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

How to convert 1000 1001 0010 1001 1100 1010 1000 1010 0101 0010 0100 1010 0100 1001 0000 1000(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 1001 0010 1001 1100 1010 1000 1010 0101 0010 0100 1010 0100 1001 0000 1000 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 1001 0010 1001 1100 1010 1000 1010 0101 0010 0100 1010 0100 1001 0000 1000 - 1 = 1000 1001 0010 1001 1100 1010 1000 1010 0101 0010 0100 1010 0100 1001 0000 0111


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


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

    1

  • 257

    1

  • 256

    0

  • 255

    1

  • 254

    1

  • 253

    0

  • 252

    1

  • 251

    0

  • 250

    1

  • 249

    1

  • 248

    0

  • 247

    0

  • 246

    0

  • 245

    1

  • 244

    1

  • 243

    0

  • 242

    1

  • 241

    0

  • 240

    1

  • 239

    0

  • 238

    1

  • 237

    1

  • 236

    1

  • 235

    0

  • 234

    1

  • 233

    0

  • 232

    1

  • 231

    1

  • 230

    0

  • 229

    1

  • 228

    0

  • 227

    1

  • 226

    1

  • 225

    0

  • 224

    1

  • 223

    1

  • 222

    0

  • 221

    1

  • 220

    1

  • 219

    0

  • 218

    1

  • 217

    0

  • 216

    1

  • 215

    1

  • 214

    0

  • 213

    1

  • 212

    1

  • 211

    0

  • 210

    1

  • 29

    1

  • 28

    0

  • 27

    1

  • 26

    1

  • 25

    1

  • 24

    1

  • 23

    1

  • 22

    0

  • 21

    0

  • 20

    0

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

0111 0110 1101 0110 0011 0101 0111 0101 1010 1101 1011 0101 1011 0110 1111 1000(2) =

(0 × 263 + 1 × 262 + 1 × 261 + 1 × 260 + 0 × 259 + 1 × 258 + 1 × 257 + 0 × 256 + 1 × 255 + 1 × 254 + 0 × 253 + 1 × 252 + 0 × 251 + 1 × 250 + 1 × 249 + 0 × 248 + 0 × 247 + 0 × 246 + 1 × 245 + 1 × 244 + 0 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 0 × 239 + 1 × 238 + 1 × 237 + 1 × 236 + 0 × 235 + 1 × 234 + 0 × 233 + 1 × 232 + 1 × 231 + 0 × 230 + 1 × 229 + 0 × 228 + 1 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 1 × 223 + 0 × 222 + 1 × 221 + 1 × 220 + 0 × 219 + 1 × 218 + 0 × 217 + 1 × 216 + 1 × 215 + 0 × 214 + 1 × 213 + 1 × 212 + 0 × 211 + 1 × 210 + 1 × 29 + 0 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 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 + 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 + 4 503 599 627 370 496 + 0 + 1 125 899 906 842 624 + 562 949 953 421 312 + 0 + 0 + 0 + 35 184 372 088 832 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 0 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 0 + 17 179 869 184 + 0 + 4 294 967 296 + 2 147 483 648 + 0 + 536 870 912 + 0 + 134 217 728 + 67 108 864 + 0 + 16 777 216 + 8 388 608 + 0 + 2 097 152 + 1 048 576 + 0 + 262 144 + 0 + 65 536 + 32 768 + 0 + 8 192 + 4 096 + 0 + 1 024 + 512 + 0 + 128 + 64 + 32 + 16 + 8 + 0 + 0 + 0)(10) =

(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 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 + 4 503 599 627 370 496 + 1 125 899 906 842 624 + 562 949 953 421 312 + 35 184 372 088 832 + 17 592 186 044 416 + 4 398 046 511 104 + 1 099 511 627 776 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 17 179 869 184 + 4 294 967 296 + 2 147 483 648 + 536 870 912 + 134 217 728 + 67 108 864 + 16 777 216 + 8 388 608 + 2 097 152 + 1 048 576 + 262 144 + 65 536 + 32 768 + 8 192 + 4 096 + 1 024 + 512 + 128 + 64 + 32 + 16 + 8)(10) =

8 563 090 521 033 389 816(10)

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

1000 1001 0010 1001 1100 1010 1000 1010 0101 0010 0100 1010 0100 1001 0000 1000(2) = -8 563 090 521 033 389 816(10)

The number 1000 1001 0010 1001 1100 1010 1000 1010 0101 0010 0100 1010 0100 1001 0000 1000(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
1000 1001 0010 1001 1100 1010 1000 1010 0101 0010 0100 1010 0100 1001 0000 1000(2) = -8 563 090 521 033 389 816(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.