GATE CSE 1999 | Question: 2.17

Question

Zero has two representations in

• A. Sign-magnitude
• B. $2's$ complement
• C. $1's$ complement
• D. None of the above

Comments

SM and 1's complement only

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Range of Binary Numbers in ALL Representations: https://youtu.be/JxK_KfSa4GY

Answer 1

A and C.

Sign Magnitude

• $+0 = 0000$
• $-0 = 1000$

$1$'s complement

• $+0 = 0000$
• $-0 = 1111$

Comments

although we know the answer one thing must be noted here:-

any odd's complement like 1's, 7's, 9's etc has two representations of zero and also sign magnitude

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Any references @prateek?

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@Prateek kumar

I don’t think this is true. Complement of a number is relative to the base being used. 2’s complement of for a number in base 2 is different from 2’s complement of a number in base 3.

For eg:- 2’s complement of $010100_{2} = 101100_{2}$

whereas 2’s complement of $010100_{3} = 212122_{3} $

Clearly 2’s complement for base 3 will have +0 and -0 but for base 2 there will only one representation.

The point is for a number in base n, (n-1)’s complement will have two representations for 0 (+0 and -0) but n’s complement will have 1 representation only.

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any odd's complement like 1's, 7's, 9's etc has two representations of zero and also sign magnitude
ref :- https://g.co/gemini/share/7d222d2e6809

Answer 2

Detailed Video Explanation of Range of Binary Numbers in ALL Representations: https://youtu.be/JxK_KfSa4GY

$\color{red}{\text{Range of Binary Numbers in Various Representations}}$, using $n-bits$:

1. Unsigned Binary Numbers:

• Min: $000 \dots 0$ = $(0)_{10}$
• Max: $111 \dots 1$ = $(2^n - 1)_{10}$
• Range of integer values: $0$ to $2^n -1$
• The number of distinct integers that can be represented is: $2^n$
• Every number that can be represented, has a unique representation.

2. Sign-Magnitude Representation of Signed Binary Numbers:

• Min: $111 \dots 1$ = $-(2^{n-1} -1)_{10}$
• Max: $011 \dots 1$ = $(2^{n-1} - 1)_{10}$
• Range of integer values: $-(2^{n-1} -1)$ to $(2^{n-1} -1)$
• The number of distinct integers that can be represented is: $2^n -1$
• $0$ has two different representations in Sign-Magnitude Representation: $000 \dots 0$ & $100 \dots 0$
• Every number except $0$, has a unique representation.

3. 1's Complement Representation of Signed Binary Numbers:

• Min: $100 \dots 0$ = $-(2^{n-1} -1)_{10}$
• Max: $011 \dots 1$ = $(2^{n-1} - 1)_{10}$
• Range of integer values: $-(2^{n-1} -1)$ to $(2^{n-1} -1)$
• The number of distinct integers that can be represented is: $2^n -1$
• $0$ has two different representations in Sign-Magnitude Representation: $000 \dots 0$ & $111 \dots 1$
• Every number except $0$, has a unique representation.

4. 2's Complement Representation of Signed Binary Numbers:

• Min: $100 \dots 0$ = $-(2^{n-1})_{10}$
• Max: $011 \dots 1$ = $(2^{n-1} - 1)_{10}$
• Range of integer values: $-(2^{n-1})$ to $(2^{n-1} -1)$
• The number of distinct integers that can be represented is: $2^n$
• Every number that can be represented, has a unique representation.

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Detailed Video Explanation of Range of Binary Numbers in ALL Representations: https://youtu.be/JxK_KfSa4GY

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Related Important Lectures:

Sign Extension Concept in ALL Representations: https://youtu.be/dkoaDtyiz9k

Answer 3

-0 and 0 in 1's complement that is why the range is -2^n-1 to 2^n-1

Comments

sign magnitude ..has also two representation  for zero 

+0 = 0000
-0 = 1000

Answer 4

In sign magnitude and 1s complement we have 2 representaion of 2 therefore we prefer 2s complement generally

A and C is correct