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Binary (2)
The binary numeral system represents numeric values using two symbols, typically 0 and 1. More specifically, binary is a positional notation with a radix of two. Owing to its relatively straightforward implementation in electronic circuitry, the binary system is used internally by virtually all modern computers.
The modern binary number system was first fully documented
by Gottfried Leibniz in the 17th century in his article Explication de
l'Arithmétique Binaire. Leibniz's uses 0 and 1, like the modern binary numeral
system.
In 1854, British mathematician George Boole published a
landmark paper detailing a system of logic that would become known as Boolean
algebra. His logical system proved instrumental in the development of the
binary system, particularly in its implementation in electronic circuitry.
In 1937, Claude Shannon produced his master's thesis at MIT
that implemented Boolean algebra and binary arithmetic using electronic relays
and switches for the first time in history. Entitled A Symbolic Analysis of
Relay and Switching Circuits,
In November of 1937, George Stibitz, then working at Bell
Labs, completed a relay-based computer he dubbed the "Model K" (for
"kitchen", where he had assembled it), which calculated using binary
addition. Bell Labs thus authorized a full research program in late 1938 with
Stibitz at the helm. Their Complex Number Computer, completed January 8, 1940,
was able to calculate complex numbers. In a demonstration to the American
Mathematical Society conference at
A binary number can be represented by any sequence of bits binary
digits), which in turn may be represented by any mechanism capable of being in
two mutually exclusive states. The following sequences of symbols could all be
interpreted as different binary numeric values:
11010011
on on off on off off on on
| | - | - - | |
o o x o x x o o
yes yes no yes no no yes yes
y y n y n n y y
true true false true false false true true
The numeric value represented in each case is dependent upon
the value assigned to each symbol. In a computer, the numeric values may be
represented by two different voltages ; on a magnetic
disk, magnetic polarities may be used. A "positive", "yes",
or "on" state is not necessarily equivalent to the numerical value of
one; it depends on the architecture in use.
In keeping with customary representation of numerals using
arabic numerals, binary numbers are commonly written using the symbols 0 and 1.
When written, binary numerals are often subscripted or suffixed in order to
indicate their base, or radix. The following notations are equivalent:
100101 binary (explicit
statement of format)
100101b (a suffix indicating
binary format)
bin 100101 (a prefix indicating
binary format)
1001012 (a subscript indicating
base-2 notation)
When spoken, binary numerals are usually pronounced by
pronouncing each individual digit, in order to distinguish them from decimal
numbers. For example, the binary numeral "100" is pronounced
"one zero zero", rather than "one hundred", to make its
binary nature explicit, and for purposes of correctness. Since the binary
numeral "100" is equal to the decimal value four, it would be
confusing, and numerically incorrect, to refer to the numeral as "one
hundred."
Counting in binary
Counting in binary is similar to counting in any other
number system. Beginning with a single digit, counting proceeds through each
symbol, in increasing order. Decimal counting uses the symbols 0 through 9,
while binary only uses the symbols 0 and 1.
When the symbols for the first digit are exhausted, the
next-higher digit (to the left) is incremented, and counting starts over at 0.
In decimal, counting proceeds like so:
00, 01, 02, ... 07, 08, 09
(rightmost digit starts over, and the 0 is incremented)
10, 11, 12, ... 17, 18, 19
(rightmost digit starts over, and the 1 is incremented)
20, 21, 22, ...
When the rightmost digit reaches 9, counting returns to 0, and
the second digit is incremented. In binary, counting is similar, with the
exception that only the two symbols 0 and 1 are used. When 1 is reached,
counting begins at 0 again, with the digit to the left being incremented:
000, 001 (rightmost digit starts
over, and the second 0 is incremented)
010, 011 (middle and rightmost
digits start over, and the first 0 is incremented)
100, 101 (rightmost digit starts
over again, middle 0 is incremented)
110, 111...
Arithmetic
in binary is much like arithmetic in other numeral systems. Addition,
subtraction, multiplication, and division can be performed on binary numerals.
The
simplest arithmetic operation in binary is addition. Adding two single-digit
binary numbers is relatively simple:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (the 1 is carried)
Adding
two "1" values produces the value "10", equivalent to the
decimal value 2. This is similar to what happens in decimal when certain
single-digit numbers are added together; if the result exceeds the value of the
radix (10), the digit to the left is incremented:
5 + 5 = 10
7 + 9 = 16
This is known as carrying in most numeral systems. When the
result of an addition exceeds the value of the radix, the procedure is to
"carry the one" to the left, adding it to the next positional value.
Carrying works the same way in binary:
1 1 1 1 1 (carry)
0 1 1 0 1
+ 1 0 1 1 1
-------------
= 1 0 0 1 0 0
In this example, two numerals are being added together:
01101 (13 decimal) and 10111 (23 decimal). The top row shows the carry bits
used. Starting in the rightmost column, 1 + 1 = 10. The 1 is carried to the
left, and the 0 is written at the bottom of the rightmost column. The second
column from the right is added: 1 + 0 + 1 = 10 again; the 1 is carried, and 0
is written at the bottom. The third column: 1 + 1 + 1 = 11. This time, a 1 is
carried, and a 1 is written in the bottom row. Proceeding like this gives the
final answer 100100.
Subtraction
works in much the same way:
0 - 0 = 0
0 - 1 = 1 (with borrow)
1 - 0 = 1
1 - 1 = 0
One binary numeral can be subtracted from another as
follows:
* * * *
(starred columns are borrowed from)
1 1 0 1 1 1 0
- 1 0 1 1 1
----------------
= 1 0 1 0 1 1 1
Subtracting a positive number is equivalent to adding a negative number of equal absolute; computers typically use the two’s complement notation to represent negative values. This notation eliminates the need for a separate "subtract" operation.
Multiplication
in binary is similar to its decimal counterpart. Two numbers A and B
can be multiplied by partial products: for each digit in B, the product
of that digit in A is calculated and written on a new line, shifted
leftward so that its rightmost digit lines up with the digit in B that
was used. The sum of all these partial products gives the final result.
Since
there are only two digits in binary, there are only two possible outcomes of
each partial multiplication:
For example, the
binary numbers 1011 and 1010 are multiplied as follows:
1 0 1
1 (A)
× 1 0 1
0 (B)
---------
0 0 0
0 ← Corresponds to a zero in B
1 0 1 1 ← Corresponds to a one in B
0 0 0 0
+ 1 0 1 1
---------------
= 1 1 0 1 1 1 0
Division
Binary division is again similar to its decimal counterpart:
__________
1 0 1 | 1 1 0 1 1
Here, the divisor is 101, or 5 decimal, while the dividend
is 11011, or 27 decimal. The procedure is the same as that of decimal long
division; here, the divisor 101 goes into the first three digits 110 of the
dividend one time, so a "1" is written on the top line. This result
is multiplied by the divisor, and subtracted from the first three digits of the
dividend; the next digit (a "1") is included to obtain a new
three-digit sequence:
1
__________
1 0 1 | 1 1 0 1 1
- 1 0 1
-----
0 1 1
The procedure is then repeated with the new sequence,
continuing until the digits in the dividend have been exhausted:
1 0 1
__________
1 0 1 | 1 1 0 1 1
- 1 0 1
-----
0 1 1
- 0 0 0
-----
1 1 1
- 1 0 1
-----
1 0
Thus, the quotient of 11011 divided by 101 is 1012, as shown
on the top line, while the remainder, shown on the bottom line, is 102. In
decimal, 27 divided by 5 is 5, with a remainder of 2.
Though
not directly related to the numerical interpretation of binary symbols,
sequences of bits may be manipulated using Boolean Logical Operator. When a
string of binary symbols is manipulated in this way, it is called a bitwise
operation; the logical operators AND, OR and XOR may be performed on
corresponding bits in two binary numerals provided as input. The logical NOT operation may be performed on
individual bits in a single binary numeral provided as input. Sometimes, such
operations may be used as arithmetic short-cuts, and may have other
computational benefits as well. For example, discarding the last bit of a
binary number (also known as binary shifting), is the decimal equivalent of
division by two.
This
method works for conversion from any base, but there are better methods for
bases which are powers of two, such as octal and hexadecimal given below.
In
place-value numeral systems, digits in successively lower, or less significant,
positions represent successively smaller powers of the radix. The starting
exponent is one less than the number of digits in the number. A five-digit
number would start with an exponent of four. In the decimal system, the radix
is 10 (ten), so the left-most digit of a five-digit number represents the 104
(ten thousands) position. Consider:
9735210 is equal to:
9 times 104 (9 × 10000 = 90000)
plus
7 times 103 (7 × 1000 = 7000)
plus
3 times 102 (3 × 100 = 300) plus
5 times 101 (5 × 10 = 50) plus
2 times 100 (2 × 1 = 2)
Multiplication
by the radix is simple. The digits are shifted left, and a 0 is appended to the
right end of the number. For example, 9735 times 10 is equal to 97350.
So one way to interpret a string of digits is as the last digit added to the
radix times all but the last digit. 97352 equals 9735 times 10
plus 2. An example in binary is 11011001112 equals 1101100112
times 2 plus 1. This is the essence of the conversion method. At each
step, write the number to be converted as 2*k + 0 or 2*k + 1 for an integer k,
which becomes the new number to be converted.
11810 equals
59 x 2 + 0
(29 x 2 + 1) x 2 + 0
((14 x 2 + 1) x 2 + 1) x 2 + 0
(((7 x 2 + 0) x 2 + 1) x 2 + 1)
x 2 + 0
((((3 x 2 + 1) x 2 + 0) x 2 +
1) x 2 + 1) x 2 + 0
(((((1 x 2 + 1) x 2 + 1) x 2
+ 0) x 2 + 1) x 2 + 1) x 2 + 0
1 x 26 + 1 x 25 + 1
x 24 + 0 x 23 + 1 x 22 + 1
x 21 + 0 x 20
11101102
So in
the algorithm to convert from an integer decimal numeral to its binary
equivalent, the number is divided by two, and the remainder written in the ones-place.
The result is again divided by two, its remainder written in the next place to
the left. This process repeats until the number becomes zero.
For
example, 11810, in binary, is:
|
Operation |
Remainder |
|
118/2 = 59 |
0 |
|
59/2 = 29 |
1 |
|
29/2 = 14 |
1 |
|
14/2 = 7 |
0 |
|
7/2 = 3 |
1 |
|
3/2 = 1 |
1 |
|
1/2 = 0 |
1 |
Reading
the sequence of remainders from the bottom up gives the binary numeral 11101102.
To
convert from binary to decimal is the reverse algorithm. Starting from the left,
double the result and add the next digit until there are no more. For example
to convert 1100101011012 to decimal:
|
Result |
Remaining digits |
|
0 |
110010101101 |
|
0*2 + 1 = 1 |
10010101101 |
|
1*2 + 1 = 3 |
0010101101 |
|
3*2 + 0 = 6 |
010101101 |
|
6*2 + 0 = 12 |
10101101 |
|
12*2 + 1 = 25 |
0101101 |
|
25*2 + 0 = 50 |
101101 |
|
50*2 + 1 = 101 |
01101 |
|
101*2 + 0 = 202 |
1101 |
|
202*2 + 1 = 405 |
101 |
|
405*2 + 1 = 811 |
01 |
|
811*2 + 0 = 1622 |
1 |
|
1622*2 + 1 = 3245 |
|
and
the result is 324510.
The
fractional parts of a numbers are converted with similar methods. They are
again based on the equivalence of shifting with doubling or halving.
In a
fractional binary number such as .110101101012, the first digit is
1/2, the second 1/22, etc. So if there is a 1 in the first place
after the decimal, then the number is at least 1/2, and vice versa. Double that
number is at least 1. This suggests the algorithm: Repeatedly double the number
to be converted, record if the result is at least 1, and then throw away the
integer part.
For
example, (1/3)10, in binary, is:
|
Converting |
Result |
|
1/3 |
0. |
|
1/3 * 2 = 2/3 < 1 |
0.0 |
|
2/3 * 2 = 1 1/3 ≥ 1 |
0.01 |
|
1/3 * 2 = 2/3 < 1 |
0.010 |
|
2/3 * 2 = 1 1/3 ≥ 1 |
0.0101 |
which
is the repeating fraction 0.0101...2
Or for
example, 0.110, in binary, is:
|
Converting |
Result |
|
0.1 |
0. |
|
0.1 * 2 = 0.2 < 1 |
0.0 |
|
0.2 * 2 = 0.4 < 1 |
0.00 |
|
0.4* 2 = 0.8 < 1 |
0.000 |
|
0.8* 2 = 1.6 ≥ 1 |
0.0001 |
|
0.6 * 2 = 1.2 ≥ 1 |
0.00011 |
|
0.2 * 2 = 0.4 < 1 |
0.000110 |
|
0.4 * 2 = 0.8 < 1 |
0.0001100 |
|
0.8 * 2 = 1.6 ≥ 1 |
0.00011001 |
|
0.6 * 2 = 1.2 ≥ 1 |
0.000110011 |
|
0.2 * 2 = 0.4 < 1 |
0.0001100110 |
which
is also a repeating fraction 0.000110011...2 It may come as a
surprise that terminating decimal fractions can have repeating expansions in
binary. It is for this reason that many are surprised to discover that 0.1 +
... + 0.1, (10 additions) differs from 1 in floating point arithmetic. In fact,
the only binary fractions with terminating expansions are of the form of an
integer divided by a power of 2, which 1/10 is not.
The
final conversion is from binary to decimal fractions. The only difficulty
arises with repeating fractions, but otherwise the method is to shift the
fraction to an integer, convert it as above, and then divide by the appropriate
power of two in the decimal base. For example,
|
x |
= |
1100 |
.101110011100... |
|
x times 26 |
= |
1100101110 |
.0111001110... |
|
x times 2 |
= |
11001 |
.0111001110... |
|
x times (26 - 2) |
= |
1100010101 |
|
|
x |
= |
(789/62)10 |
|
Another
way, perhaps quicker and more efficient than the previous, of converting from
binary to decimal, is to do so indirectly- first converting (x binary) or (x
decimal) to (x hexadecimal) and then converting (x hexidecimal) to the opposite
of the former, respectively.
Binary
may be converted to and from hexadecimal somewhat more easily. This is due to
the fact that the radix of the hexadecimal system (16) is a power of the radix
of the binary system (2). More specifically, 16 = 24, so it takes
exactly four digits of binary to represent one digit of hexadecimal.
The
following table shows each hexadecimal digit along with the equivalent
four-digit binary sequence:
|
|
|
|
To
convert a hexadecimal number into its binary equivalent, simply substitute the
corresponding binary digits:
3A16 = 0011 10102
E716 = 1110 01112
To
convert a binary number into its hexadecimal equivalent, divide it into groups
of four bits. If the number of bits isn't a multiple of four, simply insert
extra 0 bits at the left (called padding). For example:
10100102 = 0101 0010 grouped with
padding = 5216
110111012 = 1101 1101 grouped = DD16
Binary
is also easily converted to the octal numeral system, since octal uses a radix
of 8, which is a power of two (namely, 23, so it takes exactly three
binary digits to represent an octal digit). The correspondence between octal
and binary numerals is the same as for the first eight digits of hexadecimal in
the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is
equivalent to octal 7, and so on.
|
|
Converting
from octal to decimal proceeds in the same fashion as it does for hexadecimal:
658 = 110 1012
178 = 001 1112
And
from binary to octal:
1011002 = 101 1002 grouped =
548
100112 = 010 0112 grouped
with padding = 238
Non-integers
can be represented by using negative powers, which are set off from the other
digits by means of a radix point (called
a decimal point in the decimal system). For example, the binary number 11.012
thus means:
1 times 21 (1 × 2 = 2) plus
1 times 20 (1 × 1 = 1) plus
0 times 2-1 (0 × (1/2) = 0) plus
1 times 2-2 (1 × (1/4) = 0.25)
For a
total of 3.25 decimal.
All
dyadic rational numbers p/2a have a terminating binary
numeral -- the binary representation has only finitely many terms after the
radix point. Other rational numbers have binary representation, but instead of
terminating, they recur, with a finite sequence of digits repeating
indefinitely. For instance
1/310 = 1/112 =
0.0101010101...2
1210/1710 = 11002
/ 100012 = 0.10110100 10110100 10110100...2
The
phenomenon that the binary representation of any rational is either terminating
or recurring also occurs in other radix-based numeral systems. See, for
instance, the explanation in Decimal. Another similarity is the existence of
alternative representations for any terminating representation, relying on the
fact that 0.111111... is the sum of the geometric series 2-1 + 2-2
+ 2-3 + ... which is 1.
Binary
numerals which neither terminate nor recur represent irrational numbers. For
instance,