Convert 1000 1100 0101 1001 0111 0011 1010 0110 0101 0111 0010 0000 1111 0100 1101 1101 Signed Binary in Two's (2's) Complement Representation on 64 Bit to Decimal

How to convert 1000 1100 0101 1001 0111 0011 1010 0110 0101 0111 0010 0000 1111 0100 1101 1101(2), signed binary in two's (2's) complement representation, to decimal

What are the steps to convert the signed binary in two's (2's) complement representation to an integer in decimal system (in base ten)?

1. Is this a positive or a negative number?

1000 1100 0101 1001 0111 0011 1010 0110 0101 0111 0010 0000 1111 0100 1101 1101 is the binary representation of a negative integer, on 64 bits (8 Bytes).


  • In a signed binary in two's complement representation, the first bit (the leftmost) indicates the sign, 1 = negative, 0 = positive.

2. Get the binary representation in one's complement.

* Run this step only if the number is negative

  • Note on binary subtraction rules:
  • 11 - 1 = 10; 10 - 1 = 01; 1 - 0 = 1; 1 - 1 = 0.

  • Subtract 1 from the initial binary number.

  • 1000 1100 0101 1001 0111 0011 1010 0110 0101 0111 0010 0000 1111 0100 1101 1101 - 1 = 1000 1100 0101 1001 0111 0011 1010 0110 0101 0111 0010 0000 1111 0100 1101 1100


3. 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:

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


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

  • 263

    0

  • 262

    1

  • 261

    1

  • 260

    1

  • 259

    0

  • 258

    0

  • 257

    1

  • 256

    1

  • 255

    1

  • 254

    0

  • 253

    1

  • 252

    0

  • 251

    0

  • 250

    1

  • 249

    1

  • 248

    0

  • 247

    1

  • 246

    0

  • 245

    0

  • 244

    0

  • 243

    1

  • 242

    1

  • 241

    0

  • 240

    0

  • 239

    0

  • 238

    1

  • 237

    0

  • 236

    1

  • 235

    1

  • 234

    0

  • 233

    0

  • 232

    1

  • 231

    1

  • 230

    0

  • 229

    1

  • 228

    0

  • 227

    1

  • 226

    0

  • 225

    0

  • 224

    0

  • 223

    1

  • 222

    1

  • 221

    0

  • 220

    1

  • 219

    1

  • 218

    1

  • 217

    1

  • 216

    1

  • 215

    0

  • 214

    0

  • 213

    0

  • 212

    0

  • 211

    1

  • 210

    0

  • 29

    1

  • 28

    1

  • 27

    0

  • 26

    0

  • 25

    1

  • 24

    0

  • 23

    0

  • 22

    0

  • 21

    1

  • 20

    1

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

0111 0011 1010 0110 1000 1100 0101 1001 1010 1000 1101 1111 0000 1011 0010 0011(2) =

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

(0 + 4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 0 + 0 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 0 + 9 007 199 254 740 992 + 0 + 0 + 1 125 899 906 842 624 + 562 949 953 421 312 + 0 + 140 737 488 355 328 + 0 + 0 + 0 + 8 796 093 022 208 + 4 398 046 511 104 + 0 + 0 + 0 + 274 877 906 944 + 0 + 68 719 476 736 + 34 359 738 368 + 0 + 0 + 4 294 967 296 + 2 147 483 648 + 0 + 536 870 912 + 0 + 134 217 728 + 0 + 0 + 0 + 8 388 608 + 4 194 304 + 0 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 0 + 0 + 0 + 0 + 2 048 + 0 + 512 + 256 + 0 + 0 + 32 + 0 + 0 + 0 + 2 + 1)(10) =

(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 9 007 199 254 740 992 + 1 125 899 906 842 624 + 562 949 953 421 312 + 140 737 488 355 328 + 8 796 093 022 208 + 4 398 046 511 104 + 274 877 906 944 + 68 719 476 736 + 34 359 738 368 + 4 294 967 296 + 2 147 483 648 + 536 870 912 + 134 217 728 + 8 388 608 + 4 194 304 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 2 048 + 512 + 256 + 32 + 2 + 1)(10) =

8 333 502 477 208 849 187(10)

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

1000 1100 0101 1001 0111 0011 1010 0110 0101 0111 0010 0000 1111 0100 1101 1101(2) = -8 333 502 477 208 849 187(10)

The number 1000 1100 0101 1001 0111 0011 1010 0110 0101 0111 0010 0000 1111 0100 1101 1101(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
1000 1100 0101 1001 0111 0011 1010 0110 0101 0111 0010 0000 1111 0100 1101 1101(2) = -8 333 502 477 208 849 187(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 two's complement representation from binary system to decimal

To understand how to convert a signed binary number in two's complement representation from the binary system to decimal (base ten), the easiest way is to do it by an example - convert binary, 1101 1110, to base ten:

  • In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive. The first bit is 1, so our number is negative.
  • Get the signed binary representation in one's complement, subtract 1 from the initial number:
    1101 1110 - 1 = 1101 1101
  • 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:
    !(1101 1101) = 0010 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, increasing each corresonding power of 2 by exactly one unit:
  • powers of 2: 7 6 5 4 3 2 1 0
    digits: 0 0 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:

    0010 0010(2) =

    (0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =

    (0 + 0 + 32 + 0 + 0 + 0 + 2 + 0)(10) =

    (32 + 2)(10) =

    34(10)

  • Signed binary number in two's complement representation, 1101 1110 = -34(10), a signed negative integer in base 10.