Convert 0100 0000 0110 0011 0001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 Signed Binary in One's (1's) Complement Representation on 64 Bit to Decimal

How to convert 0100 0000 0110 0011 0001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101(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?

0100 0000 0110 0011 0001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 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

    1

  • 261

    0

  • 260

    0

  • 259

    0

  • 258

    0

  • 257

    0

  • 256

    0

  • 255

    0

  • 254

    1

  • 253

    1

  • 252

    0

  • 251

    0

  • 250

    0

  • 249

    1

  • 248

    1

  • 247

    0

  • 246

    0

  • 245

    0

  • 244

    1

  • 243

    0

  • 242

    1

  • 241

    1

  • 240

    1

  • 239

    1

  • 238

    1

  • 237

    1

  • 236

    1

  • 235

    1

  • 234

    1

  • 233

    1

  • 232

    1

  • 231

    1

  • 230

    1

  • 229

    1

  • 228

    1

  • 227

    1

  • 226

    1

  • 225

    1

  • 224

    1

  • 223

    1

  • 222

    1

  • 221

    1

  • 220

    1

  • 219

    1

  • 218

    1

  • 217

    1

  • 216

    1

  • 215

    1

  • 214

    1

  • 213

    1

  • 212

    1

  • 211

    1

  • 210

    1

  • 29

    1

  • 28

    1

  • 27

    1

  • 26

    1

  • 25

    1

  • 24

    1

  • 23

    1

  • 22

    1

  • 21

    0

  • 20

    1

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

0100 0000 0110 0011 0001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101(2) =

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

(0 + 4 611 686 018 427 387 904 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 0 + 0 + 0 + 562 949 953 421 312 + 281 474 976 710 656 + 0 + 0 + 0 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 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 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 0 + 1)(10) =

(4 611 686 018 427 387 904 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 562 949 953 421 312 + 281 474 976 710 656 + 17 592 186 044 416 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 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 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 1)(10) =

4 639 578 429 400 809 469(10)

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

0100 0000 0110 0011 0001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101(2) = 4 639 578 429 400 809 469(10)

The number 0100 0000 0110 0011 0001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101(2), signed binary in one's (1's) complement representation, converted and written as an integer in decimal system (base ten):
0100 0000 0110 0011 0001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101(2) = 4 639 578 429 400 809 469(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