Review on Number Systems

1
Review on Number Systems

• Decimal, Binary, and Hexadecimal

2
Base-N Number System

• Base N
• N Digits 0, 1, 2, 3, 4, 5, , N-1
• Example 1045N
• Positional Number System

• Digit do is the least significant digit (LSD).
• Digit dn-1 is the most significant digit (MSD).

3
Decimal Number System

• Base 10
• Ten Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Example 104510
• Positional Number System
• Digit d0 is the least significant digit (LSD).
• Digit dn-1 is the most significant digit (MSD).

4
Binary Number System

• Base 2
• Two Digits 0, 1
• Example 10101102
• Positional Number System
• Binary Digits are called Bits
• Bit bo is the least significant bit (LSB).
• Bit bn-1 is the most significant bit (MSB).

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Definitions

• nybble 4 bits
• byte 8 bits
• (short) word 2 bytes 16 bits
• (double) word 4 bytes 32 bits
• (long) word 8 bytes 64 bits
• 1K (kilo or kibi) 1,024
• 1M (mega or mebi) (1K)(1K) 1,048,576
• 1G (giga or gibi) (1K)(1M) 1,073,741,824

6
Hexadecimal Number System

• Base 16
• Sixteen Digits 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
• Example EF5616
• Positional Number System

7
Binary Addition

• Single Bit Addition Table

0 0 0
0 1 1
1 0 1
1 1 10 Note carry
8
Hex Addition

• 4-bit Addition

4 4 8
4 8 C
8 7 F
F E 1D Note carry
9
Hex Digit Addition Table
10
1s Complements

• 1s complement (or Ones Complement)
• To calculate the 1s complement of a binary
  number just flip each bit of the original
  binary number.
• E.g. 0 ? 1 , 1 ? 0
• 01010100100 ? 10101011011

11
Why choose 2s complement?
12
2s Complements

• 2s complement
• To calculate the 2s complement just calculate
  the 1s complement, then add 1.
• 01010100100 ? 10101011011 1
• 10101011100
• Handy Trick Leave all of the least significant
  0s and first 1 unchanged, and then flip the
  bits for all other digits.
• Eg 01010100100 -gt 10101011100

13
Complements

• Note the 2s complement of the 2s complement is
  just the original number N
• EX let N 01010100100
• (2s comp of N) M 10101011100
• (2s comp of M) 01010100100 N

14
Twos Complement Representation for Signed Numbers

• Lets introduce a notation for negative digits
• For any digit d, define d -d.
• Notice that in binary, where d ? 0,1, we have
• Twos complement notation
• To encode a negative number, we implicitly negate
  the leftmost (most significant) bit
• E.g., 1000 (-1)000 -123 022 021
  020 -8

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Negating in Twos Complement

• Theorem To negatea twos complementnumber,
  just complement it and add 1.
• Proof (for the case of 3-bit numbers XYZ)

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Signed Binary Numbers

• Two methods
• First method sign-magnitude
• Use one bit to represent the sign
• 0 positive, 1 negative
• Remaining bits are used to represent the
  magnitude
• Range - (2n-1 1) to 2n-1 - 1
• where nnumber of digits
• Example Let n4 Range is 7 to 7 or
• 1111 to 0111

17
Signed Binary Numbers

• Second method Twos-complement
• Use the 2s complement of N to represent
• -N
• Note MSB is 0 if positive and 1 if negative
• Range - 2n-1 to 2n-1 -1
• where nnumber of digits
• Example Let n4 Range is 8 to 7
• Or 1000 to 0111

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Signed Numbers 4-bit example

• Decimal 2s comp Sign-Mag
• 7 0111 0111
• 6 0110 0110
• 5 0101 0101
• 4 0100 0100
• 3 0011 0011
• 2 0010 0010
• 1 0001 0001
• 0 0000 0000

Pos 0
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Signed Numbers-4 bit example

• Decimal 2s comp Sign-Mag
• -8 1000 N/A
• -7 1001 1111
• -6 1010 1110
• -5 1011 1101
• -4 1100 1100
• -3 1101 1011
• -2 1110 1010
• -1 1111 1001
• -0 0000 ( 0) 1000

20
Signed Numbers-8 bit example
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Notes

• Humans normally use sign-magnitude
  representation for signed numbers
• Eg Positive numbers N or N
• Negative numbers -N
• Computers generally use twos-complement
  representation for signed numbers
• First bit still indicates positive or negative.
• If the number is negative, take 2s complement to
  determine its magnitude
• Or, just add up the values of bits at their
  positions, remembering that the first bit is
  implicitly negative.

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Examples

• Let N4 twos-complement
• What is the decimal equivalent of
• 01012
• Since MSB is 0, number is positive
• 01012 41 510
• What is the decimal equivalent of
• 11012
• Since MSB is one, number is negative
• Must calculate its 2s complement
• 11012 -(00101) - 00112 or -310

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Very Important!!! Unless otherwise stated,
assume twos-complement numbers for all problems,
quizzes, HWs, etc.The first digit will not
necessarily be explicitly underlined.
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Arithmetic Subtraction

• Borrow Method
• This is the technique you learned in grade school
• For binary numbers, we have

0 - 0 0
1 - 0 1
1 - 1 0
1
0 - 1 1 with a borrow
25
Binary Subtraction

• Note
• A (B) A (-B)
• A (-B) A (-(-B)) A (B)
• In other words, we can subtract B from A by
  adding B to A.
• However, -B is just the 2s complement of B, so
  to perform subtraction, we
• 1. Calculate the 2s complement of B
• 2. Add A (-B)

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Binary Subtraction - Example

• Let n4, A01002 (410), and
• B00102 (210)
• Lets find AB, A-B and B-A

0 1 0 0 0 0 1 0
? (4)10
AB
? (2)10
0 11 0 6
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Binary Subtraction - Example

0 1 0 0 - 0 0 1 0
? (4)10
A-B
? (2)10
0 1 0 0 1 1 1 0
? (4)10
A (-B)
? (-2)10
10 0 1 0 2
Throw this bit away since n4
28
Binary Subtraction - Example

0 0 1 0 - 0 1 0 0
? (2)10
B-A
? (4)10
0 0 1 0 1 1 0 0
? (2)10
B (-A)
? (-4)10
1 1 1 0 -2
1 1 1 02 - 0 0 1 02 -210
29
16s Complement method

• The 16s complement of a 16 bit Hexadecimal
  number is just
• 1000016 N16
• Q What is the decimal equivalent of B2CE16 ?

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16s Complement

• Since sign bit is one, number is negative. Must
  calculate the 16s complement to find magnitude.
• 1000016 B2CE16 ?
• We have
• 10000
• - B2CE

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16s Complement

• FFF10
• - B2CE

2
3
D
4
32
16s Complement

• So,
• 1000016 B2CE16 4D3216
• 44,096 13256 316 2
• 19,76210
• Thus, B2CE16 (in signed-magnitude)represents
  -19,76210.

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Why does 2s complement work?
34
Sign Extension
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Sign Extension

• Assume a signed binary system
• Let A 0101 (4 bits) and B 010 (3 bits)
• What is AB?
• To add these two values we need A and B to be of
  the same bit width.
• Do we truncate A to 3 bits or add an additional
  bit to B?

36
Sign Extension

• A 0101 and B010
• Cant truncate A! Why?
• A 0101 -gt 101
• But 0101 ltgt 101 in a signed system
• 0101 5
• 101 -3

37
Sign Extension

• Must sign extend B,
• so B becomes 010 -gt 0010
• Note Value of B remains the same
• So 0101 (5)
• 0010 (2)
• --------
• 0111 (7)

Sign bit is extended
38
Sign Extension

• What about negative numbers?
• Let A0101 and B100
• Now B 100 ? 1100

Sign bit is extended
0101 (5) 1100 (-4) ------- 10001 (1)
Throw away
39
Why does sign extension work?

• Note that (-1) 1 11 111 1111 1111
• Thus, any number of leading 1s is equivalent, so
  long as the leftmost one of them is implicitly
  negative.
• Proof 1111 -(1111) -(1000 - 111)
  -(1)
• So, the combined value of any sequence of leading
  ones is always just -1 times the position value
  of the rightmost 1 in the sequence.
• 1111000 (-1)2n

n
40
Number Conversions
41
Decimal to Binary Conversion
Method I Use repeated subtraction. Subtract
largest power of 2, then next largest, etc.
Powers of 2 1, 2, 4, 8, 16, 32, 64, 128, 256,
512, 1024, 2n Exponent 0, 1, 2, 3, 4, 5,
6, 7, 8, 9, 10 , n
210
2n
29
28
20
27
21
22
23
26
24
25
42
Decimal to Binary Conversion
Suppose x 156410
Subtract 1024 1564-1024 (210) 540 ? n10
or 1 in the (210)s position
Subtract 512 540-512 (29) 28 ? n9
or 1 in the (29)s position
28256, 27128, 2664, 2532 gt 28, so we have 0
in all of these positions
Subtract 16 28-16 (24) 12 ?
n4 or 1 in (24)s position
Subtract 8 12 8 (23) 4 ? n3 or 1
in (23)s position
Subtract 4 4 4 (22) 0 ? n2
or 1 in (22)s position
Thus 156410 (1 1 0 0 0 0 1 1 1 0 0)2
43
Decimal to Binary Conversion
Method II Use repeated division by radix.
2 1564 782 R 0
2__24_ 12 R 0
2_____ 391 R 0
2_____ 6 R 0
?
2_____ 195 R 1
2_____ 3 R 0
2_____ 97 R 1
2_____ 1 R 1
2_____ 48 R 1
2_____ 0 R 1
2_____ 24 R 0
Collect remainders in reverse order
1 1 0 0 0 0 1 1 1 0 0
44
Binary to Hex Conversion

• Divide binary number into 4-bit groups

1 1 0 0 0 0 1 1 1 0 0
0
Pad with 0s If unsigned number
2. Substitute hex digit for each group
Pad with sign bit if signed number
61C16
45
Hexadecimal to Binary ConversionExample

• Convert each hex digit to equivalent binary

(1 E 9 C)16
(0001 1110 1001 1100)2
46
Decimal to Hex Conversion
Method II Use repeated division by radix.
16 1564 97 R 12 C
16_____ 6 R 1
?
16_____ 0 R 6
N 61C 16