Convert 0000 0000 1000 1100 1000 0110 1000 0100 1000 0100 0111 0000 0110 1010 0011 1001 Base 2 Signed Binary Number on 64 Bit - To Base 10 Decimal System Integer

How to convert 0000 0000 1000 1100 1000 0110 1000 0100 1000 0100 0111 0000 0110 1010 0011 1001(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
0000 0000 1000 1100 1000 0110 1000 0100 1000 0100 0111 0000 0110 1010 0011 1001(2) to a base 10 decimal system equivalent integer?

1. Is this a positive or a negative number?

0000 0000 1000 1100 1000 0110 1000 0100 1000 0100 0111 0000 0110 1010 0011 1001 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 1000 1100 1000 0110 1000 0100 1000 0100 0111 0000 0110 1010 0011 1001 = 000 0000 1000 1100 1000 0110 1000 0100 1000 0100 0111 0000 0110 1010 0011 1001


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

    1

  • 254

    0

  • 253

    0

  • 252

    0

  • 251

    1

  • 250

    1

  • 249

    0

  • 248

    0

  • 247

    1

  • 246

    0

  • 245

    0

  • 244

    0

  • 243

    0

  • 242

    1

  • 241

    1

  • 240

    0

  • 239

    1

  • 238

    0

  • 237

    0

  • 236

    0

  • 235

    0

  • 234

    1

  • 233

    0

  • 232

    0

  • 231

    1

  • 230

    0

  • 229

    0

  • 228

    0

  • 227

    0

  • 226

    1

  • 225

    0

  • 224

    0

  • 223

    0

  • 222

    1

  • 221

    1

  • 220

    1

  • 219

    0

  • 218

    0

  • 217

    0

  • 216

    0

  • 215

    0

  • 214

    1

  • 213

    1

  • 212

    0

  • 211

    1

  • 210

    0

  • 29

    1

  • 28

    0

  • 27

    0

  • 26

    0

  • 25

    1

  • 24

    1

  • 23

    1

  • 22

    0

  • 21

    0

  • 20

    1

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

000 0000 1000 1100 1000 0110 1000 0100 1000 0100 0111 0000 0110 1010 0011 1001(2) =

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

(0 + 0 + 0 + 0 + 0 + 0 + 0 + 36 028 797 018 963 968 + 0 + 0 + 0 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 0 + 0 + 140 737 488 355 328 + 0 + 0 + 0 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 0 + 549 755 813 888 + 0 + 0 + 0 + 0 + 17 179 869 184 + 0 + 0 + 2 147 483 648 + 0 + 0 + 0 + 0 + 67 108 864 + 0 + 0 + 0 + 4 194 304 + 2 097 152 + 1 048 576 + 0 + 0 + 0 + 0 + 0 + 16 384 + 8 192 + 0 + 2 048 + 0 + 512 + 0 + 0 + 0 + 32 + 16 + 8 + 0 + 0 + 1)(10) =

(36 028 797 018 963 968 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 140 737 488 355 328 + 4 398 046 511 104 + 2 199 023 255 552 + 549 755 813 888 + 17 179 869 184 + 2 147 483 648 + 67 108 864 + 4 194 304 + 2 097 152 + 1 048 576 + 16 384 + 8 192 + 2 048 + 512 + 32 + 16 + 8 + 1)(10) =

39 554 400 455 256 633(10)

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

0000 0000 1000 1100 1000 0110 1000 0100 1000 0100 0111 0000 0110 1010 0011 1001(2) = 39 554 400 455 256 633(10)

0000 0000 1000 1100 1000 0110 1000 0100 1000 0100 0111 0000 0110 1010 0011 1001(2), Base 2 signed binary number, converted and written as a base 10 decimal system equivalent integer:
0000 0000 1000 1100 1000 0110 1000 0100 1000 0100 0111 0000 0110 1010 0011 1001(2) = 39 554 400 455 256 633(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