the computer, but it is often convenient for us to talk in the hexadecimal number system (HEX), that is, in base-16.
HEX digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F (0-15)
Example: 0xAC means "A" * \(16^1\) + "C" * \(16^0\) = 10 * 16 + 12 * 1 = 172 in decimal
The "0x" indicates that what follows is in hex
Another example: 0x57 means 5 * \(16^1\) + 7 * \(16^0\) = 87 in decimal

Subsection 5.4.1 Why Use Hexadecimal?

Binary numbers can be REALLY long (used in memory location addressing for example.)
  • Need: easy shorthand for binary
  • Solution: HEX
    • Condense 4 bits (binary digits) into 1 hexadecimal digit
    • Example: 1011 binary = 11 decimal = B hexadecimal
  • 2-byte words (16 bits) can then be written as four HEX digits
We want powers of 2 (binary) for easy conversion.
Alternative?
  • Octal (digits 0-7)
  • Condenses 3 bits into one

Subsection 5.4.2 Decimal to Hexadecimal

We already know how this works: keep dividing by 16, record the remainders in reverse order! 
23,597    /16       = 1,474  R13 (D)     * 16^0   LSD
 1,474    /16       = 92     R2          * 16^1    |
    92    /16       = 5      R12 (C)     * 16^2    V
     5    /16       = 0      R5          * 16^3   MSD

23,597              ->       0x5C2D

Subsection 5.4.3 Binary to Hexadecimal and Decimal

We can go directly from binary to decimal, or via hex.
Example: 16-bit binary integer 
0101   1100   0010   1101    = 23,597    Decimal
  5    C (12)   2    D (13)  = 0x5C2D    HEX
16^3   16^2   16^1   16^0                (HEX place values)
\(5 \times 16^3 + 12 \times 16^2 + 2 \times 16^1 + 13 \times 16^0\)
\(= 5 \times 4096 + 12 \times 256 + 2 \times 16 + 13 \times 1\)
\(= 20,480 + 3072 + 32 + 13\)
\(= 23,597\) Decimal
Remember: precede hexadecimal number with 0x in order to avoid confusion with decimal number 0x5C2D
 
Hexadecimal to decimal: best done via binary!
0x15A9 2-byte hexadecimal (base-16) number
Hex-to-binary: each hex digit is 4 binary digits 
                1    5    10   9
0x15A9        0001 0101 1010 1001
MSB is 0, so decimal value is the same for unsigned, signed-magnitude, and 2’s complement representations
Binary-to-decimal:
=5545

Subsection 5.4.4 Another Example

0xB5A9 2-byte hexadecimal (base-16) number
Hex-to-binary: 
            11    5   10    9
0xB5A9     1011 0101 1010 1001
Now MSB is 1, so decimal value depends on which representation we use...
Binary-to-decimal:
Unsigned: \(2^{15} + 2^{13} + 2^{12} + 2^{10} + 2^8 + 2^7 + 2^5 + 2^3 + 2^0 = 46,505\)
Signed magnitude: \(-(2^{13} + 2^{12} + 2^{10} + 2^8 + 2^7 + 2^5 + 2^3 + 2^0) = -13,737\)
2’s complement: \(-(2^{14} + 2^{11} + 2^9 + 2^6 + 2^4 + 2^2 + 2^1 + 1) = -19,031\)
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