Convert 0100 0010 0010 0100 1000 1001 0100 1001 0010 0100 1001 0101 0100 1010 1010 0111 Signed Binary in One's (1's) Complement Representation on 64 Bit to Decimal

How to convert 0100 0010 0010 0100 1000 1001 0100 1001 0010 0100 1001 0101 0100 1010 1010 0111(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 0010 0010 0100 1000 1001 0100 1001 0010 0100 1001 0101 0100 1010 1010 0111 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

    1

  • 256

    0

  • 255

    0

  • 254

    0

  • 253

    1

  • 252

    0

  • 251

    0

  • 250

    1

  • 249

    0

  • 248

    0

  • 247

    1

  • 246

    0

  • 245

    0

  • 244

    0

  • 243

    1

  • 242

    0

  • 241

    0

  • 240

    1

  • 239

    0

  • 238

    1

  • 237

    0

  • 236

    0

  • 235

    1

  • 234

    0

  • 233

    0

  • 232

    1

  • 231

    0

  • 230

    0

  • 229

    1

  • 228

    0

  • 227

    0

  • 226

    1

  • 225

    0

  • 224

    0

  • 223

    1

  • 222

    0

  • 221

    0

  • 220

    1

  • 219

    0

  • 218

    1

  • 217

    0

  • 216

    1

  • 215

    0

  • 214

    1

  • 213

    0

  • 212

    0

  • 211

    1

  • 210

    0

  • 29

    1

  • 28

    0

  • 27

    1

  • 26

    0

  • 25

    1

  • 24

    0

  • 23

    0

  • 22

    1

  • 21

    1

  • 20

    1

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

0100 0010 0010 0100 1000 1001 0100 1001 0010 0100 1001 0101 0100 1010 1010 0111(2) =

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

(0 + 4 611 686 018 427 387 904 + 0 + 0 + 0 + 0 + 144 115 188 075 855 872 + 0 + 0 + 0 + 9 007 199 254 740 992 + 0 + 0 + 1 125 899 906 842 624 + 0 + 0 + 140 737 488 355 328 + 0 + 0 + 0 + 8 796 093 022 208 + 0 + 0 + 1 099 511 627 776 + 0 + 274 877 906 944 + 0 + 0 + 34 359 738 368 + 0 + 0 + 4 294 967 296 + 0 + 0 + 536 870 912 + 0 + 0 + 67 108 864 + 0 + 0 + 8 388 608 + 0 + 0 + 1 048 576 + 0 + 262 144 + 0 + 65 536 + 0 + 16 384 + 0 + 0 + 2 048 + 0 + 512 + 0 + 128 + 0 + 32 + 0 + 0 + 4 + 2 + 1)(10) =

(4 611 686 018 427 387 904 + 144 115 188 075 855 872 + 9 007 199 254 740 992 + 1 125 899 906 842 624 + 140 737 488 355 328 + 8 796 093 022 208 + 1 099 511 627 776 + 274 877 906 944 + 34 359 738 368 + 4 294 967 296 + 536 870 912 + 67 108 864 + 8 388 608 + 1 048 576 + 262 144 + 65 536 + 16 384 + 2 048 + 512 + 128 + 32 + 4 + 2 + 1)(10) =

4 766 085 252 904 209 063(10)

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

0100 0010 0010 0100 1000 1001 0100 1001 0010 0100 1001 0101 0100 1010 1010 0111(2) = 4 766 085 252 904 209 063(10)

The number 0100 0010 0010 0100 1000 1001 0100 1001 0010 0100 1001 0101 0100 1010 1010 0111(2), signed binary in one's (1's) complement representation, converted and written as an integer in decimal system (base ten):
0100 0010 0010 0100 1000 1001 0100 1001 0010 0100 1001 0101 0100 1010 1010 0111(2) = 4 766 085 252 904 209 063(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