Summary: This document introduces a variety of mechanisms
for representing integers in binary (as a sequence of 0's and 1's).
Internally, most computers represent most values as a sequence of 0's and
1's. The number system in which all numbers are sequences of 0's and 1's
is called the binary system.
To understand binary numbers, begin by recalling elementary school math.
When we first learned about numbers, we were taught that, in the
decimal system, things are organized into columns, such that H
is the hundreds column, T
is the tens column, and O
is
the ones column. Hence, we know that number 293
is 2-hundreds
plus 9-tens plus 3-ones.
Years later, we learned that the ones column meant 100
, the tens column meant 101, the hundreds column
102 and so on. Hence, we started to think about 293 as
(2*102)+(9*101)+(3*100). In columnar
form,
As you know, the decimal system uses the digits 0-9 to represent numbers.
If we wanted to put a larger number in column 10n (e.g.,
10), we would have to multiply 10*10n, which would give
10n+1, and be carried a column to the left. For example,
putting ten in the 100 column is impossible, so we put a
1 in the 101 column, and a 0 in the 100 column,
thus using two columns. Twelve would be 12*100, or 100
(10+2), or 101+2*100, which also uses an
additional column to the left (12).
This binary system works under the exact same principles as the decimal system, only it operates in base 2 rather than base 10. In other words, instead of columns being
they are
Instead of using the digits 0-9, we only use 0-1 (again, if we used
anything larger it would be like multiplying 2*2n and getting
2n+1, which would not fit in the 2n column.
Therefore, it would shift you one column to the left. For example,
"3" cannot be put into one column in binary represnation. The first
column we fill is the right-most column, which is 20, or 1.
Since 3>1, we need to use an extra column to the left, and indicate
3 as 11 in binary (1*21) + (1*20).
Consider the addition of decimal numbers:
We begin by adding 3+8=11. Since 11 is greater than 10, a one is put
into the 10's column (carried), and a 1 is recorded in the one's column
of the sum. Next, add 2+4+1 (the one is from the carry)=7, which
is put in the 10's column of the sum. Thus, the answer is 71.
Unsigned binary addition works on the same principle, but the numerals are
different. We begin with one-bit binary addition:
0 0 1
+ 0 + 1 + 0
___ ____ ___
0 1 1
1+1 carries us into the next column. In decimal form, 1+1=2. In binary,
any digit higher than 1 puts us a column to the left (as would 10 in
decimal notation). The decimal number 2
is written in binary
notation as 10
(1*21)+(0*2 0). Record the
0 in the ones column, and carry the 1 to the twos column to get an answer
of 10
. In our vertical notation,
The process is the same for multiple-bit binary numbers:
-
Column 20: 0+1=1.
Record the 1.
Temporary Result: 1; Carry: 0
-
Column 21: 1+1=10.
Record the 0, carry the 1.
Temporary Result: 01; Carry: 1
-
Column 22: 1+0=1 Add 1 from carry: 1+1=10.
Record the 0, carry the 1.
Temporary Result: 001; Carry: 1
-
Column 23: 1+1=10. Add 1 from carry: 10+1=11.
Record the 11.
Final result: 11001
Pictorially,,
11 (carry)
1010
+ 1111
______
11001
Always remember
Single-digit multiplication in the binary system works the same way as
in the decimal system:
Multiple-digit binary multiplication can be performed just as multiple
digit decimal multiplication.
101
* 11
_____
101
101
_____
1111
Note that multiplying by two is extremely easy. To multiply by two,
just add a 0 on the end.
For binary division, we can also follow the same kinds of rules as in decimal
division. For the sake of simplicity, we'll throw away any remainder.
As an example, let's compute 111011/11
10011 r 10
_______
11)111011
-11
______
101
-11
______
101
11
______
10
Our discussion so far has focused on unsigned integers, integers
which we always assume to be positive. However, in practice, we need both
positive and negative integers. Such integers are called signed
integers. In the decimal system, we use an extra symbol,
- (the minus sign
) to indicate that a value is negative.
We use an optional extra symbol, + (the plus sign
) to
indicate that a value is positive. In binary, all we have are 0's and 1's.
It turns out, there are many possible mechanisms for representing signed
numbers, include signed magnitude,
one's complement,
twos's complement,
and
excess 2m-1.
Before we investigate signed numbers, we note that the computer uses a
fixed number of bits
(short for binary digits), for most numbers.
For example, an 8-bit number is 8 digits long and a 16-bit number is 16
digits long. Although most languages represent integers with 32 or 64
bits, for this section, we will work with 8 bits.
The simplest way to indicate negation is signed magnitude.
The key idea in signed magnitude is to use the first bit as the sign,
just as the leftmost thing
is a sign (plus or minus) in signed
decimal numbers. In binary signed magnitude, the left-most bit is not
actually part of the number, but is just the equivalent of a +/- sign.
0
indicates that the number is positive, "1" indicates negative.
In 8 bits, 00001100 would be 12 (break this down into
(1*23) + (1*22)). To indicate -12, we would simply put a 1
rather than a 0
as the first bit: 10001100.
In eight bit signed-magnitude representation, the largest possible number
is 01111111, or 127, and the smallest possible numbers is 11111111,
or -127.
It turns out that the signed-magnitude representation is not appropriate
for all applications. For example, there are two ways to represent the
value 0: 00000000 and 10000000. There are many other drawbacks, some
of which we discuss later. Because of these drawbacks,
computer scientists have designed a number of other representations.
In one's complement notation, positive numbers are represented
as usual in unsigned binary. However, negative numbers are represented
differently. To negate a number, replace all zeros with ones, and ones
with zeros: flip the bits. Thus, 12 would be 00001100, and -12 would
be 11110011. As in signed magnitude, the leftmost bit indicates the
sign (1 is negative, 0 is positive). To compute the value of a negative
number, flip the bits and translate as before.
In eight bit one's complement representation, the largest possible number
is 01111111, or 127, and the smallest possible numbers is 10000000,
or -127.
Once again, we have the drawback of having two different representations
for zero. However, there are some advantages of the one's complement
representation, which we discuss later.
The two's complement representation is a slight variation
of one's complement, with a twist. To represent a number in two's
complement, begin with the number in one's complement. If the number
is negative, add 1 using unsigned binary addition and throw away the
leftmost bit if the number ends up using nine bits.
In two's complement notation, twelve would be represented as 00001100,
and negative twelve as 11110100. To verify this, let's subtract 1 from
11110110, to get 11110011. If we flip the bits, we get 00001100, or 12
in decimal.
This notation does not seem very natural to many people, but it does
have the advantage that negative 0 is 0. Start with 00000000. Flip
all the bits: 11111111. Add 1: 100000000. Throw away the leftmost bit:
00000000.
Because of this single representation, we can represent one more number.
The largest number is still 01111111, or 127. The smallest is 10000000,
or -128.
In this notation, m
indicates the total number of bits.
If we're working with an 8-bit representation, we would use excess
27 notation. To represent a number (positive or negative)
in excess 27, add 27 (or 128) to that number and
then represent the result in unsigned binary notation.
For example, to represent 7 would be 128 + 7=135, or
27+22+21+20,
and, in binary, 10000111. We would represent -7 as 128-7=121, and,
in binary, 01111001.
The largest possible number in excess 27 representation is
1111111, or 127. The smallest possible number is -128, or 00000000.
There is only one representation for 0, 10000000.
Because computer scientists have developed these alternative representations,
in order to interpret a sequence of bits, you need to know which representation
has been used. The sequence 10100011 can be:
- 163, in unsigned binary,
- -35, in signed-magnitude,
- -92, in one's complement,
- -93, in two's complement,
- 35, in excess 27
How do we select a representation to use? To see the advantages and
disadvantages of each method, you might try working with them. Here's a
way to start: Using the regular algorithm for binary adition, add (5+12),
(-5+12), (-12+-5), and (12+-12) in each system. Then, convert back to
decimal numbers.
Signed Magnitude:
5+12 -5+12 -12+-5 12+-12
00000101 10000101 10001100 00001100
+ 00001100 + 00001100 + 10000101 + 10001100
__________ __________ __________ ___________
00010001 10010001 00010000 10011000
17 -17 16 -24
One's Complement:
5+12 -5+12 -12+-5 12+-12
00000101 11111010 11110011 00001100
+ 00001100 + 00001100 + 11111010 + 11110011
___________ __________ __________ __________
00010001 00000110 11101101 11111111
17 6 -18 0
Two's Complement:
5+12 -5+12 -12+-5 12+-12
00000101 11111011 11110100 00001100
+ 00001100 + 00001100 + 11111011 + 11110100
__________ __________ __________ __________
00010001 00000111 11101111 00000000
17 7 -17 0
Excess 27
5+12 -5+12 -12+-5 12+-12
10000101 01111011 01110100 10001100
+ 10001100 + 10001100 + 01111011 + 01110100
__________ __________ __________ __________
00010001 00000111 11101111 00000000
-111 -121 111 -128
As you can see, this technique seems to work best for two's
complement, and works very strangely for excess 27.
However, it's easy to fix the results for excess 27:
simply flip the leftmost bit.
Excess 27
5+12 -5+12 -12+-5 12+-12
10000101 01111011 01110100 10001100
+ 10001100 + 10001100 + 01111011 + 01110100
__________ __________ __________ __________
00010001 00000111 11101111 00000000
Flip
10010001 10000111 01101111 10000000
17 7 -17 0
Because addition is easiest for two's complement and excess
2m-1, most computers use one of the those two
representations.
Friday, 31 January 2003 [Samuel A. Rebelsky]
- Created.
- Much of the content was taken from
The Binary System: A Pretty damn clear guide to a quite confusing
concept by Christine R. Wright, with some help from Samuel A.
Rebelsky. The only version of that document I can currently find is
at
http://www.cs.grinnell.edu/~rebelsky/Courses/CS152/1997F/Readings/student-binary.html.
- Dropped a section on converting between representations.
- Dropped problems.
- Reformatted.
- Changed the phrases
negative numbers
and
potentially-negative numbers
to signed numbers
.
- Lots and lots of other small changes.
Sunday, 2 February 2003 [Samuel A. Rebelsky]
- Corrected a few typos.
- Verified that HTML is valid.
Tuesday, 4 February 2003 [Samuel A. Rebelsky]
;
;
Check with Bobby