Number Systems

Question 1
P, Q, and R are the decimal integers corresponding to the 4-bit binary number 1100 considered in signed magnitude, 1's complement, and 2's complement representations, respectively. The 6-bit 2's complement representation of (P+Q+R) is
A
110101
B
110010
C
111101
D
111001
GATE EC 2020   Digital Circuits
Question 1 Explanation: 
Given, binary number 1100
1's complement of 1100 = -3
Sign magnitude of 1100 = -4
2's complement of 1100 = -4
P + Q + R = -4 - 3 - 4 = -11
The 6 digit 2's complement of (-11) = 110101
Question 2
The number of bytes required to represent the decimal number 1856357 in packed BCD (Binary Coded Decimal) form is_______.
A
2
B
3
C
4
D
5
GATE EC 2014-SET-2   Digital Circuits
Question 2 Explanation: 
To represent decimal number into BCD number each decimal number is represented in 4 -bits while converting in BCD numbers, as
\begin{array}{ll}1 \rightarrow 0001 & 6 \rightarrow 0110 \\ 8 \rightarrow 1000 & 3 \rightarrow 0011 \\ 5 \rightarrow 0101 & 5 \rightarrow 0101 \\ & 7 \rightarrow 0111\end{array}


So total 4 bytes are required.
Question 3
The two numbers represented in signed 2's complement form are P = 11101101 and Q = 11100110. If Q is subtracted from P, the value obtained in signed 2's complement is
A
1000001111
B
00000111
C
11111001
D
111111001
GATE EC 2008   Digital Circuits
Question 3 Explanation: 
\because Signed 2 's complement of
\begin{aligned} P&=11101101\\ \therefore \text{No.} \quad P&=00010011 \end{aligned}
\because Signed 2 's complement of
\begin{aligned} Q &=11100110 \\ P-Q=P+(2 ' s& \text { complement of }Q) \\ &=00010011 \\ &\frac{+11100110}{11111001} \\ \text{2's complement of }\\ (P-Q)&=00000111 \end{aligned}
Question 4
X = 01110 and Y =11001 are two 5-bit binary numbers represented in two's complement format. The sum of X and Y represented in two's complement format using 6 bits is
A
100111
B
001000
C
000111
D
101001
GATE EC 2007   Digital Circuits
Question 4 Explanation: 
\begin{array}{cccccccc} x &=&&0&1&1&1&0 \\ Y &=&&1&1&0&0&1 \\ x+y &=&1&0&0&1&1&1 \end{array}
Carry is discarded in the addition of numbers represented in 2's complement form. X + Y in 6 bits is 000111.
Question 5
A new Binary Coded Pentary (BCP) number system is proposed in which every digit of a base-5 number is represented by its corresponding 3-bit binary code. For example, the base-5 number 24 will be represented by its BCP code 010100. In this numbering system, the BCP code 10001001101 corresponds of the following number is base-5 system
A
423
B
1324
C
2201
D
4321
GATE EC 2006   Digital Circuits
Question 5 Explanation: 
100010011001 \rightarrow 4231
Question 6
Decimal 43 in Hexadecimal and BCD number system is respectively
A
B2, 0100 0011
B
2B, 0100 0011
C
2B, 0011 0100
D
B2, 0100 0100
GATE EC 2005   Digital Circuits
Question 6 Explanation: 
(43)_{10}\rightarrow \begin{array}{c|c|c}16&43&\\\hline16&2&B\\\hline&0&2\\\hline\end{array}
\therefore (2B)_{H} (43)_{10}=(01000011)_{BCD}
Question 7
11001, 1001, 111001 correspond to the 2's complement representation of which one of the following sets of number
A
25, 9, and 57 respectively
B
-6, -6, and -6 respectively
C
-7, -7 and -7 respectively
D
-25, -9 and -57 respectively
GATE EC 2004   Digital Circuits
Question 7 Explanation: 
11001 \rightarrow 00111(+7)
1001 \rightarrow 0111(+7)
111001 \rightarrow 000111(+7)
\therefore Numbers given in question in 2 's complement correspond to -7
Question 8
The range of signed decimal numbers that can be represented by 6-bits 1's complement number is
A
-31 to +31
B
-63 to +63
C
-64 to +63
D
-32 to +31
GATE EC 2004   Digital Circuits
Question 8 Explanation: 
\begin{aligned} \text { Range } &=-\left(2^{n-1}-1\right) \text { to }+\left(2^{n-1}-1\right) \\ &=-\left(2^{6-1}-1\right) \text { to }+\left(2^{6-1}-1\right) \\ &=-31 \text { to }+31 \end{aligned}
Question 9
4-bit 2's complement representation of a decimal number is 1000. The number is
A
8
B
0
C
-7
D
-8
GATE EC 2002   Digital Circuits
Question 9 Explanation: 
1000
MSB is 1 so, -ve number
Take 2's complement for magnitude.
+\begin{array}{cccc} 0&1&1&1\\ &&&1\\ \hline1&0&0&0 \end{array}=8
Question 10
The 2's complement representation of -17 is
A
101110
B
101111
C
111110
D
110001
GATE EC 2001   Digital Circuits
Question 10 Explanation: 
\begin{aligned} 17 &=010001 \\ -17 &=101111(2 \text { 's complement }) \end{aligned}
There are 10 questions to complete.
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