Digital Logic · Unit 1
Number Systems and Conversions
Exam-focused notes for Number Systems and Conversions (Digital Logic, BIT103): what the TU syllabus asks and how it has actually been tested, with 13 solved past questions from this unit.
What this unit covers
- Binary decimal octal hexadecimal conversions
- Fractional number conversions
- Complements of decimal numbers
- Binary arithmetic operations
- Binary subtraction using complements
Binary decimal octal hexadecimal conversions
Convert (257)₈ into hexadecimal and decimal number system. [5]
- Number: $(257)8$ (octal, base 8) - Required: convert to hexadecimal (base 16) and decimal (base 10) --- Multiply each digit by its positional weight (power of 8): $$ (257)8 = 2 \times 8^2 + 5 \times 8^1 + 7 \times 8^0 $$ $$ = 2 \times 64 + 5 \times 8 + 7 ...
Full solved answer →Convert $(591.62)_{10}$ into hexadecimal and octal number system. [5]
- Decimal number: $(591.62){10}$ - Target bases: 16 (hex) and 8 (octal) --- Division Quotient Remainder Digit -------------------------------------- 591 ÷ 16 36 15 F 36 ÷ 16 2 4 4 2 ÷ 16 0 2 2 Reading bottom to top: $(591){10} = (24F){16}$ Multiplication Re...
Full solved answer →Convert $(110.101)_8$ into binary and decimal number system. [5]
Given data: - Number to convert: $(110.101)8$ (octal, base 8) - Target systems: Binary (base 2) and Decimal (base 10) All required data present. --- Each octal digit maps to a 3-bit binary group. Octal Digit Binary :-----------::------: 1 001 1 001 0 000 . ...
Full solved answer →Convert 51966.57 decimal number system into octal number system and hexadecimal number system. [5]
- Decimal number: $51966.57$ - Integer part: $51966$ - Fractional part: $0.57$ - Target bases: Octal (8) and Hexadecimal (16) --- Division Quotient Remainder ------------------------------- $51966 \div 8$ 6495 6 $6495 \div 8$ 811 7 $811 \div 8$ 101 3 $101 \...
Full solved answer →Binary arithmetic operations
Perform following arithmetic operation: a) $101101 + 011011$ b) $101111 - 010101$ [5]
- a) $101101 + 011011$ - b) $101111 - 010101$ --- Rules: $0+0=0$, $0+1=1$, $1+0=1$, $1+1=10$, $1+1+1=11$ Step-by-step (right to left): Position A B Carry In Sum Carry Out ------------------ 1 (LSB) 1 1 0 0 1 2 0 1 1 0 1 3 1 0 1 0 1 4 1 1 1 1 1 5 0 1 1 0 1 6...
Full solved answer →Binary subtraction using complements
Subtract $(739.57)_{10}$ - $(78.35)_2$ using both 10's and 9's complement. [5]
- Minuend: $(739.57){10}$ - Subtrahend: written as $(78.35)2$ Data issue: The value $78.35$ contains the digits $7$ and $8$ and $3$ and $5$, none of which are valid binary digits (base 2 only allows $0$ and $1$). Therefore $(78.35)2$ cannot be a genuine bin...
Full solved answer →Subtract $(111000.110)_2 - (110100.101)_2$ using both 2's and 1's complement. [5]
- Minuend $A = (111000.110)2$ - Subtrahend $B = (110100.101)2$ - Format: 6 integer bits, 3 fractional bits. --- - $A = 111000.1102 = 56 + 0.75 = 56.75$ - $B = 110100.1012 = 52 + 0.625 = 52.625$ - $A - B = 4.125 = (100.001)2 = (000100.001)2$ So the expected ...
Full solved answer →Substract (1011.11- 1010.10) using 2's and l's complement. [5]
- Minuend: $1011.112$ - Subtrahend: $1010.102$ - Operation: $1011.11 - 1010.10$ Decimal check of inputs: - $1011.11 = 8 + 2 + 1 + 0.5 + 0.25 = 11.75$ - $1010.10 = 8 + 2 + 0.5 = 10.50$ - Expected difference $= 11.75 - 10.50 = 1.25$ --- Step 1: 1's complement...
Full solved answer →Perform $A - B$ with the given binary numbers using 1’s complement. $A = 1010100$, $B = 1000100$. [5]
- $A = 1010100$ - $B = 1000100$ - Operation: $A - B$ using 1's complement Decimal check of inputs: - $A = 10101002 = 64+16+4 = 84$ - $B = 10001002 = 64+4 = 68$ Invert every bit of $B$: Bit-by-bit (right to left): - $0+1 = 1$ - $0+1 = 1$ - $1+0 = 1$ - $0+1 =...
Full solved answer →Fractional number conversions
Convert (1011.110) into decimal. and hexadecimal [5]
- Binary number: $(1011.110)2$ - Integer part: $1011$ - Fractional part: $.110$ --- Multiply each bit by its positional weight (power of 2) and sum. $$1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11$$ $$1 \times 2^{-1} + 1 \ti...
Full solved answer →Convert (1011.110) into decimal and hexadecimal. [5]
Given data: - Binary number: $1011.1102$ - Integer part: $1011$ - Fractional part: $110$ Required conversions: to decimal (base 10) and to hexadecimal (base 16). --- Integer part $1011$: $$1\times2^3 + 0\times2^2 + 1\times2^1 + 1\times2^0 = 8 + 0 + 2 + 1 = ...
Full solved answer →Perform the following conversion: (a) $(0.625)_{10}$ to binary. (b) $(173)_8$ to decimal. [2.5+2.5]
- (a) Convert $(0.625){10}$ to binary - (b) Convert $(173)8$ to decimal --- Method: Multiply the fractional part by 2 repeatedly, recording the integer part each time, reading top to bottom. Step Fraction × 2 Result Bit ---------------------------------- 1 ...
Full solved answer →Complements of decimal numbers
a) Obtain the 9's and 10's complement of i) 13579 ii) 90090 decimal number. b) Convert 6524275 octal to hexadecimal [5]
Formulas: - 9's complement: subtract each digit from 9 - 10's complement = 9's complement + 1 9's complement: subtract each digit from 9 $9-1=8,\ 9-3=6,\ 9-5=4,\ 9-7=2,\ 9-9=0$ $$\text{9's complement} = 86420$$ 10's complement: $86420 + 1 = 86421$ 9's compl...
Full solved answer →Make Unit 1 stick
Practice BIT103 with flashcards & quizzes