Convert 1001 0001 1110 0001 0001 0111 0000 0111 Signed Binary in Two's (2's) Complement Representation on 32 Bit to Decimal

How to convert 1001 0001 1110 0001 0001 0111 0000 0111(2), signed binary in two's (2's) complement representation, to decimal

What are the steps to convert the signed binary in two's (2's) complement representation to an integer in decimal system (in base ten)?

1. Is this a positive or a negative number?

1001 0001 1110 0001 0001 0111 0000 0111 is the binary representation of a negative integer, on 32 bits (4 Bytes).


  • In a signed binary in two's complement representation, the first bit (the leftmost) indicates the sign, 1 = negative, 0 = positive.

2. Get the binary representation in one's complement.

* Run this step only if the number is negative

  • Note on binary subtraction rules:
  • 11 - 1 = 10; 10 - 1 = 01; 1 - 0 = 1; 1 - 1 = 0.

  • Subtract 1 from the initial binary number.

  • 1001 0001 1110 0001 0001 0111 0000 0111 - 1 = 1001 0001 1110 0001 0001 0111 0000 0110


3. Get the binary representation of the positive (unsigned) number.

* Run this step only if the number is negative

Flip all the bits of the signed binary in one's complement representation (reverse the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:

!(1001 0001 1110 0001 0001 0111 0000 0110) = 0110 1110 0001 1110 1110 1000 1111 1001


4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:

  • 231

    0

  • 230

    1

  • 229

    1

  • 228

    0

  • 227

    1

  • 226

    1

  • 225

    1

  • 224

    0

  • 223

    0

  • 222

    0

  • 221

    0

  • 220

    1

  • 219

    1

  • 218

    1

  • 217

    1

  • 216

    0

  • 215

    1

  • 214

    1

  • 213

    1

  • 212

    0

  • 211

    1

  • 210

    0

  • 29

    0

  • 28

    0

  • 27

    1

  • 26

    1

  • 25

    1

  • 24

    1

  • 23

    1

  • 22

    0

  • 21

    0

  • 20

    1

5. Multiply each bit by its corresponding power of 2 and add all the terms up.

0110 1110 0001 1110 1110 1000 1111 1001(2) =

(0 × 231 + 1 × 230 + 1 × 229 + 0 × 228 + 1 × 227 + 1 × 226 + 1 × 225 + 0 × 224 + 0 × 223 + 0 × 222 + 0 × 221 + 1 × 220 + 1 × 219 + 1 × 218 + 1 × 217 + 0 × 216 + 1 × 215 + 1 × 214 + 1 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20)(10) =

(0 + 1 073 741 824 + 536 870 912 + 0 + 134 217 728 + 67 108 864 + 33 554 432 + 0 + 0 + 0 + 0 + 1 048 576 + 524 288 + 262 144 + 131 072 + 0 + 32 768 + 16 384 + 8 192 + 0 + 2 048 + 0 + 0 + 0 + 128 + 64 + 32 + 16 + 8 + 0 + 0 + 1)(10) =

(1 073 741 824 + 536 870 912 + 134 217 728 + 67 108 864 + 33 554 432 + 1 048 576 + 524 288 + 262 144 + 131 072 + 32 768 + 16 384 + 8 192 + 2 048 + 128 + 64 + 32 + 16 + 8 + 1)(10) =

1 847 519 481(10)

6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:

1001 0001 1110 0001 0001 0111 0000 0111(2) = -1 847 519 481(10)

The number 1001 0001 1110 0001 0001 0111 0000 0111(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
1001 0001 1110 0001 0001 0111 0000 0111(2) = -1 847 519 481(10)

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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How to convert signed binary numbers in two's complement representation from binary system to decimal

To understand how to convert a signed binary number in two's complement representation from the binary system to decimal (base ten), the easiest way is to do it by an example - convert binary, 1101 1110, to base ten:

  • In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive. The first bit is 1, so our number is negative.
  • Get the signed binary representation in one's complement, subtract 1 from the initial number:
    1101 1110 - 1 = 1101 1101
  • Get the binary representation of the positive number, flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:
    !(1101 1101) = 0010 0010
  • Write bellow the positive binary number representation in base two, and above each bit that makes up the binary number write the corresponding power of 2 (numeral base) that its place value represents, starting with zero, from the right of the number (rightmost bit), walking to the left of the number, increasing each corresonding power of 2 by exactly one unit:
  • powers of 2: 7 6 5 4 3 2 1 0
    digits: 0 0 1 0 0 0 1 0
  • Build the representation of the positive number in base 10, by taking each digit of the binary number, multiplying it by the corresponding power of 2 and then adding all the terms up:

    0010 0010(2) =

    (0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =

    (0 + 0 + 32 + 0 + 0 + 0 + 2 + 0)(10) =

    (32 + 2)(10) =

    34(10)

  • Signed binary number in two's complement representation, 1101 1110 = -34(10), a signed negative integer in base 10.