Convert 1010 0101 0011 1010 0100 1010 0111 0001 Signed Binary in One's (1's) Complement Representation on 32 Bit to Decimal

How to convert 1010 0101 0011 1010 0100 1010 0111 0001(2), signed binary in one's (1's) complement representation, to decimal

What are the steps to convert the signed binary in one's (1's) complement representation to an integer in decimal system (in base ten)?

1. Is this a positive or a negative number?

1010 0101 0011 1010 0100 1010 0111 0001 is the binary representation of a negative integer, on 32 bits (4 Bytes).


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

2. 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:

!(1010 0101 0011 1010 0100 1010 0111 0001) = 0101 1010 1100 0101 1011 0101 1000 1110


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

  • 231

    0

  • 230

    1

  • 229

    0

  • 228

    1

  • 227

    1

  • 226

    0

  • 225

    1

  • 224

    0

  • 223

    1

  • 222

    1

  • 221

    0

  • 220

    0

  • 219

    0

  • 218

    1

  • 217

    0

  • 216

    1

  • 215

    1

  • 214

    0

  • 213

    1

  • 212

    1

  • 211

    0

  • 210

    1

  • 29

    0

  • 28

    1

  • 27

    1

  • 26

    0

  • 25

    0

  • 24

    0

  • 23

    1

  • 22

    1

  • 21

    1

  • 20

    0

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

0101 1010 1100 0101 1011 0101 1000 1110(2) =

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

(0 + 1 073 741 824 + 0 + 268 435 456 + 134 217 728 + 0 + 33 554 432 + 0 + 8 388 608 + 4 194 304 + 0 + 0 + 0 + 262 144 + 0 + 65 536 + 32 768 + 0 + 8 192 + 4 096 + 0 + 1 024 + 0 + 256 + 128 + 0 + 0 + 0 + 8 + 4 + 2 + 0)(10) =

(1 073 741 824 + 268 435 456 + 134 217 728 + 33 554 432 + 8 388 608 + 4 194 304 + 262 144 + 65 536 + 32 768 + 8 192 + 4 096 + 1 024 + 256 + 128 + 8 + 4 + 2)(10) =

1 522 906 510(10)

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

1010 0101 0011 1010 0100 1010 0111 0001(2) = -1 522 906 510(10)

The number 1010 0101 0011 1010 0100 1010 0111 0001(2), signed binary in one's (1's) complement representation, converted and written as an integer in decimal system (base ten):
1010 0101 0011 1010 0100 1010 0111 0001(2) = -1 522 906 510(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 one's complement representation from binary system to decimal

To understand how to convert a signed binary number in one's complement representation from binary system to decimal (base ten), the easiest way is to do it through an example - convert binary, 1001 1101, to base ten:

  • In a signed binary one's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive. The first bit is 1, so our number is negative.
  • 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:
    !(1001 1101) = 0110 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 by increasing each corresonding power of 2 by exactly one unit:
  • powers of 2: 7 6 5 4 3 2 1 0
    digits: 0 1 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:

    0110 0010(2) =

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

    (0 + 64 + 32 + 0 + 0 + 0 + 2 + 0)(10) =

    (64 + 32 + 2)(10) =

    98(10)

  • Signed binary number in one's complement representation, 1001 1110 = -98(10), a signed negative integer in base 10