Hex, Octal, and Negative Numbers
CS 301 Lecture, Dr. Lawlor
Base Conversion
It's pretty easy to write code to extract the bits of an integer, and print them out.
/* Print the low 32 bits of this number */
void print_binary(int v)
{
for (int bit=31;bit>=0;bit) {
if ((v&(1<<bit))!=0) {
std::cout<<"1";
} else {
std::cout<<"0";
}
}
std::cout<<" binary\n";
}
int foo(void) {
print_binary(6);
return 0;
}
(executable NetRun link)
It's also pretty easy to stick together a set of bits (say, in an array) into an integer:
const int bits[]={1,1,0};
int nbits=sizeof(bits)/sizeof(bits[0]); /*< funky trick to find size of static array!*/
int value=0;
for (int bit=0;bit<nbits;bit++) {
if (bits[nbits1bit]) /*< gotta index the bit array starting at the end */
value = value  (1<<bit);
}
return value;
(Try this in NetRun now!)
Or you can do the same binarytointeger conversion using digits from cin:
void print_binary(int v) {
for (int i=0;i<32;i++)
{
int mask=1<<(321i);
if ((v&mask)==0) std::cout<<"0";
else std::cout<<"1";
}
std::cout<<endl;
}
int read_binary(void) {
int v=0;
while (std::cin) {
char c='?';
std::cin>>c;
//std::cout<<"I just read the character '"<<c<<"'."<<endl;
if (c=='0') {
/* zero digit in that place */
v=v<<1;
} else if (c=='1') {
/* one digit in that place */
v=(v<<1)+1;
}
else {
break;
}
}
return v;
}
int foo(void) {
return read_binary();
}
(Try this in NetRun now!)
What's the deal with all this hex?
Humans have used the "place/value" number system for a long timethe Sumerians
used base60 in 4000BC! (Base60 still shows up in our measurement of
time (hours have 60 minutes, which have 60 seconds) and angles (360
degrees)). The Maya used base 20. The world standard, though, is now base 10 using Arabic numerals. For example, I'm 32 = 3 * 10 + 2 years old.
But every computer built today uses binary1's and 0'sto do its
work. The reason is electrical0 is no voltage, 1 is
voltage. Having just two states makes it easy to build cheap and reliable
circuits; for example, a transistor will threshold the input value and
either conduct or not conduct. A single zero or one is called a
"bit".
OK, so we got 1's and 0's: how to we build bigger numbers? The modern standard
method is using "binary", which is just the placevalue system using
base 2. In binary, 1 means 1; 10 (base 2) means 2 (base 10); 100
(base 2) means 4 (base 10); 1000 (base 4) means 8 (base 10);
10000 (base 2) means 16 (base 10); and so on. Every machine
produced today supports direct binary arithmetic.
Sadly, for a human writing or reading binary is really painful and
errorprone for large numbers. For example, one million is
11110100001001000000 (base 2), which is painful to write or read.
So instead, we often use a larger base.
Back in the 1970's, it
was pretty common to use octal (base 8), but the modern standard is
hexadecimalbase 16. Base 16's gotta use 16 different digits,
and there are only 10 arabic numerals, so we use normal alphabet
letters for the remaining digits. For example, 15 (base 10) is
just F (base 16); an one million in hex is F4240 (base 16).
You've got to be careful about the base, thoughthe string "11" would
be interpreted as having the value 1*2+1=3 if it was base 2, the usual
1*10+1=11 if it was base 10, or 1*16+1=17 in base 16!
Place/bit Number  i
 ...
 4
 3
 2
 1
 0

Decimal: Base10

10^{i}

...

10000

1000

100

10

1

Binary: Base2

2^{i} 
...

16 = 2^{4}

8 = 2^{3}

4 = 2^{2}

2

1

Octal: Base8

8^{i}

...

4096=8^{4} 
512=8^{3} 
64=8^{2} 
8

1

Hex: Base16

16^{i} 
...

65536 = 2

4096 = 16^{3}

256 = 16^{2}

16

1

Basen

n^{i} 
...

n^{4}

n^{3} 
n^{2} 
n

1 = n^{0}

Number bases used throughout time:
 Base 0: not counting at all. (only representable number is zero)
 Base 1: counting on your fingers. (inefficient use of fingers, though...)
 Base 2: binary. Used by all known computers.
 Base 8: octal. Popular back in 1970s, 1980s. In C++,
prefix the number with zero to make it octal, like "070" (==7*8 = 56).
 Base 10: decimal. Matches number of human fingers. In C++, numbers are decimal by default.
 Base 16: hexadecimal. Fits nicely with bytes. In C++,
prefix the number with "0x" (zerox) to make it hex, like "0xF3".
 Base 20, used by Mayans. Matches number of human fingers and toes.
 Base 60, used by Sumerians six thousand years ago.
Decimal

Hex 
Binary

0

0 
0

1

1 
1

2

2 
10

3

3 
11

4

4 
100

5

5 
101

6

6 
110

7

7 
111

8

8 
1000

9

9 
1001

10

A 
1010

11

B 
1011

12

C 
1100

13

D 
1101

14

E 
1110

15

F 
1111

16

10

10000

Hexadecimaltodecimal and binary table.
Note that a single digit in base 16 corresponds to exactly 4 bits,
since 16=2*2*2*2. This means it's easy to convert from a binary
number to a hex number: take groups of 4 bits and convert to a hex
digitor back again: take each hex digit and expand out to the 4
bits it represents. For example, 0xF036 is, in binary,
1111000000110100, because you can match up the placevalues like this:
Hex PlaceValue

16^{3}

16^{2}

16

1=16^{0}

Hex Digit

F

0

3

6

Binary Digit

1

1

1

1

0

0

0

0

0

0

1

1

0

1

1

0

Binary PlaceValue

2^{15}

2^{14}

2^{13}

2^{12}

2^{11}

2^{10}

2^{9}

2^{8}

2^{7}

2^{6}

2^{5}

2^{4}

2^{3}

2^{2}

2^{1}

2^{0}

Hex really is the only true universal in assembly and
disassembly. For example, here's some random disassembled code
(produced using "objdump drC M intel /bin/ls" on a Linux machine):
804a516: 80 fa 3f cmp dl,0x3f
804a519: 0f 84 7c 01 00 00 je 804a69b <exit@plt+0xc2b>
Note that every single number
is listed in hexthe addresses, on the left; the machine code, in the
middle; and the constants in the assembly, on the right. A binary
file display tool is called a "hex dump". A binary file editor is
called a "hex editor". That's how common hex is, so for the rest
of the class to make sense, you've gotta learn it!
Hex & Bitwise Operations
Remember that every hex digit represents four bits. So if you
shift a hex constant by four bits, it shifts by one entire hex digit:
0xf0d<<4 == 0xf0d0
0xf0d>>4 == 0xf0
Bitwise operators make perfect sense working with hex digits, because they operate on the underlying bits of those digits:
0xff0 & 0x0ff == 0x0f0
0xff0  0x0ff == 0xfff
0xff0 ^ 0x0ff == 0xf0f
You can use these bitwise operators to peel off the hex digits of a number, to print out stuff in hex:
int v=1024+15;
for (int digit=7;digit>=0;digit) {
char *digitTable="0123456789abcdef";
int d=(v>>(digit*4))&0xF;
std::cout<<digitTable[d];
}
std::cout<<std::endl;
return v;
(Try this in NetRun now!)
Arithmetic In Binary
We can do arithmetic in binary or hex by hand just like we do in
decimal. To add, line up the digits, add corresponding digits, and if
the perdigit sum is greater than the base, carry to the next digit. Easy,
right? To subtract, do the exact same thing, but "borrow" from
the next digit if the perdigit difference is less than 0.
For example, in binary 01+01=10 because the first digit overflows, and
the extra "1" is carried to the next digit. Similarly,
1111+1=10000 because all the digits overflow in sequence. In
general, adding a "1" will cause carries to ripple up through ones,
flipping them to zeros, until it finally reaches a zero, which it will
flip to a one.
Addition in hex is exactly the sameit's a bit tricker to add the
larger digits involved, but carries work exactly the same.
So 0x8+0x8=0x10, and 0xffff+0x1=0x10000.
Subtraction in binary seems really bizarre until you remember than
101=1 in binarythat is, "1" is the onelessthanzero digit in
binary; just like "9" is the corresponding onebelowzero digit in decimal. So when
you have to borrow, you just flip the zeros to ones until you hit a
one.
Subtraction actually shows you how to represent negative numbers in
binary. Consider 1==01: you just keep on flipping zeros to ones
forever, "borrowing" against the future that never comes (just like the US banking system!). So 1
is represented (in principle) by an infinite number of ones, just like
+1 is represented (in principle) by an infinite number of zeros
followed by a one. In practice, we only store a finite number
(usually 32 or 64) of bits, so on a 32bit machine we define 32 ones as
1. The highest bit, then, can be thought of as a "sign
bit"negative numbers have that bit set, positive numbers have that
bit clear. Suprisingly, it make sense to think of the sign bit in
digit n as having value 2^{n}, instead of value 2^{n} for unsigned numbers. What's weirder is that addition and subtraction are exactly the same
for signed or unsigned numberstry it! (All other operations are
different, though: comparison, multiplication, division, etc.)
Signed versus unsigned numbers are clearer if you draw them out for a 3bit machine:
Bit Pattern

Unsigned Value

Signed Value

000

0

0

001

1

1

010

2

2

011

3

3

100

4

4

101

5

3

110

6

2

111

7

1

Note that everything but Java supports "signed" as well as "unsigned"
integers explicitly. In C/C++/C#, typecasting to an unsigned
value does nothing to the bits, and is hence a zeroclock
operation. One way to speed up a "zero to n" bounds test on a
signed value is to turn it into an unsigned value and just compare
against nthat is, "if ((x>=0)&&(x<n)) ..." is the same
as "if (((unsigned int)x)<n) ..." do exactly the same thing, because
negative signed x's will turn into really huge unsigned x's!
In hex, 1 (in binary, all ones) is written in hex as 0xFFffFFff (on a 32bit machine).
Another way to think about 1 is that it's like "99999" on a car's
odometerif you drive one more mile, you'll be at "00000", so you're
really at mile 1!
Storage Sizes
Eight bits make a "byte" (note: it's pronounced exactly like "bite",
but always spelled with a 'y'), although in some rare networking
manuals (and in French) the same eight bits would be called an
"octet" (hard drive sizes are in "Go", Gigaoctets, when sold in French).
There are some rarelyused, quasijoke measurements like a
fourbit "nybble", but these are quite rare, and basically just jokes.
In DOS and Windows programming, 16 bits is a "WORD", 32 bits is
a
"DWORD" (double word), and 64 bits is a "QWORD"; but in other contexts
"word" means the machine's
natural binary processing size, which ranges from 32 to 64 bits
nowadays. "word" should be considered ambiguous. Giving an
actual bit count is the best approach ("The file begins with a 32bit
binary integer describing...").
Object

Overflow Value

Bits

Hex Digits (4 bits each)
 Bytes (8 bits each)  Octal Digits (3 bits each)

Bit

2

1

less than 1
 less than 1  less than 1

Byte, char

256

8

2
 1
 two and two thirds

"short" (or Windows WORD)

65536

16

4
 2
 five and one third

"int" (Windows DWORD)

>4 billion

32

8
 4
 ten and two thirds

"long" (or sometimes "long long")

>16 quadrillion

64

16
 8
 twentyone and onethird
