Convert 1101 0101 1110 0111 1100 1000 1101 0100 1101 0001 1100 0011 1100 1001 0010 1111 Signed Binary in One's (1's) Complement Representation on 64 Bit to Decimal

How to convert 1101 0101 1110 0111 1100 1000 1101 0100 1101 0001 1100 0011 1100 1001 0010 1111(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?

1101 0101 1110 0111 1100 1000 1101 0100 1101 0001 1100 0011 1100 1001 0010 1111 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:

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


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

    0

  • 259

    1

  • 258

    0

  • 257

    1

  • 256

    0

  • 255

    0

  • 254

    0

  • 253

    0

  • 252

    1

  • 251

    1

  • 250

    0

  • 249

    0

  • 248

    0

  • 247

    0

  • 246

    0

  • 245

    1

  • 244

    1

  • 243

    0

  • 242

    1

  • 241

    1

  • 240

    1

  • 239

    0

  • 238

    0

  • 237

    1

  • 236

    0

  • 235

    1

  • 234

    0

  • 233

    1

  • 232

    1

  • 231

    0

  • 230

    0

  • 229

    1

  • 228

    0

  • 227

    1

  • 226

    1

  • 225

    1

  • 224

    0

  • 223

    0

  • 222

    0

  • 221

    1

  • 220

    1

  • 219

    1

  • 218

    1

  • 217

    0

  • 216

    0

  • 215

    0

  • 214

    0

  • 213

    1

  • 212

    1

  • 211

    0

  • 210

    1

  • 29

    1

  • 28

    0

  • 27

    1

  • 26

    1

  • 25

    0

  • 24

    1

  • 23

    0

  • 22

    0

  • 21

    0

  • 20

    0

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

0010 1010 0001 1000 0011 0111 0010 1011 0010 1110 0011 1100 0011 0110 1101 0000(2) =

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

(0 + 0 + 2 305 843 009 213 693 952 + 0 + 576 460 752 303 423 488 + 0 + 144 115 188 075 855 872 + 0 + 0 + 0 + 0 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 0 + 0 + 0 + 0 + 0 + 35 184 372 088 832 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 0 + 0 + 137 438 953 472 + 0 + 34 359 738 368 + 0 + 8 589 934 592 + 4 294 967 296 + 0 + 0 + 536 870 912 + 0 + 134 217 728 + 67 108 864 + 33 554 432 + 0 + 0 + 0 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 0 + 0 + 0 + 0 + 8 192 + 4 096 + 0 + 1 024 + 512 + 0 + 128 + 64 + 0 + 16 + 0 + 0 + 0 + 0)(10) =

(2 305 843 009 213 693 952 + 576 460 752 303 423 488 + 144 115 188 075 855 872 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 35 184 372 088 832 + 17 592 186 044 416 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 137 438 953 472 + 34 359 738 368 + 8 589 934 592 + 4 294 967 296 + 536 870 912 + 134 217 728 + 67 108 864 + 33 554 432 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 8 192 + 4 096 + 1 024 + 512 + 128 + 64 + 16)(10) =

3 033 235 007 632 848 592(10)

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

1101 0101 1110 0111 1100 1000 1101 0100 1101 0001 1100 0011 1100 1001 0010 1111(2) = -3 033 235 007 632 848 592(10)

The number 1101 0101 1110 0111 1100 1000 1101 0100 1101 0001 1100 0011 1100 1001 0010 1111(2), signed binary in one's (1's) complement representation, converted and written as an integer in decimal system (base ten):
1101 0101 1110 0111 1100 1000 1101 0100 1101 0001 1100 0011 1100 1001 0010 1111(2) = -3 033 235 007 632 848 592(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