Question

If the word length is 4 binary bits (including sign), what decimal
number does 1001 represent in
a. in sign and magnitude?
b. in 2’s complement?

Answer 1

Sign and Magnitude Representation:

It is most basic method of representing both Positive and Negative Number in Binary Format. In this we represent the differentiate Positive and Negative Numbers placing a Sign bit before the data bits.

• If the MSB(Left Most Bit) is 0 then the number is Positive
• If the MSB(Left Most Bit) is 1 then the number is Negative.

And the Magnitude of the Number is determined by converting the remaining bits to Decimal representation using Positional Weight Method

A) The given word is 1001 and length is 4 binary bits (including sign),

The MSB bit of 1 001 is 1 .

Therefore It is a Negative Number.

Now finding the magnitude of the data bits (001) using Positional Weight Method

Positional Weights Method: (exclude the MSB because it is sign bit)

Positional Weight	22	21	20
Data Bits	0	0	1
Value	0	0	1

Total value = ( 0 × 2² + 0 × 2¹ + 1 × 2⁰)

= 1

Magnitude of 001 is 1 and sign is negative

Therefore 1001 represent the decimal number - 1

Hence ( 1001 )₂ = ( -1 )₁₀

2's Complement Forms: It is the enhanced form of representing the Negative Numbers in the binary format. which rectify the drawback of the sign Magnitude form.

Converting a 2's Complement number into Decimal Format:

1. Observe the Sign Bit or MSB bit if it is 0 then the number is Positive .So directly convert the binary number into decimal using Positional Weight Method.
2. If the sign Bit or MSB bit is 1 then it Negative number .So find 1's complement form(just alter 0's with 1's vice versa) of the given number and add 1 to result. Then convert the final result into Decimal using Positional Weight Method.

B) The given word is 1001 and length is 4 binary bits (including sign),

The MSB bit of 1 001 is 1 .

Therefore It is a Negative Number.

2' Complement	1	0	0	1
1' Complement	0	1	1	0
Adding 1	+			1
Final Result	0	1	1	1

Now convert the final result into Decimal using positional weight Method.

Positional Weights Method: (do not exclude the MSB because it is sign bit)

Positional Weight	23	22	21	20
Data Bits	0	1	1	1
Value	0	4	2	1

Total value = ( 0 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰)

= 4 + 2 + 1 = 7

Magnitude of 1001 is 7 and the Sign is Negative

Hence ( 1001 )₂ = ( -7 )₁₀

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