Activity 5.6.
Please convert the decimal number 20,000 into hexadecimal.
Answer.
0x4E20
Section 5.4 Hexadecimal Numbers
And finally, a brief excursion into hexadecimal numbers:
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)
The "0x" indicates that what follows is in hex
Subsection 5.4.1 Why Use Hexadecimal?
Binary numbers can be REALLY long (used in memory location addressing for example.)
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Need: easy shorthand for binary
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Solution: HEX
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Condense 4 bits (binary digits) into 1 hexadecimal digit
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Example: 1011 binary = 11 decimal = B hexadecimal
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2-byte words (16 bits) can then be written as four HEX digits
We want powers of 2 (binary) for easy conversion.
Alternative?
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Octal (digits 0-7)
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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: \(-(2^{14} + 2^{11} + 2^9 + 2^6 + 2^4 + 2^2 + 2^1 + 1) = -19,031\)
