One's Complement: Binary ↘ Integer: 1010 1001 0011 0010 0001 0111 1111 0001 1100 1001 1011 0101 1001 0100 0001 0111 Signed Binary Number in One's Complement Representation, Converted and Written as a Decimal System Integer (in Base Ten)

Signed binary in one's complement representation 1010 1001 0011 0010 0001 0111 1111 0001 1100 1001 1011 0101 1001 0100 0001 0111(2) converted to an integer in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

1010 1001 0011 0010 0001 0111 1111 0001 1100 1001 1011 0101 1001 0100 0001 0111 is the binary representation of a negative 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:

!(1010 1001 0011 0010 0001 0111 1111 0001 1100 1001 1011 0101 1001 0100 0001 0111) = 0101 0110 1100 1101 1110 1000 0000 1110 0011 0110 0100 1010 0110 1011 1110 1000


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

  • 263

    0
  • 262

    1
  • 261

    0
  • 260

    1
  • 259

    0
  • 258

    1
  • 257

    1
  • 256

    0
  • 255

    1
  • 254

    1
  • 253

    0
  • 252

    0
  • 251

    1
  • 250

    1
  • 249

    0
  • 248

    1
  • 247

    1
  • 246

    1
  • 245

    1
  • 244

    0
  • 243

    1
  • 242

    0
  • 241

    0
  • 240

    0
  • 239

    0
  • 238

    0
  • 237

    0
  • 236

    0
  • 235

    1
  • 234

    1
  • 233

    1
  • 232

    0
  • 231

    0
  • 230

    0
  • 229

    1
  • 228

    1
  • 227

    0
  • 226

    1
  • 225

    1
  • 224

    0
  • 223

    0
  • 222

    1
  • 221

    0
  • 220

    0
  • 219

    1
  • 218

    0
  • 217

    1
  • 216

    0
  • 215

    0
  • 214

    1
  • 213

    1
  • 212

    0
  • 211

    1
  • 210

    0
  • 29

    1
  • 28

    1
  • 27

    1
  • 26

    1
  • 25

    1
  • 24

    0
  • 23

    1
  • 22

    0
  • 21

    0
  • 20

    0

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

0101 0110 1100 1101 1110 1000 0000 1110 0011 0110 0100 1010 0110 1011 1110 1000(2) =


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


(0 + 4 611 686 018 427 387 904 + 0 + 1 152 921 504 606 846 976 + 0 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 0 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 0 + 0 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 0 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 0 + 8 796 093 022 208 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 0 + 0 + 0 + 536 870 912 + 268 435 456 + 0 + 67 108 864 + 33 554 432 + 0 + 0 + 4 194 304 + 0 + 0 + 524 288 + 0 + 131 072 + 0 + 0 + 16 384 + 8 192 + 0 + 2 048 + 0 + 512 + 256 + 128 + 64 + 32 + 0 + 8 + 0 + 0 + 0)(10) =


(4 611 686 018 427 387 904 + 1 152 921 504 606 846 976 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 8 796 093 022 208 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 536 870 912 + 268 435 456 + 67 108 864 + 33 554 432 + 4 194 304 + 524 288 + 131 072 + 16 384 + 8 192 + 2 048 + 512 + 256 + 128 + 64 + 32 + 8)(10) =


6 254 910 605 225 520 104(10)

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

1010 1001 0011 0010 0001 0111 1111 0001 1100 1001 1011 0101 1001 0100 0001 0111(2) = -6 254 910 605 225 520 104(10)

The signed binary number in one's complement representation 1010 1001 0011 0010 0001 0111 1111 0001 1100 1001 1011 0101 1001 0100 0001 0111(2) converted and written as an integer in decimal system (base ten):
1010 1001 0011 0010 0001 0111 1111 0001 1100 1001 1011 0101 1001 0100 0001 0111(2) = -6 254 910 605 225 520 104(10)

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

The latest binary numbers in one's complement representation converted to signed integers numbers written in decimal system (base ten)

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