What are the steps to convert the base 2 signed binary number
0011 0011 0100 1110 0100 0001 0011 0000 0011 0000 0011 0100 0011 0000 0111 1111(2) to a base 10 decimal system equivalent integer?
1. Is this a positive or a negative number?
0011 0011 0100 1110 0100 0001 0011 0000 0011 0000 0011 0100 0011 0000 0111 1111 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:
0011 0011 0100 1110 0100 0001 0011 0000 0011 0000 0011 0100 0011 0000 0111 1111 = 011 0011 0100 1110 0100 0001 0011 0000 0011 0000 0011 0100 0011 0000 0111 1111
3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
262
0 261
1 260
1 259
0 258
0 257
1 256
1 255
0 254
1 253
0 252
0 251
1 250
1 249
1 248
0 247
0 246
1 245
0 244
0 243
0 242
0 241
0 240
1 239
0 238
0 237
1 236
1 235
0 234
0 233
0 232
0 231
0 230
0 229
1 228
1 227
0 226
0 225
0 224
0 223
0 222
0 221
1 220
1 219
0 218
1 217
0 216
0 215
0 214
0 213
1 212
1 211
0 210
0 29
0 28
0 27
0 26
1 25
1 24
1 23
1 22
1 21
1 20
1
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
011 0011 0100 1110 0100 0001 0011 0000 0011 0000 0011 0100 0011 0000 0111 1111(2) =
(0 × 262 + 1 × 261 + 1 × 260 + 0 × 259 + 0 × 258 + 1 × 257 + 1 × 256 + 0 × 255 + 1 × 254 + 0 × 253 + 0 × 252 + 1 × 251 + 1 × 250 + 1 × 249 + 0 × 248 + 0 × 247 + 1 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 0 × 242 + 0 × 241 + 1 × 240 + 0 × 239 + 0 × 238 + 1 × 237 + 1 × 236 + 0 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 0 × 231 + 0 × 230 + 1 × 229 + 1 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 0 × 223 + 0 × 222 + 1 × 221 + 1 × 220 + 0 × 219 + 1 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 1 × 213 + 1 × 212 + 0 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20)(10) =
(0 + 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 + 0 + 18 014 398 509 481 984 + 0 + 0 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 0 + 0 + 70 368 744 177 664 + 0 + 0 + 0 + 0 + 0 + 1 099 511 627 776 + 0 + 0 + 137 438 953 472 + 68 719 476 736 + 0 + 0 + 0 + 0 + 0 + 0 + 536 870 912 + 268 435 456 + 0 + 0 + 0 + 0 + 0 + 0 + 2 097 152 + 1 048 576 + 0 + 262 144 + 0 + 0 + 0 + 0 + 8 192 + 4 096 + 0 + 0 + 0 + 0 + 0 + 64 + 32 + 16 + 8 + 4 + 2 + 1)(10) =
(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 + 18 014 398 509 481 984 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 70 368 744 177 664 + 1 099 511 627 776 + 137 438 953 472 + 68 719 476 736 + 536 870 912 + 268 435 456 + 2 097 152 + 1 048 576 + 262 144 + 8 192 + 4 096 + 64 + 32 + 16 + 8 + 4 + 2 + 1)(10) =
3 696 964 019 340 718 207(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0011 0011 0100 1110 0100 0001 0011 0000 0011 0000 0011 0100 0011 0000 0111 1111(2) = 3 696 964 019 340 718 207(10)
0011 0011 0100 1110 0100 0001 0011 0000 0011 0000 0011 0100 0011 0000 0111 1111(2), Base 2 signed binary number, converted and written as a base 10 decimal system equivalent integer:
0011 0011 0100 1110 0100 0001 0011 0000 0011 0000 0011 0100 0011 0000 0111 1111(2) = 3 696 964 019 340 718 207(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.