Hacking the Xbox

Chapter 2 - Thinking Inside the Box 43
Binary and Hexadecimal Numbers
Digital circuits use 1’s and 0’s to represent numbers. This binary, or
“base-2”, notation is a reflection of the way electrical signals are
used to represent numbers: two ranges of voltage levels are used
to define one logic state or the other. It is possible to build electrical
systems that represent information using more than two voltage
levels, but only at the cost of power and complexity. Modern modems,
for example, use multiple voltage levels and phase information
to represent multiple bits of data in a single time unit.
Number composition and arithmetic in binary follows the same
rules as our familiar decimal (“base-10”) representation. In decimal,
0’s are used as placeholders to remember when a digit has overflowed.
For example, 1 more than 9 leads to an overflow because
there is no single digit bigger than 9. Hence, the number 10 records
that we had one overflow of the right-most decimal location. Likewise,
in binary, 1 more than 1 is 10, since the largest single digit in
binary is 1.
Thus, in decimal, the value of a four-digit decimal number d d d d 4 3 2 1
can be broken down as:
d * 10 4 3 + d * 10 3 2 + d * 10 2 1 + d * 10 1 0 = d * 1000 + d * 100 + d * 10 + d * 1
4 3 2 1
Likewise, a four-digit binary number b b b b can be broken down
4 3 2 1
as:
b * 2 4 3 + b * 2 3 2 + b * 2 2 1 + b * 2 1 0 = b * 8 4 + b * 4 3 + b * 2 2 + b * 1 1
For example, the number 1010 = 1*8 + 0*4 + 1*2 + 0*1 = 10 decimal.
Keeping track of numbers in straight binary can become cumbersome
quickly; for example, to represent decimal 968, you need ten
binary digits. To save on screen space, binary numbers are converted
to octal or hexadecimal. The octal format, or “base-8”, was
popular in the early days of computers, but has since become a
rarity. Hexadecimal, or “base-16”, is the de-facto numbering system.
There are 16 digits in hexadecimal, so the hex digits that correspond
to decimal numbers 10 through 15 are represented by the
letters A through F. Table 2-2 summarizes the conversion between
binary, decimal, and hexadecimal for the first 16 positive integers.
In order to differentiate hexadecimal from decimal numbers, many
people use the C language convention where 0x[number] represents a
hexadecimal number, and [number] is implicitly a decimal number. Binary
numbers have no similar standard to draw from in C, so some
people use the Verilog standard, [digits]’b[number], where [digits] is the
number of digits in the binary number. The suffix “b” after a string of 1’s
and 0’s, such as 1010.1100.1110b is also used to denote a binary number.
Notice how a “.” was used to group the binary digits into sets of four;
this assists in mentally translating the binary number into hexadecimal:
0xACE.
(continued)