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 | 2^{2} |
2^{1} |
2^{0} |

Data Bits | 0 |
0 |
1 |

Value | 0 |
0 |
1 |

Total value = ( 0 × 2^{2} + 0 × 2^{1} + 1 ×
2^{0})

= 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:**

- 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. - 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 | 2^{3} |
2^{2} |
2^{1} |
2^{0} |

Data Bits | 0 |
1 |
1 |
1 |

Value | 0 |
4 |
2 |
1 |

Total value = ( 0 × 2^{3} + 1 × 2^{2} + 1 ×
2^{1} + 1 × 2^{0})

= 4 + 2 + 1 **= 7**

**Magnitude of 1001 is 7 and the Sign is
Negative**

**Hence ( 1001 ) _{2} = ( -7
)_{10}**

If You Have Any Doubts. Please Ask Using Comments.

_{Have A Great Day!}

Part a)
For an unsigned number that represent binary with n bits, what
is the range of value for that?
Part b)
How many rows appear in a truth table with n input
variables?
Part c)
How many memory location are there if the memory address is n
bits?

Sign extension means that any given signed number can be
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numbers as well. For example, 1011 = -5 and 11011 = -5. Also,
11111011 = -5. Using finite summation notation, prove that sign
extension can be applied to any 2's complement binary integer, B,
with b number of bits

Find the 2's complement of the following decimal numbers (You
must convert the number to binary and the answer has to be in
binary with the same number of bits as the conversion)?
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b. 15
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