Computer Science 1000
Digital Information

Arithmetic


we have seen how to represent numbers in other
bases
how about manipulating numbers in other bases


for example, how can we add two binary numbers?
as it turns out, we can, using the same technique
as you learned in grade school


do a digit-by-digit addition of both numbers
be sure to add in carry digit when necessary

Example: Add 1287 to 1432

Step 1: Add the last two digits
1287
+1432

Example: Add 1287 to 1432

Step 2: Add the next two digits

note that this results in a carry digit
1287
+1432
9

Example: Add 1287 to 1432

Step 3: Add the next two digits

be sure to add in the carry digit
1
1287
+1432
19

Example: Add 1287 to 1432

Step 4: Add the final two digits

be sure to add in the carry digit
1
1287
+1432
719

Example: Add 1287 to 1432

Step 4: Add the final two digits

be sure to add in the carry digit
1
1287
+1432
2719

Adding Binary Numbers

given the process for adding two decimal numbers
1
1287
+1432
2719

can we do the same thing with binary numbers?

Adding Binary Numbers

yes, we can


add each two corresponding digits, plus carry (if any)
propagate new carry digit (if any)
1011
+0010

Adding Binary Numbers

in order to add single bits together, we need
to remember a few conversions:
Dec
0
1
2
3
Bin
0
1
10
11

Example:

Step 1: Add last two digits (in binary)


1+0=1
no carry bit
1011
+0010
1
Dec
0
1
2
3
Bin
0
1
10
11

Example:

Step 2: Add second-last two digits (in binary)


1 + 1 = 10
there is a carry bit
1
1011
+0010
01
1
Dec
0
1
2
3
Bin
0
1
10
11

Example:

Step 3: Add second two digits (plus carry)


1+0+0=1
there is no carry bit
1
1011
+0010
101
01
1
Dec
0
1
2
3
Bin
0
1
10
11

Example:

Step 4: Add first two digits


1+0=1
there is no carry bit
1
1011
+0010
1101
101
01
1
Dec
0
1
2
3
Bin
0
1
10
11

Check:
1011
+0010
_____
1101

table shows correct
computation of 11 + 2 = 13
Decimal
Binary
Decimal
Binary
0
0000
8
1000
1
0001
9
1001
2
0010
10
1010
3
0011
11
1011
4
0100
12
1100
5
0101
13
1101
6
0110
14
1110
7
0111
15
1111

Signed Integers



our discussion so far has focused on non-negative
integers
computers permit the storage and manipulation of
signed numbers, which includes both positive and
negative numbers
different ways for representing signed numbers


sign-magnitude
two's complement

Signed Integers


let's first consider the decimal case
recall: integer is a sequence of digits

padded with leading zeroes, if necessary
00001234

how do we represent negative numbers?

Sign-Magnitude

instead of storing a digit in the first symbol of the sequence,
let's store a special character



our previous example becomes:


+0001234
to store a negative number, switch the symbol


+ : number is positive
- : number is negative
-0001234
note that if the number of symbols being stored remains the
same, then we must give one digit up to store the negative
sign

Sign-Magnitude

the previously described representation is often
referred to as sign-magnitude



first symbol: stores sign of number (+/-)
remaining symbols: stores the magnitude (size) of number
one of the primary advantages of sign-magnitude is its
familiarity


people typically represent negative numbers using a leading –
although the leading + is often omitted when writing numbers, it
is still familiar (e.g. most people know what +56 means)

Sign-Magnitude

Question: Given x symbols (digits and sign), what is the
largest number that can be represented using signmagnitude?

Consider: an x-symbol, sign-magnitude number has two parts



first part: sign (+/-) : 1 symbol
second part: magnitude: x-1 digits
to make the largest number possible, set the sign to +, and the digits
to 9s


4 symbols:
6 symbols:
+999
+99999

Sign-Magnitude

Question: Given x symbols (digits and sign), what is the
largest number that can be represented using signmagnitude?

Answer: 10x-1 - 1


this follows from our previous discussion, as we now only have x-1 bits
with which to represent the magnitude of the number
this also means that the smallest number we can store is – (10x-1 - 1), by
flipping the + to a -

Binary Sign-Magnitude


sign magnitude works the same in binary
however, when using binary, a simplifying assumption is
typically made




positive number: set left-most bit to zero
negative number: set left-most bit to one
this assumption means that the number still only requires 1s
and 0s
example: store 6 and -6 in binary (4 bits)


6: 0110
-6: 1110

Binary Sign-Magnitude



limits on number similar to decimal case
Given x symbols:
Base
Largest
Smallest
Decimal (10)
10x-1 -1
-(10x-1 -1)
Binary (2)
2x-1 -1
-(2x-1 -1)
Example:


3 bits: -3 ... 3
5 bits: -15... 15

Example: Given the sequence of binary digits 1010,
what is the equivalent number in decimal if this is (a)
an unsigned integer and (b) a signed integer?

Solution (a): Recall our algorithm for converting
binary digits to its decimal number
1010 = 1 x 23
+ 0 x 22
+ 1 x 21
+ 0 x 20 = 10

Example: Given the sequence of binary digits 1010,
what number is being represented if this represents (a)
an unsigned integer and (b) a signed integer?

Solution (b): Use the algorithm from before, but ignore the leftmost bit
1010 = 0 x 22
+ 1 x 21
+ 0 x 20 = 2

then use the left-most bit to determine the sign
 since it's 1, we know that the number is negative:
-2

Sign-Magnitude

although sign-magnitude is intuitive, it does suffer from
several drawbacks

1) Two representations for zero:


4 bit: 0000 and 1000
this requires special case functionality in certain circumstances

e.g. 0000 and 1000 must be determined to be equal

Sign-Magnitude

although sign-magnitude is intuitive, it does suffer from
several drawbacks

2) Special code needed for negative arithmetic

e.g. addition



our previous addition function would not work correctly if the number
is negative
additional code would need to be written
this may seem trivial, except that low-level arithmetic is often built
into hardware

Two's Complement



a different way to represent signed numbers
unlike sign-magnitude, all digits in two's
complement represent a coefficient to be multiplied
to a power of 2
in fact, the representation is nearly identical to the
positive (unsigned) version, with one simple
exception:

the first power is assumed to be negative

Recall our algorithm for converting binary digits to its
actual number
1010 = 1 x 23
+ 0 x 22
+ 1 x 21
+ 0 x 20 = 10

in two's complement, the first power is negative
1010 = 1 x –(23)
+ 0 x 22
+ 1 x 21
+ 0 x 20 = -6

Example: What is the decimal representation of the
binary number 1111, assuming two's complement?
1111 = 1 x –(23)
+ 1 x 22
+ 1 x 21
+ 1 x 20 = -1

Two's Complement – Observations

for positive numbers, the sign-magnitude and two's
complement representation are identical


negative numbers always have their left-most bit set to one in
both representations


e.g. 610 = 01102 for both representations
remember: for two's complement, the left-most bit represents magnitude
as well
for negative numbers, a sequence of all 1s represents -1 in
two's complement, or the largest negative number


this is the opposite of sign-magnitude, where all 1s represented the
smallest negative number (11112 = -710)
see previous slide for example

Two's Complement

given x bits, what is the largest two's
complement number that we can represent?

Answer: 2x-1 - 1
previous slide says that positive sign-magnitude
and two's complement numbers are identical
 therefore, largest sign-magnitude number and
two's complement number are the same


Two's Complement

given x bits, what is the smallest two's
complement number that we can represent?
the answer is perhaps not as clear, as two's
complement and sign-magnitude differ
considerably in representation of negative
numbers
 let's consider a specific example


Recall our breakdown of a two's complement number
1111 = 1 x –(23)
+ 1 x 22
+ 1 x 21
+ 1 x 20 = -1

what should our digits be set to, to produce the smallest
number?

set left-most bit to one, and all others to zero

Two's Complement

given x bits, what is the smallest two's
complement number that we can represent?

Answer:
-(2x-1)
accomplished by setting first bit to one, and all
others to zero
 hence, range of two's complement number with
x bits is -(2x-1) ... 2x-1 - 1


Example: given an integer of size 5 bits, what
would the range of the integer be if it is:

a) unsigned


b) signed (sign-magnitude)


0 ... 2x -1 = { 0 ... 31}
- (2x-1 -1) .. (2x-1 -1) = { -15... 15}
c) signed (two's complement)

- (2x-1) .. (2x-1 -1) = { -16 ... 15}

Sign Extension

recall with unsigned numbers, we could
increase number of bits of any number by
adding leading zeroes

e.g. 6



using 4 bits:
using 8 bits:
0110
00000110
what do we do with signed numbers?

Sign Extension

sign-magnitude
with sign-magnitude, we can also add leading
zeroes, but only after the sign bit
 e.g. -6



4 bit:
8 bit:
1110
10000110

Sign Extension

two's complement

we pad the left side of the number with the left-most bit




this is often referred to as sign-extension
e.g. 6



positive: pad with zeroes
negative: pad with ones
4 bit: 0110
8 bit: 00000110
e.g. -6


4 bit: 1010
8 bit: 11111010

Check:
11111010 =
1 x –(27)
+ 1 x 26
+ 1 x 25
+ 1 x 24
+ 1 x 23
+ 0 x 22
+ 1 x 21
+ 0 x 20 = -6