Basic Definition
REPRESENTING
INTEGER DATA
• An integer is a number which has no
fractional part.
Examples:
-2022
-213
0
1
514
323434565232
Englander Ch. 4
ITEC 1011
Unsigned Binary and
Binary-Coded Decimal
Introduction to Information Technologies
BCD: Binary-Coded Decimal
• Each decimal digit individually converted to
binary
– Requires 4 bits per digit
8-bit location hold 2 BCD digits — 00 to 99
6810 ≡ 0110 1000BCD
• Hexa: 4 bits can hold 16 ≠ values, 0 to F
• A to F not used in BCD
1
Ranges for Data Formats
No. of bits
Binary
1
0–1
2
0–3
3
0–7
4
0 – 15
5
0 – 31
6
0 – 63
7
0 – 127
8
0 – 255
BCD
In General (binary)
ASCII
Binary
No. of bits
0–9
n
0 – 99
9
0 – 511
16
0 - 65,535
0 – 9999
0 – 99
24
0 – 16,777,215
0 – 999999
0 – 999
0
2n - 1
Remember !!
Introduction to Information Technologies
ITEC 1011
Introduction to Information Technologies
Sign-Magnitude
Signed Integers
• Previous examples were for “unsigned
integers” (positive values only!)
• Must also have a mechanism to represent
“signed integers” (positive and negative
values!)
• E.g., -510 = ?2
• Two common schemes: 1) sign-magnitude
2) two’s complement
ITEC 1011
Max
0–9
Etc.
ITEC 1011
Min
Introduction to Information Technologies
• Extra bit on left to represent sign
• 0 = positive value
• 1 = negative value
• E.g., 6-bit sign-magnitude representation of
+5 and –5:
+5:
0 0 0 1 0 1
+
ITEC 1011
5
-5:
1 0 0 1 0 1
-
5
Introduction to Information Technologies
2
Ranges (revisited)
No. of bits
1
2
3
4
5
6
Etc.
ITEC 1011
In General (revisited)
Binary
Unsigned
Sign-magnitude
Min
Max
Min
Max
0
1
0
3
-1
1
0
7
-3
3
0
15
-7
7
0
31
-15
15
0
63
-31
31
Introduction to Information Technologies
Binary
No. of bits
n
ITEC 1011
• 0: 000000
• 0: 100000
• Arithmetic is awkward!
ITEC 1011
Min
0
Max
n
Sign-magnitude
Min
Max
n-1
n-1
2 - 1 -(2
- 1) 2
-1
Introduction to Information Technologies
One’s complement
Difficulties with Sign-Magnitude
• Two representations of zero
Unsigned
• Principle: Invert bits (0
1 and 1
0)
• 6: 000110
• -6: 111001
• Range
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3
Add / Sub in 1’s complement
Two’s Complement
Overflow
•
•
•
Most common scheme of representing
negative numbers in computers
Affords natural arithmetic (no special rules!)
To represent a negative number in 2’s
complement notation…
1.
2.
3.
4.
Overflow sign of result ≠ sign both operands
Decide upon the number of bits (n)
Find the binary representation of the positive value in n-bits
Flip all the bits (change 1’s to 0’s and vice versa)
Add 1
Learn!
ITEC 1011
Introduction to Information Technologies
4
Two’s Complement Example
Two’s complement representation
•
Represent –5 in binary using 2’s
complement notation
1. Decide on the number of bits, for example: 6
2. Find the binary representation of the positive (5)
value in 6 bits
000101
+5
3. Flip all the bits
111010
+
1
111011
4. Add 1
ITEC 1011
111010
-5
Introduction to Information Technologies
Sign Bit
“Complementary” Notation
• In 2’s complement notation, the MSB is the
sign bit (as with sign-magnitude notation)
• Conversions between positive and negative
numbers are easy
• For binary (base 2)…
• 0 = positive value
• 1 = negative value
2’s C
+5:
0 0 0 1 0 1
-5:
1 1 1 0 1 1
+
+
ITEC 1011
5
-
? (previous slide)
Introduction to Information Technologies
2’s C
ITEC 1011
Introduction to Information Technologies
5
Example
ITEC 1011
Detail for -20
+5
0 0 0 1 0 1
2’s C
1 1 1 0 1 0
+
1
-5
1 1 1 0 1 1
2’s C
0 0 0 1 0 0
+
1
+5
0 0 0 1 0 1
-2010: Positive Value =
00010100
“Flip”:
11101011
(One’s complement)
Add 1:
+
1
11101100
ITEC 1011
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Detail for 1100011
11101100
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Range for 2’s Complement
- 29
• For example, 6-bit 2’s complement notation
2’s Complement:
1100011
“Flip”: (One’s complement)
1011100
Add One:
+
100001
-32
-31
111111
000000
000001
011111
1
1011101
Converts to:
100000
=
- 29
...
Negative, sign bit = 1
-1
0
1
...
31
Zero or positive, sign bit = 0
kc
ITEC 1011
Introduction to Information Technologies
ITEC 1011
Introduction to Information Technologies
6
Ranges (revisited)
In General (revisited)
Binary
No. of
bits
1
2
3
4
5
6
Etc.
ITEC 1011
Unsigned
Min
Max
1
0
3
0
7
0
15
0
31
0
63
0
Sign-magnitude 2’s complement
Min
Max
Min
Max
-1
-3
-7
-15
-31
1
3
7
15
31
-2
-4
-8
-16
-32
1
3
7
15
31
Binary
No. of Unsigned
bits
Min Max
n
0
n
Sign-magnitude 2’s complement
Min
2 - 1 -(2
n-1
Max Min
n-1
- 1) 2 -1 -2
n-1
Max
2
n-1
-1
To remember
Introduction to Information Technologies
ITEC 1011
Introduction to Information Technologies
What is -5 plus +5?
2’s Complement Addition
• Zero, of course, but let’s see
• Easy
• No special rules
• Just add
Sign-magnitude
Two’s-complement
1 1 1 1 1 1 1 1
-5:
+5:
ITEC 1011
Introduction to Information Technologies
ITEC 1011
10000101
+00000101
10001010
-5:
+5:
11111011
+00000101
00000000
Introduction to Information Technologies
7
What is 10 subtract 3?
2’s Complement Subtraction
• Easy
• No special rules
• Just subtract, well … actually … just add!
A – B = A + (-B)
add
ITEC 1011
10 – 3 = 10 + (-3) = 7
+3: 000011
2’s complement of B
1s C: 111100
+1:
1
-3: 111101
ITEC 1011
Introduction to Information Technologies
What is 10 subtract -3?
(-(-3))
• 13, of course, but…
• Let’s do it (we’ll use 6-bit values)
• 7, of course, but…
• Let’s do it (we’ll use 6-bit values)
001010
+111101
000111
Introduction to Information Technologies
Overflows and Carries
= 3
10 – (-3) = 10 + (-(-3)) = 13
-3: 111101
1s C: 000010
+1:
1
+3: 000011
ITEC 1011
001010
+000011
001101
Introduction to Information Technologies
8