Number System

Question

A signed number in $2′s$ compliment notation with $16−$bit register $P=DFA0$

What is the value of $P∗4$?

I have tried $P∗4=0111111010000000$ but sign bit changed from $1$ to $0$. What is wrong with my solution?

Comments

x=DFA0=1101 1111 1010 0000

y=4=0100

x*y=110111111010000000 (18-bit)

but, since it is 16-bit register so, 0111111010000000 will be stored.

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but sign bit changed.

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$P = DFA0 = 1101 \ 1111 \ 1010 \ 0000$

$4 * P =0011 \ 0111 \ 1110 \ 1000 \ 0000 = 37E80$

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Use 2's complement sign extension

Answer 1

See P= DFA0 = 1101 1111 1010 0000 = -8288(Decimal equivalent)

P*4 = -8288*4= -33152

but as you know that 2's complement number have range between $-2^{n-1}$ to $2^{n-1} - 1$

so in 16 bit you can represent last negative number as $-2^{16-1}$ = $-2^{15}$ = -32768 to 32767

and multiplication is going out of the range so it can't be represent in 16 bit register. to get the result you need to store in 32 bit register and answer will be

11111111111111110111111010000000

Comments

can we do left shift?

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@Vishnathan, yes multiplication in 2's complement can be done by left shift but you should be take care of that, Is result coming in given range or exceeding?

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i think it is a case of overflow because input is negative number and output is positive number.correct me if my interpretation is wrong.