Number Systems 1/30/12

Decimal number system

Binary number system

Special bit sequences

A position's weight is rk, where k is the number of positions from the decimal point.

To the left of the decimal point, k takes values 0, 1, 2... ,

and to the right, it takes values -1,-2,...

Octal number system

Hexadecimal number system

Examples

Base-r representation of a decimal integer

1. Let the given integer be denoted N.

2. By the Division Algorithm, there exist unique N1 and a0 such that

N = N1 r + a0, 0 ≤ a0 < r.

a0 is the first digit (rightmost digit) of base-r representation.

3. Applying the Division Algorithm on N1, we get

N1 = N2 r + a1, 0 ≤ a1 < r.

a1 is the next digit (on the left) in base-r representation.

4. Repeat this until quotient is zero.

If the Division Algorithm is applied k times, then

ak-1 … a1 a0

is the base-r representation of N.

Converting a decimal fraction to base-r notation

Let i=-1, the fractional part be denoted as 0.f

The digits in the fractional part of base-r notation are denoted 0.a-1 a-2 a-3 …

1. Multiply 0.f with r

Let the result be x.y, where x is the integer part and y is the new fractional part

2. Use x as the ith digit in the fractional part of base-r representation

3. If y is not zero, set i = i-1 repeat steps 1 and 2 with 0.y

Example: Convert 0.62510 to equivalent binary representation.

Binary to Octal Conversion

Binary radix = 2 Octal radix = 8 = 23

Direction conversion is feasible. Starting with the binary point,

(i) group 3 bits at a time, and use their decimal value as an octal digit, and

(ii) pad the left of integer part and the right of fractional part with 0s as needed to makes groups of 3 bits

Octal 0 1 2 3 4 5 6 7

Binary 000 001 010 011 100 101 110 111

Octal to Binary Conversion: Expand each octal digit into the equivalent binary code.

Verify:

A byte (8-bit number) is given by two hex digits.

Decimal

Value

Binary

Code

Hex

Code

0 0 0 0 0 0

1 0 0 0 1 1

2 0 0 1 0 2

3 0 0 1 1 3

4 0 1 0 0 4

5 0 1 0 1 5

6 0 1 1 0 6

7 0 1 1 1 7

8 1 0 0 0 8

9 1 0 0 1 9

10 1 0 1 0 A

11 1 0 1 1 B

12 1 1 0 0 C

13 1 1 0 1 D

14 1 1 1 0 E

15 1 1 1 1 F

Memorize this table

Binary to Hexadecimal Conversion

Binary radix = 2 Hexadecimal radix = 16 = 24

· Hexadecimal system uses 16=24 as the radix.

· Group 4 bits of binary number to obtain the corresponding hexadecimal digit. See the table below.

· Pad the left of integer part and the right of fractional part as needed to make groups of 4 bits.

Example:

1 0 0 0 1 1 1 0 0 . 0 1 0 0 0 12 =

First hex digit

2nd

hex

digit

Bits can be used to represent everything

7-bit ASCII codes to represent characters in a computer

Prefixes

Source:http://ascii-table.com

Source: Slides for the Computer Organization course, Fall 07, from UC Berkely.

Decimal

Value

Binary

Code

Hex

Code

0 0 0 0 0 0

1 0 0 0 1 1

2 0 0 1 0 2

3 0 0 1 1 3

4 0 1 0 0 4

5 0 1 0 1 5

6 0 1 1 0 6

7 0 1 1 1 7

8 1 0 0 0 8

9 1 0 0 1 9

10 1 0 1 0 A

11 1 0 1 1 B

12 1 1 0 0 C

13 1 1 0 1 D

14 1 1 1 0 E

15 1 1 1 1 F

Binary-coded decimal (BCD) system

See the book for algorithms on adding, multiplying, modular arithmetic of binary integers.

Section 4.2, # 31.

Show that a positive integer is divisible by 3 if and only if the sum of its

decimal digits is divisible by 3.

1/30/12

Section 4.1, # 40

Also, try #38, section 4.1

Extra: not covered in class