Signed: Binary ↘ Integer: 1010 0100 1101 1100 1100 0111 0001 1000 1011 1011 0011 0001 0011 0010 1010 1001 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 1010 0100 1101 1100 1100 0111 0001 1000 1011 1011 0011 0001 0011 0010 1010 1001(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

1010 0100 1101 1100 1100 0111 0001 1000 1011 1011 0011 0001 0011 0010 1010 1001 is the binary representation of a negative 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:


1010 0100 1101 1100 1100 0111 0001 1000 1011 1011 0011 0001 0011 0010 1010 1001 = 010 0100 1101 1100 1100 0111 0001 1000 1011 1011 0011 0001 0011 0010 1010 1001


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

    0
  • 258

    1
  • 257

    0
  • 256

    0
  • 255

    1
  • 254

    1
  • 253

    0
  • 252

    1
  • 251

    1
  • 250

    1
  • 249

    0
  • 248

    0
  • 247

    1
  • 246

    1
  • 245

    0
  • 244

    0
  • 243

    0
  • 242

    1
  • 241

    1
  • 240

    1
  • 239

    0
  • 238

    0
  • 237

    0
  • 236

    1
  • 235

    1
  • 234

    0
  • 233

    0
  • 232

    0
  • 231

    1
  • 230

    0
  • 229

    1
  • 228

    1
  • 227

    1
  • 226

    0
  • 225

    1
  • 224

    1
  • 223

    0
  • 222

    0
  • 221

    1
  • 220

    1
  • 219

    0
  • 218

    0
  • 217

    0
  • 216

    1
  • 215

    0
  • 214

    0
  • 213

    1
  • 212

    1
  • 211

    0
  • 210

    0
  • 29

    1
  • 28

    0
  • 27

    1
  • 26

    0
  • 25

    1
  • 24

    0
  • 23

    1
  • 22

    0
  • 21

    0
  • 20

    1

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

010 0100 1101 1100 1100 0111 0001 1000 1011 1011 0011 0001 0011 0010 1010 1001(2) =


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


(0 + 2 305 843 009 213 693 952 + 0 + 0 + 288 230 376 151 711 744 + 0 + 0 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 0 + 0 + 140 737 488 355 328 + 70 368 744 177 664 + 0 + 0 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 0 + 0 + 0 + 68 719 476 736 + 34 359 738 368 + 0 + 0 + 0 + 2 147 483 648 + 0 + 536 870 912 + 268 435 456 + 134 217 728 + 0 + 33 554 432 + 16 777 216 + 0 + 0 + 2 097 152 + 1 048 576 + 0 + 0 + 0 + 65 536 + 0 + 0 + 8 192 + 4 096 + 0 + 0 + 512 + 0 + 128 + 0 + 32 + 0 + 8 + 0 + 0 + 1)(10) =


(2 305 843 009 213 693 952 + 288 230 376 151 711 744 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 140 737 488 355 328 + 70 368 744 177 664 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 68 719 476 736 + 34 359 738 368 + 2 147 483 648 + 536 870 912 + 268 435 456 + 134 217 728 + 33 554 432 + 16 777 216 + 2 097 152 + 1 048 576 + 65 536 + 8 192 + 4 096 + 512 + 128 + 32 + 8 + 1)(10) =


2 656 216 789 275 456 169(10)

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

1010 0100 1101 1100 1100 0111 0001 1000 1011 1011 0011 0001 0011 0010 1010 1001(2) = -2 656 216 789 275 456 169(10)

The number 1010 0100 1101 1100 1100 0111 0001 1000 1011 1011 0011 0001 0011 0010 1010 1001(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
1010 0100 1101 1100 1100 0111 0001 1000 1011 1011 0011 0001 0011 0010 1010 1001(2) = -2 656 216 789 275 456 169(10)

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

The latest signed binary numbers converted and written as signed integers in decimal system (in base ten)

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