decimal number system

©zaher elsir Sudan Academy for Banking & Financial Sciences

Decimal Number SystemBase (Radix) 10Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9e.g. 747510

The magnitude represented by a digit is decided by the position of the digit within the number.

For example the digit 7 in the left-most position of 7475 counts for 7000 and the digit 7 in the second position from the right counts for 70.

7

1000 100

4 7 5

110


Binary Number SystemBase (Radix) 2Digits 0, 1e.g. 11102

The digit 1 in the third position from the right represents the value 4 and the digit 1 in the fourth position from the right represents the value 8.

1

8=23

1 1 0

4=22 2=21 1=20


Octal Number SystemBase (Radix) 8Digits 0, 1, 2, 3, 4, 5, 6, 7e.g. 16238

The digit 2 in the second position from the right represents the value 16 and the digit 1 in the fourth position from the right represents the value 512.

1

512=83

6

64=82

2

8=81

3

1=80


Hexadecimal Number SystemBase (Radix) 16

Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

e.g. 2F4D 16

The digit F in the third position from the right represents the value 3840 and the digit D in the first position from the right represents the value 1.

2

4096=163

F

256=162

4

16=161

D

1=160


Binary Arithmetic

• Addition

•Complements

•Subtraction


Binary Addition 0+ 0 0

0+ 1 1

1+ 0 1

1+ 11 0

Carry Bit

(a) (b)

(c) (d)


Binary Addition Example 1

1 1 0 1 1 1

+ 0 1 1 1 0 0

–1

–1

–1–1–1

–0–1–0–0 –1–1

–Example 1: Add

–binary 110111 to 11100

– Col 1) Add 1 + 0 = 1– Write 1

– Col 2) Add 1 + 0 = 1– Write 1

– Col 3) Add 1 + 1 = 2 (10 in binary)– Write 0, carry 1

– Col 4) Add 1+ 0 + 1 = 2– Write 0, carry 1

– Col 6) Add 1 + 1 + 0 = 2– Write 0, carry 1

– Col 5) Add 1 + 1 + 1 = 3 (11 in binary)– Write 1, carry 1

– Col 7) Bring down the carried 1– Write 1


Binary Addition VerificationVerification

1101112 5510

+0111002 + 2810

8310

64 32 16 8 4 2 1

1 0 1 0 0 1 1

= 64 + 16 + 2 +1

= 8310

1 1 0 1 1 1

+ 0 1 1 1 0 0

–1–0–1–0–0 –1–1

–You can always check your –answer by converting the figures to decimal, doing the addition, and comparing the answers.


Binary Addition Examples

10011001+ 101100 11000101

101 + 1001 1110

1011 + 101 10000

1010 + 100 1110

1011 + 1100 10111

(a) (b) (c)

(d) (e)


Binary Substraction 0

- 0 0

0- 1 1

1- 0 1

1- 1 0

Borrow Bit

(a) (b)

(c) (d)


Binary Subtraction Example 1

1 1 0 0 1 1

- 1 1 1 0 0

–Example 1: Subtract–binary 11100 from 110011

–2–0–0–2

–1–2

–1–1–0–1

Col 1) Subtract 1 – 0 = 1

Col 5) Try to subtract 0 – 1 can’t.

Must borrow from next column.


Col 3) Try to subtract 0 – 1 can’t.

Must borrow 2 from next column.

But next column is 0, so must go to

column after next to borrow. Add the borrowed 2 to the 0 on the right.

Now you can borrow from this column

(leaving 1 remaining).


Add the borrowed 2 to the original 0.

Then subtract 2 – 1 = 1

–1 Add the borrowed 2 to the remaining 0.

Then subtract 2 – 1 = 1Col 6) Remaining leading 0 can be ignored.


Binary Subtraction Verification

Verification

1100112 5110

- 111002 - 2810

2310

64 32 16 8 4 2 1

1 0 1 1 1

= 16 + 4 + 2 + 1

= 2310

1 1 0 0 1 1

- 1 1 1 0 0

–2–0–0–2

–1–2

–1–1–0–1 –1

–Subtract binary –11100 from 110011:


Binary SubtractionExample 2

1 0 1 0 0 1

- 1 0 1 0 0

–Example 2: Subtract–binary 10100 from 101001

–2–0 –0–2

–1–1–0–1 –0

Verification

1010012 4110

- 101002 - 2010

2110

64 32 16 8 4 2 1

1 0 1 0 1

= 16 + 4 + 1

= 2110


Example

1 1 0 0 1 0 1 1 0

0 0 1 1 0 1 0 0 1

Binary Complement

(1s Complement) Operation

1 00 1


Two’s ComplementThe Two’s complement of a binary number is obtained by first complementing the number and then adding 1 to the result.1001110

0110001+ 1

0110010

One’s Complement

Two’s Complement


Binary SubtractionBinary subtraction is implemented by adding the Two’s complement of the number to be subtracted.

Example

1101 1101 -1001 +0111

10100

If there is a carry then it is ignored. Thus, the answer is 0100.

Two’s complement of 1001


Binary Codes

• BCD – Binary Coded Decimal

• ASCII – American Standard Code for Information Interchange

A binary code is a group of n bits that assume up to 2n distinct combinations of 1’s and 0’s with each combination representing one element of the set that is being coded.


BCD – Binary Coded Decimal Decimal BCD

Number Number

0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001

When the decimal numbers are represented in BCD, each decimal digit is represented by the equivalent BCD code.

Example :BCD Representation of Decimal 6349

6 3 4 9

0110 0011 0100 1001


ASCII

001100001

01100012

01100103

01100114

01101005

01101016

01101107

01101118

01110009

0111001

Number ASCII Letter ASCII

A1000001B

1000010C

1000011D

1000100E

1000101F

1000110G

1000111H

1001000I

1001001


ASCII Continued.

J1001010K

1001011L

1001100M

1001101N

1001110O

1001111P

1010000Q

1010001R

1010010

Letter ASCII Letter ASCIIS

1010011T

1010100U

1010101V

1010110W

1010111X

1011000Y

1011001Z

1011010


Logic Gates• Binary information is represented in

digital computers by physical quantities called signals.

• Two different electrical voltage levels such as 3 volts and 0.5 volts may be used to represent binary 1 and 0.

• Binary logic deals with binary variables and with operations that assume a logical meaning.


Logic Gates Contd…

• A particular logic operation can be described in an algebraic or tabular form.

• The manipulation of binary information is done by the circuits called logic gates which are blocks of hardware that produce signals of binary 1 or 0 when input logic requirements are satisfied.


Logic Gates Contd…

• Each gate has a distinct graphics symbol and it’s operation can be described by means of an algebraic expression or in a form of a table called the truth table.

• Each gate has one or more binary inputs and one binary output.

–© zaher elsir –Sudan Academy for Banking & Financial Sciences

Logic Gates ANDOR NOT (Inverter)NAND (Not AND)NOR (Not OR)XOR (Exclusive-OR)Exclusive-NOR


Logic Gates Cont.

AND Logic Gate Truth Table

A

Bx

A B x0 0 00 1 01 0 01 1 1A, B Binary Input Variables

x Binary Output Variable

x = A . B


Logic Gates Cont. OR Logic Gate Truth Table

A B x0 0 00 1 11 0 11 1 1

A

Bx

x = A + B

This is read as x equals A or B.


Logic Gates Cont.

NOTLogic Gate Truth Table

A x 0 1 1 0

xA

x = A


Logic Gates Cont. NAND Logic Gate Truth

Table

A B x0 0 10 1 11 0 11 1 0

A

Bx

x = A . B


Logic Gates Cont. NORLogic Gate Truth

TableA B x0 0 10 1 01 0 01 1 0

A

Bx

x = A + B


Logic Gates Cont. XORLogic Gate Truth

TableA B x0 0 00 1 11 0 11 1 0

A

Bx

x = A + B


Logic Gates Cont.

Exclusive-NOR Logic Gate Truth Table

A B x0 0 10 1 01 0 01 1 1

A

Bx

x = A + B

decimal number system

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