What are the steps to convert the base 2 signed binary number
0000 0000 0000 0000 0111 1111 0001 0011 1001 1010 0001 0111 0100 0001 1011 0010(2) to a base 10 decimal system equivalent integer?
1. Is this a positive or a negative number?
0000 0000 0000 0000 0111 1111 0001 0011 1001 1010 0001 0111 0100 0001 1011 0010 is the binary representation of a positive integer, on 64 bits (8 Bytes).
- 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:
0000 0000 0000 0000 0111 1111 0001 0011 1001 1010 0001 0111 0100 0001 1011 0010 = 000 0000 0000 0000 0111 1111 0001 0011 1001 1010 0001 0111 0100 0001 1011 0010
3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
262
0 261
0 260
0 259
0 258
0 257
0 256
0 255
0 254
0 253
0 252
0 251
0 250
0 249
0 248
0 247
0 246
1 245
1 244
1 243
1 242
1 241
1 240
1 239
0 238
0 237
0 236
1 235
0 234
0 233
1 232
1 231
1 230
0 229
0 228
1 227
1 226
0 225
1 224
0 223
0 222
0 221
0 220
1 219
0 218
1 217
1 216
1 215
0 214
1 213
0 212
0 211
0 210
0 29
0 28
1 27
1 26
0 25
1 24
1 23
0 22
0 21
1 20
0
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
000 0000 0000 0000 0111 1111 0001 0011 1001 1010 0001 0111 0100 0001 1011 0010(2) =
(0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 0 × 249 + 0 × 248 + 0 × 247 + 1 × 246 + 1 × 245 + 1 × 244 + 1 × 243 + 1 × 242 + 1 × 241 + 1 × 240 + 0 × 239 + 0 × 238 + 0 × 237 + 1 × 236 + 0 × 235 + 0 × 234 + 1 × 233 + 1 × 232 + 1 × 231 + 0 × 230 + 0 × 229 + 1 × 228 + 1 × 227 + 0 × 226 + 1 × 225 + 0 × 224 + 0 × 223 + 0 × 222 + 0 × 221 + 1 × 220 + 0 × 219 + 1 × 218 + 1 × 217 + 1 × 216 + 0 × 215 + 1 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 0 × 210 + 0 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 0 + 0 + 0 + 68 719 476 736 + 0 + 0 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 0 + 0 + 268 435 456 + 134 217 728 + 0 + 33 554 432 + 0 + 0 + 0 + 0 + 1 048 576 + 0 + 262 144 + 131 072 + 65 536 + 0 + 16 384 + 0 + 0 + 0 + 0 + 0 + 256 + 128 + 0 + 32 + 16 + 0 + 0 + 2 + 0)(10) =
(70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 68 719 476 736 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 268 435 456 + 134 217 728 + 33 554 432 + 1 048 576 + 262 144 + 131 072 + 65 536 + 16 384 + 256 + 128 + 32 + 16 + 2)(10) =
139 722 166 321 586(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 0000 0000 0111 1111 0001 0011 1001 1010 0001 0111 0100 0001 1011 0010(2) = 139 722 166 321 586(10)
0000 0000 0000 0000 0111 1111 0001 0011 1001 1010 0001 0111 0100 0001 1011 0010(2), Base 2 signed binary number, converted and written as a base 10 decimal system equivalent integer:
0000 0000 0000 0000 0111 1111 0001 0011 1001 1010 0001 0111 0100 0001 1011 0010(2) = 139 722 166 321 586(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.