0001 0000 0000 1000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111 Base 2 Signed Binary Number on 64 Bit Converted To Base 10 Decimal System Integer

How to convert 0001 0000 0000 1000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111(2), the base 2 signed binary number on 64 bit, to a base 10 decimal system integer

What are the steps to convert the base 2 signed binary number
0001 0000 0000 1000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111(2) to a base 10 decimal system equivalent integer?

1. Is this a positive or a negative number?

0001 0000 0000 1000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111 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:

0001 0000 0000 1000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111 = 001 0000 0000 1000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111


3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:

  • 262

    0

  • 261

    0

  • 260

    1

  • 259

    0

  • 258

    0

  • 257

    0

  • 256

    0

  • 255

    0

  • 254

    0

  • 253

    0

  • 252

    0

  • 251

    1

  • 250

    0

  • 249

    0

  • 248

    0

  • 247

    0

  • 246

    0

  • 245

    0

  • 244

    0

  • 243

    1

  • 242

    1

  • 241

    1

  • 240

    1

  • 239

    1

  • 238

    1

  • 237

    1

  • 236

    1

  • 235

    1

  • 234

    1

  • 233

    1

  • 232

    1

  • 231

    1

  • 230

    1

  • 229

    1

  • 228

    1

  • 227

    1

  • 226

    1

  • 225

    1

  • 224

    1

  • 223

    1

  • 222

    1

  • 221

    1

  • 220

    1

  • 219

    1

  • 218

    1

  • 217

    1

  • 216

    1

  • 215

    1

  • 214

    1

  • 213

    1

  • 212

    1

  • 211

    1

  • 210

    1

  • 29

    1

  • 28

    1

  • 27

    1

  • 26

    1

  • 25

    1

  • 24

    0

  • 23

    0

  • 22

    1

  • 21

    1

  • 20

    1

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

001 0000 0000 1000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111(2) =

(0 × 262 + 0 × 261 + 1 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 1 × 251 + 0 × 250 + 0 × 249 + 0 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 1 × 243 + 1 × 242 + 1 × 241 + 1 × 240 + 1 × 239 + 1 × 238 + 1 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 1 × 232 + 1 × 231 + 1 × 230 + 1 × 229 + 1 × 228 + 1 × 227 + 1 × 226 + 1 × 225 + 1 × 224 + 1 × 223 + 1 × 222 + 1 × 221 + 1 × 220 + 1 × 219 + 1 × 218 + 1 × 217 + 1 × 216 + 1 × 215 + 1 × 214 + 1 × 213 + 1 × 212 + 1 × 211 + 1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 1 × 20)(10) =

(0 + 0 + 1 152 921 504 606 846 976 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 251 799 813 685 248 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 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 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 0 + 0 + 4 + 2 + 1)(10) =

(1 152 921 504 606 846 976 + 2 251 799 813 685 248 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 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 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 4 + 2 + 1)(10) =

1 155 190 896 606 576 615(10)

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

0001 0000 0000 1000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111(2) = 1 155 190 896 606 576 615(10)

0001 0000 0000 1000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111(2), Base 2 signed binary number, converted and written as a base 10 decimal system equivalent integer:
0001 0000 0000 1000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111(2) = 1 155 190 896 606 576 615(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 from binary system to decimal (base ten)

To understand how to convert a signed binary number from binary system to decimal (base ten), the easiest way is to do it through an example - convert the binary number, 1001 1110, to base ten:

  • In a signed binary, the first bit (leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value (value without sign). The first bit is 1, so our number is negative.
  • Write bellow the binary number 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 and increasing each corresonding power of 2 by exactly one unit, but ignoring the very first bit (the leftmost, the one representing the sign):
  • powers of 2:   6 5 4 3 2 1 0
    digits: 1 0 0 1 1 1 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, but also taking care of the number sign:

    1001 1110 =

    - (0 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20)(10) =

    - (0 + 0 + 16 + 8 + 4 + 2 + 0)(10) =

    - (16 + 8 + 4 + 2)(10) =

    -30(10)

  • Binary signed number, 1001 1110 = -30(10), signed negative integer in base 10