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 = 1; 1 - 0 = 1; 1 - 1 = 0.
Subtract 1 from the initial binary number.
1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 0111 - 1 = 1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 0110
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:
!(1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 0110) = 0011 1101 1101 1100 1100 0101 1101 1111 1100 0110 0100 0111 1011 0101 0101 1001
4. 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
1 259
1 258
1 257
0 256
1 255
1 254
1 253
0 252
1 251
1 250
1 249
0 248
0 247
1 246
1 245
0 244
0 243
0 242
1 241
0 240
1 239
1 238
1 237
0 236
1 235
1 234
1 233
1 232
1 231
1 230
1 229
0 228
0 227
0 226
1 225
1 224
0 223
0 222
1 221
0 220
0 219
0 218
1 217
1 216
1 215
1 214
0 213
1 212
1 211
0 210
1 29
0 28
1 27
0 26
1 25
0 24
1 23
1 22
0 21
0 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0011 1101 1101 1100 1100 0101 1101 1111 1100 0110 0100 0111 1011 0101 0101 1001(2) =
(0 × 263 + 0 × 262 + 1 × 261 + 1 × 260 + 1 × 259 + 1 × 258 + 0 × 257 + 1 × 256 + 1 × 255 + 1 × 254 + 0 × 253 + 1 × 252 + 1 × 251 + 1 × 250 + 0 × 249 + 0 × 248 + 1 × 247 + 1 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 1 × 239 + 1 × 238 + 0 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 1 × 232 + 1 × 231 + 1 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 1 × 226 + 1 × 225 + 0 × 224 + 0 × 223 + 1 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 1 × 218 + 1 × 217 + 1 × 216 + 1 × 215 + 0 × 214 + 1 × 213 + 1 × 212 + 0 × 211 + 1 × 210 + 0 × 29 + 1 × 28 + 0 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 0 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 0 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 0 + 0 + 140 737 488 355 328 + 70 368 744 177 664 + 0 + 0 + 0 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 0 + 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 + 0 + 0 + 0 + 67 108 864 + 33 554 432 + 0 + 0 + 4 194 304 + 0 + 0 + 0 + 262 144 + 131 072 + 65 536 + 32 768 + 0 + 8 192 + 4 096 + 0 + 1 024 + 0 + 256 + 0 + 64 + 0 + 16 + 8 + 0 + 0 + 1)(10) =
(2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 140 737 488 355 328 + 70 368 744 177 664 + 4 398 046 511 104 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 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 + 67 108 864 + 33 554 432 + 4 194 304 + 262 144 + 131 072 + 65 536 + 32 768 + 8 192 + 4 096 + 1 024 + 256 + 64 + 16 + 8 + 1)(10) =
4 457 655 296 084 915 545(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 0111(2) = -4 457 655 296 084 915 545(10)
The signed binary number in two's complement representation 1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 0111(2) converted and written as an integer in decimal system (base ten):
1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 0111(2) = -4 457 655 296 084 915 545(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.