The Chinese University of Hong Kong

ELE2120 Digital Circuits and Systems

Tutorial Note 1

Prepared by Wilson YU

Outline

1. Convert base-r system to Decimal system1. Convert base-r system to Decimal system

2. Convert Decimal system to base-r system2. Convert Decimal system to base-r system

3. 1s and 2s complement3. 1s and 2s complement

4. Error Detection – Gray code & Parity Codes4. Error Detection – Gray code & Parity Codes

3. Conversion between base-2, 8 and 16 system3. Conversion between base-2, 8 and 16 system

Convert base-r system to Decimal system

A decimal number such as 168.34 represents a quantity equal to 1 hundred, plus 6 tens, plus 8 units, plus 3 one tenths, plus 4 one hundredths.i.e.

In general, a number expressed in a base-r system has coefficients multiplied by powers of r:i.e.

2101210 104103108106101)34.168( −− ×+×+×+×+×=

102

21

1

00

11

22

11

210121

)...

...(

).......(

mm

nn

nn

rmnn

rarara

rarararara

aaaaaaaa

−−

−−

−−

−−

−−−−

×++×+×+

×+×+×++×+×=---(1)

The base of the number

Binary to Decimal Number

ExerciseConvert (1011.1)2 to decimal Number

SolutionStep 1: Find the base of Binary number, r = 2.Step 2: Substitute r = 2 and the coefficient into (1)Step 3: The following equation would be found.

Step 4: (1011.1)2= (11.5)10

1010123

2 )2121212021()1.1011( −×+×+×+×+×=

Hexadecimal to Decimal Number

ExerciseConvert (19.C)16 to decimal Number

SolutionStep 1: Find the base of Binary number, r = 16.Step 2: Substitute r = 16 and the coefficient into (1)Step 3: The following equation would be found.

Step 4: (19.C)16= (25.75)10

10101

16 )1612169161().19( −×+×+×=C

For Hexadecimal Number, A = 10, B = 11, C =12D = 13, E = 14, F = 15

Convert Decimal system to base-r system

ExerciseConvert (25.75)10 to binary number

SolutionStep 1: Convert the integer into binary number.

10011

22222512631

Put the integer here

÷

Remainder of 25÷2

Quotient of 25÷2∴(25)10=(11001)2

Convert Decimal system to base-r system

ExerciseConvert (25.75)10 to binary number

SolutionStep 2: Convert the fraction into binary number.

Step 3: Combine the integer and factional part.(25.75)10 = (11001.11)2

0

0.5Fraction

>>>

>>>

a-2=1

a-1=1Coefficient

+1=0. 5×2

+1=0.75×2Integer

Convert Decimal system to base-r system

ExerciseConvert (25.75)10 to hexadecimal number

SolutionStep 1: Convert the integer into binary number.

91

16251

Put the integer here

÷

Remainder of 25÷2

Quotient of 25÷2∴(25)10=(19)16

Convert Decimal system to base-r system

ExerciseConvert (25.75)10 to hexadecimal number

SolutionStep 2: Convert the fraction into binary number.

Step 3: Combine the integer and factional part.(25.75)10 = (19.C)16

0Fraction

>>> a-1=12(C)Coefficient

+12=0.75×16Integer

∴(0.75)10=(0.C)16

Conversion between base-2, 8 or 16 system

To Convert Binary number to Octal number.E.g. (1001100111.01011)2

Step 1: Divide the number into group with 3 numbers. And add some zeros at both ends if necessary.

(001 001 100 111. 010 110)2

Step 2: Convert each group to octal number.(1147.26)8

To Convert Octal number to Binary numberE.g. (234.54)8

Step 1: Convert each number to 3-digit binary numbers.(010 011 100. 101 100)2

Step 2: Cancel the redundant zeros.(10011100.1011)2

Conversion between base-2, 8 or 16 system

To Convert Binary number to Hexadecimal number.E.g. (1001100111.01011)2

Step 1: Divide the number into group with 4 numbers. And add some zeros at both ends if necessary.

(0010 0110 0111. 0101 1000)2

Step 2: Convert each group to hexadecimal number.(267.58)16

To Convert Hexadecimal number to Binary numberE.g. (1CD.EF)16

Step 1: Convert each number to 4-digit binary numbers.(0001 1100 1101. 1110 1111)2

Step 2: Cancel the redundant zeros.(111001101.11101111)2

1s complement & 2s complement

To do 1s complement for binary numbers, reverse “1” to be “0” and “0” to be “1” for each bit.

Example 1s complement of (1101100)2= (0010011)2

To do 2s complement for binary numbers, do 1s complement and then add 1.

Example2s complement of (1101100)2= (0010011)2 + 1= (0010100)2

2s complement

2s complement could be used to represent signed numbers.

Example-7= 2s complement of 7= 2s complement of (0111)2

= (1000)2 + 1= (1001)2

For 4-bit signed numbers, only -8 <= x <= 7 could be presented.

2s complement

By 2s complement, we could do subtraction by adding the 2s complement of the second operand.

i.e. a – b = a + (-b)Example

15 - 9=15 + (-9)= (01111)2 + (10111)2

= (00110)2

2s complement of 9

01111+10111---------

(1)00110

Carry could be ignored.

2s complement

Detect overflowIf two 2s complement numbers add together, overflow occurs when the sum is out of range.Example: 5+6 = 11 > 7

How do you know whether overflow occur?By observing the solution, we will know that two positive numbers add together will not equal to a negative number. So, overflow occur.

0101+0110

-----------(0)1011Carry is ignored!

2s complement

ExerciseCompute (1000)2 + (1100)2, see whether overflow occurs.

SolutionTwo negative numbers add together will not equal to positive number, so overflow occurs. 1000

+1100---------10100

Error Detection - Gray Code

0000000100110010011001110101010011001101111111101010101110011000

000001011010110111101100

00011110

01

4bit3bit2 bit1bit

2. flip

4. Put “1”s here

3. Put “0”s here

1 . Copy the pattern from the previous bits

Error Detection - Parity Codes (even)

For even parity, put an extra bit called parity bit to make the sum of bits always even.

Detect error if sum ≠even.

0000

1100

0110

1010

Parity bit(even)

Bit 0Bit 1Bit 2

Error Detection - Parity Codes (odd)

For odd parity, put an extra bit called parity bit to make the sum of bits always odd.

Detect error if sum ≠odd.

1000

0100

1110

0010

Parity bit(odd)

Bit 0Bit 1Bit 2

Challenge Exercise

Astronauts on Mars discovered a Martian equation

5x2-50x+125=0

with the given solutions x = 5 and x = 8.Deduce, from this equation, how many fingers do Martian’s have?Hint: they have an odd number of fingers.

Challenge Exercise

Astronauts on Mars discovered a Martian equation5x2-50x+125=0

with the given solutions x = 5 and x = 8.Deduce, from this equation, how many fingers do Martian’s have?Hint: they have an odd number of fingers.

Solution:Step 1: Let the base of the number system be r.Step 2: Convert the base-r number system into decimal number system. And substitute the root into the equation. Then, find the solution of the equation.

i.e.

So, Martian has 13 fingers.1013013023

0)5()2()1()5)(05()5)(5(2

012001200

===+−

=×+×+×+××+×−××

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