In a signed binary, the first bit (the leftmost) is reserved for the sign,
1 = negative, 0 = positive. This bit does not count when calculating the absolute value.
2. Construct the unsigned binary number.
Exclude the first bit (the leftmost), that is reserved for the sign:
1010 1010 1010 1111 0111 1010 1010 1111 0101 0101 1101 1101 1010 1110 1010 0101 = 010 1010 1010 1111 0111 1010 1010 1111 0101 0101 1101 1101 1010 1110 1010 0101
3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
262
0 261
1 260
0 259
1 258
0 257
1 256
0 255
1 254
0 253
1 252
0 251
1 250
1 249
1 248
1 247
0 246
1 245
1 244
1 243
1 242
0 241
1 240
0 239
1 238
0 237
1 236
0 235
1 234
1 233
1 232
1 231
0 230
1 229
0 228
1 227
0 226
1 225
0 224
1 223
1 222
1 221
0 220
1 219
1 218
1 217
0 216
1 215
1 214
0 213
1 212
0 211
1 210
1 29
1 28
0 27
1 26
0 25
1 24
0 23
0 22
1 21
0 20
1
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
010 1010 1010 1111 0111 1010 1010 1111 0101 0101 1101 1101 1010 1110 1010 0101(2) =
(0 × 262 + 1 × 261 + 0 × 260 + 1 × 259 + 0 × 258 + 1 × 257 + 0 × 256 + 1 × 255 + 0 × 254 + 1 × 253 + 0 × 252 + 1 × 251 + 1 × 250 + 1 × 249 + 1 × 248 + 0 × 247 + 1 × 246 + 1 × 245 + 1 × 244 + 1 × 243 + 0 × 242 + 1 × 241 + 0 × 240 + 1 × 239 + 0 × 238 + 1 × 237 + 0 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 1 × 232 + 0 × 231 + 1 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 1 × 223 + 1 × 222 + 0 × 221 + 1 × 220 + 1 × 219 + 1 × 218 + 0 × 217 + 1 × 216 + 1 × 215 + 0 × 214 + 1 × 213 + 0 × 212 + 1 × 211 + 1 × 210 + 1 × 29 + 0 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 2 305 843 009 213 693 952 + 0 + 576 460 752 303 423 488 + 0 + 144 115 188 075 855 872 + 0 + 36 028 797 018 963 968 + 0 + 9 007 199 254 740 992 + 0 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 0 + 2 199 023 255 552 + 0 + 549 755 813 888 + 0 + 137 438 953 472 + 0 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 0 + 1 073 741 824 + 0 + 268 435 456 + 0 + 67 108 864 + 0 + 16 777 216 + 8 388 608 + 4 194 304 + 0 + 1 048 576 + 524 288 + 262 144 + 0 + 65 536 + 32 768 + 0 + 8 192 + 0 + 2 048 + 1 024 + 512 + 0 + 128 + 0 + 32 + 0 + 0 + 4 + 0 + 1)(10) =
(2 305 843 009 213 693 952 + 576 460 752 303 423 488 + 144 115 188 075 855 872 + 36 028 797 018 963 968 + 9 007 199 254 740 992 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 281 474 976 710 656 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 2 199 023 255 552 + 549 755 813 888 + 137 438 953 472 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 1 073 741 824 + 268 435 456 + 67 108 864 + 16 777 216 + 8 388 608 + 4 194 304 + 1 048 576 + 524 288 + 262 144 + 65 536 + 32 768 + 8 192 + 2 048 + 1 024 + 512 + 128 + 32 + 4 + 1)(10) =
3 075 811 963 995 795 109(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1010 1010 1010 1111 0111 1010 1010 1111 0101 0101 1101 1101 1010 1110 1010 0101(2) = -3 075 811 963 995 795 109(10)
The number 1010 1010 1010 1111 0111 1010 1010 1111 0101 0101 1101 1101 1010 1110 1010 0101(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
1010 1010 1010 1111 0111 1010 1010 1111 0101 0101 1101 1101 1010 1110 1010 0101(2) = -3 075 811 963 995 795 109(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.