comp 411-3, numbr systems
DESCRIPTION
Number SystemsTRANSCRIPT
NUMBER SYSTEMS
Number Systems
1. Binary Numbers
2. Hexadecimal Numbers
3. Octal Numbers
4. Binary and Hexadecimal Arithmetic
Neil Abalajon
Table of EquivalentsDecimal Binary Hexadecimal Octal
0 0000 0 0
1 0001 1 1
2 0010 2 2
3 0011 3 3
4 0100 4 4
5 0101 5 5
6 0110 6 6
7 0111 7 7
8 1000 8 10
9 1001 9 11
10 1010 A 12
11 1011 B 13
12 1100 C 14
13 1101 D 15
14 1110 E 16
15 1111 F 17
Neil Abalajon
1. Binary Numbers
- a number that has a base of 2.
- the coefficients of the binary number system have only two possible values: 0 and 1.
Representation:
1 1 0 1 1
LSB – Least Significant Bit
MSB – Most Significant Bit
Neil Abalajon
1.1 Decimal-Binary ConversionSteps:
a) Divide the no. by 2’s.
b) If the result has a remainder, indicate a “1” , otherwise, “0” in the right hand portion.
c) Divide the resultant whole no. by 2’s.
d) Repeat steps ( b ) & ( c ) until no. that is being divided by 2 is 1 .
* Binary result is derived by reading from the bottom.
Note: The remainder after each division is used to indicate the coefficient of the Binary no. to be formed.
Neil Abalajon
1.1 Decimal-Binary Conversion
1810 = ?2
182
9 02
4 12
2 02
1 0
1710 = 100102
Remainders
Neil Abalajon
1.2 Binary-Decimal Conversion
Example:
a) 1 1 0 1 2 =
b) 1 1 1 1 2 =
13 10
15 10
Neil Abalajon
1.2 Binary-Decimal Conversion
Solved Example:
a) 1 1 0 1 2 = 13 10
11012 = 1(23)+1(22)+0(21)+1(20)
= 810+ 410+ 010 + 110
= 1310
Neil Abalajon
1.2 Binary-Decimal Conversion
x…11012 = x(2n)…+1(23)+1(22)+0(21)+1(20)
Conversion Formula:
Where n is the number total number of 1’s and 0’s in the binary number.
n Radix is 2
Neil Abalajon
2. Hexadecimal Number System A no. system that has a base, or radix of 16.
16 diff. Symbols are used to represent nos.
The first ten digits (0 to 9) are borrowed from the Decimal no. system
the letters A, B, C, D, E & F are used for digits 10, 11, 12, 13, 14 and 15, respectively.
Example form:
3BAF 16 DEF54 16
Neil Abalajon
2.1 Decimal-Hexadecimal Conversion
Steps:
same procedure in DEC-BIN conversion, except that the number is divided by 16
digit in the remainder is expressed in HEX.
Example:
a) 245 10 =
b) 61 10 =
F516
3D16
Neil Abalajon
2.2 Hexadecimal-Decimal Conversion
Steps:
a) Convert a single HEX no. to its equivalent decimal form. (See Table of Equivalents)
b) Follow the same procedure(formula) in converting BIN-DEC, but change the base number from 2 to 16 and represent each HEX digit with its DEC equivalent
B9F16 = 11(162) + 9(161) + 15(160)
= 81610 + 14410 + 1510
= 297510Neil Abalajon
HEX-DEC Examples:
a) ABC 16 =
b) AB6 16 =
c) 3A6 16 =
d) B9F 16 =
2748 10
2742 10
934 10
2975 10
Neil Abalajon
2.3 Binary-Hexadecimal Conversion
Steps:
a) Break the binary no. into groups of four digits.
b) Convert each group of four digits according to the its corresponding HEX symbol.
c) Read each set of Binary digits starting from the right (LSB).
Example:a) 1 0 1 1 1 0 1 1 2 =
b) 1 0 0 1 0 1 0 1 2 =
c) 1 0 1 1 0 0 0 1 1 0 1 0 1 1 2 =
BB 16
95 16
2C6B 16
Neil Abalajon
3. Octal Number System
A number system that has a base, or radix of 8
Eight diff. Symbols are used to represent numbers. These are 0, 1, 2, 3, 4, 5, 6 and 7.
Example form: 17 8
57 8
Neil Abalajon
3.1 Decimal-Octal Conversion
Steps:
same procedure in DEC-BIN conversion
Decimal no. is divided by 8
Remainder is placed at the right
Division stops if the quotient is less than 8
Example:
a) 153 10 =
b) 82 10 =
231 8
122 8
Neil Abalajon
3.2 Octal-Decimal Conversion
Steps:
same steps used in BIN-DEC, except that 8 is being used instead of 2 as a radix or base.
Example:
a) 17 8 =
b) 1213 8 =
15 10
651 10
Neil Abalajon
3.3 Binary-Octal Conversion
Steps:
a) Group the binary digits into groups of 3.
b) Read each set of binary digits starting from the right (LSB).
Example:
a) 1 1 1 1 1 0 1 1 1 2 =
b) 1 1 0 0 1 1 2 =
c) 1 1 0 1 1 2 =
767 8
63 8
33 8Neil Abalajon
4. Binary and Hexadecimal Arithmetic
Arithmetic between binary and hex numbers can be done by converting them into decimal first, and convert the result back to binary or in hexadecimal as the case may be.
There will be times, however, that such procedure would be time consuming.
It would be faster to operate in binary or hexadecimal directly.
Neil Abalajon
Binary Addition
is performed in the same manner as decimal addition.
• Binary Addition Table:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 Plus a carry over of 1 to the next digit
Neil Abalajon
Binary Subtraction
inverse operation of addition.
• Binary Subtraction Table:
0 – 0 = 0
1 – 0 = 1
1 – 1 = 0
0 – 1 = 1 W/ a borrow of 1 from the next column to the left
Neil Abalajon
2 Cases of Subtraction:
Case 1:
When a small number is being subtracted from a larger number.
Case 2:
When a large number is being subtracted from a smaller number resulting to a negative result (complement of a (-) no.)
Neil Abalajon
Binary Multiplication
is performed in the same manner as in decimal multiplication.
• Binary Multiplication Table:
0 X 0 = 0
1 X 0 = 0
0 X 1 = 0
1 X 1 = 1
Neil Abalajon
Hexadecimal Addition
Rules:
1) Add the first column followed by the second column. From right to left.
2) If the sum of the two digits on the same column is 1510 or less, bring down the corresponding Hex digit.
3) If the sum of is greater than 1510, bring down the amount of the sum that exceeds 1610 & carry a “1” to the next column.
Neil Abalajon
Hexadecimal Subtraction (rules when borrowing)
Rules:
1) Subtract the subtrahend digit from F, then add 1.
2) Add the result to the minuend.
Note: Always add 1 for each column that borrowed from the next column to the left.
Neil Abalajon
BCD Format
Neil Abalajon
Binary Coded Decimal (BCD)
- means that each decimal digit is represented by a binary code of four-bits.
Example:
Convert each of the ff. decimal nos. into their BCD format:
3 9 18 65 321
Neil Abalajon
Solution:
3
0011
9
1001
18
0001 1000
65
0110 0101
321
0011 0010 0001
Neil Abalajon
BCD Format Application
One of the common uses of the BCD format is in 7-segment BCD displays.
0 0 0 00 1 0 10 0 1 00 1 1 1BCD =
Neil Abalajon
BCD Addition
Rules:
1) Add the two numbers, using the rules of binary addition.
2) If a four-bit sum is equal to or less than 9, it is a valid BCD number.
3) If a four-bit sum is greater than 9, or if a carry is generated, it is an invalid result. Add 6 (0110 2) to the 4-bit sum (excess only). If a carry results when 6 is added, simply add the carry to the next 4-bit group.
Neil Abalajon
BCD SubtractionRules:
1) Take the 2’s Complement of the subtrahend and add it to the minuend and disregard the carry in the MSB of the result
2) If the four-bit difference is less than or equal to 9 it is a valid BCD number
3) If a four-bit difference is greater than 9 (in Hex), it is an invalid result. Subtract 6 (0110 2) from each 4-bit group that is in excess of 9 (10012)
Neil Abalajon
Determining whether a number is Positive or
Negative
Neil Abalajon
Positive and Negative Hexadecimal Numbers
8000h is negative because the MSB is “1”.
100h is positive because the MSB is “0”.
7FFFh is
0FFFFh is
0FFFh is
positive
negative
positive
Neil Abalajon
- End of Presentation -
Neil Abalajon