Convert 0011 1111 1101 0001 1110 0000 0001 1111 1100 1111 1110 0001 1101 1111 1111 1001 Signed Binary in One's (1's) Complement Representation on 64 Bit to Decimal

How to convert 0011 1111 1101 0001 1110 0000 0001 1111 1100 1111 1110 0001 1101 1111 1111 1001(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?

0011 1111 1101 0001 1110 0000 0001 1111 1100 1111 1110 0001 1101 1111 1111 1001 is the binary representation of a positive integer, on 64 bits (8 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:

* Not the case - the number is positive *


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

  • 263

    0

  • 262

    0

  • 261

    1

  • 260

    1

  • 259

    1

  • 258

    1

  • 257

    1

  • 256

    1

  • 255

    1

  • 254

    1

  • 253

    0

  • 252

    1

  • 251

    0

  • 250

    0

  • 249

    0

  • 248

    1

  • 247

    1

  • 246

    1

  • 245

    1

  • 244

    0

  • 243

    0

  • 242

    0

  • 241

    0

  • 240

    0

  • 239

    0

  • 238

    0

  • 237

    0

  • 236

    1

  • 235

    1

  • 234

    1

  • 233

    1

  • 232

    1

  • 231

    1

  • 230

    1

  • 229

    0

  • 228

    0

  • 227

    1

  • 226

    1

  • 225

    1

  • 224

    1

  • 223

    1

  • 222

    1

  • 221

    1

  • 220

    0

  • 219

    0

  • 218

    0

  • 217

    0

  • 216

    1

  • 215

    1

  • 214

    1

  • 213

    0

  • 212

    1

  • 211

    1

  • 210

    1

  • 29

    1

  • 28

    1

  • 27

    1

  • 26

    1

  • 25

    1

  • 24

    1

  • 23

    1

  • 22

    0

  • 21

    0

  • 20

    1

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

0011 1111 1101 0001 1110 0000 0001 1111 1100 1111 1110 0001 1101 1111 1111 1001(2) =

(0 × 263 + 0 × 262 + 1 × 261 + 1 × 260 + 1 × 259 + 1 × 258 + 1 × 257 + 1 × 256 + 1 × 255 + 1 × 254 + 0 × 253 + 1 × 252 + 0 × 251 + 0 × 250 + 0 × 249 + 1 × 248 + 1 × 247 + 1 × 246 + 1 × 245 + 0 × 244 + 0 × 243 + 0 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 0 × 238 + 0 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 1 × 232 + 1 × 231 + 1 × 230 + 0 × 229 + 0 × 228 + 1 × 227 + 1 × 226 + 1 × 225 + 1 × 224 + 1 × 223 + 1 × 222 + 1 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 0 × 217 + 1 × 216 + 1 × 215 + 1 × 214 + 0 × 213 + 1 × 212 + 1 × 211 + 1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20)(10) =

(0 + 0 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 0 + 0 + 0 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 0 + 0 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 0 + 0 + 0 + 0 + 65 536 + 32 768 + 16 384 + 0 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 0 + 0 + 1)(10) =

(2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 65 536 + 32 768 + 16 384 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 1)(10) =

4 598 703 121 758 281 721(10)

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

0011 1111 1101 0001 1110 0000 0001 1111 1100 1111 1110 0001 1101 1111 1111 1001(2) = 4 598 703 121 758 281 721(10)

The number 0011 1111 1101 0001 1110 0000 0001 1111 1100 1111 1110 0001 1101 1111 1111 1001(2), signed binary in one's (1's) complement representation, converted and written as an integer in decimal system (base ten):
0011 1111 1101 0001 1110 0000 0001 1111 1100 1111 1110 0001 1101 1111 1111 1001(2) = 4 598 703 121 758 281 721(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