Signed binary number 1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101 converted to an integer in base ten

Signed binary 1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101(2) to an integer in decimal system (in base 10) = ?

1. Is this a positive or a negative number?


In a signed binary, first bit (the leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value.

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101 is the binary representation of a negative integer, on 64 bits (8 Bytes).


2. Construct the unsigned binary number, exclude the first bit (the leftmost), that is reserved for the sign:

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101 = 100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101

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

    • 262

      1
    • 261

      0
    • 260

      0
    • 259

      0
    • 258

      0
    • 257

      0
    • 256

      0
    • 255

      0
    • 254

      1
    • 253

      0
    • 252

      1
    • 251

      1
    • 250

      0
    • 249

      1
    • 248

      1
    • 247

      1
    • 246

      0
    • 245

      1
    • 244

      1
    • 243

      1
    • 242

      0
    • 241

      1
    • 240

      1
    • 239

      1
    • 238

      0
    • 237

      1
    • 236

      1
    • 235

      1
    • 234

      0
    • 233

      1
    • 232

      0
    • 231

      0
    • 230

      1
    • 229

      0
    • 228

      0
    • 227

      0
    • 226

      1
    • 225

      0
    • 224

      1
    • 223

      1
    • 222

      0
    • 221

      1
    • 220

      0
    • 219

      0
    • 218

      0
    • 217

      0
    • 216

      0
    • 215

      0
    • 214

      1
    • 213

      0
    • 212

      1
    • 211

      1
    • 210

      0
    • 29

      1
    • 28

      0
    • 27

      1
    • 26

      1
    • 25

      1
    • 24

      1
    • 23

      0
    • 22

      1
    • 21

      0
    • 20

      1

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

100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101(2) =


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


(4 611 686 018 427 387 904 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 281 474 976 710 656 + 140 737 488 355 328 + 0 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 0 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 0 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 0 + 8 589 934 592 + 0 + 0 + 1 073 741 824 + 0 + 0 + 0 + 67 108 864 + 0 + 16 777 216 + 8 388 608 + 0 + 2 097 152 + 0 + 0 + 0 + 0 + 0 + 0 + 16 384 + 0 + 4 096 + 2 048 + 0 + 512 + 0 + 128 + 64 + 32 + 16 + 0 + 4 + 0 + 1)(10) =


(4 611 686 018 427 387 904 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 562 949 953 421 312 + 281 474 976 710 656 + 140 737 488 355 328 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 8 589 934 592 + 1 073 741 824 + 67 108 864 + 16 777 216 + 8 388 608 + 2 097 152 + 16 384 + 4 096 + 2 048 + 512 + 128 + 64 + 32 + 16 + 4 + 1)(10) =


4 637 506 650 014 505 717(10)

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

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101(2) = -4 637 506 650 014 505 717(10)

Number 1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101(2) converted from signed binary to an integer in decimal system (in base 10):
1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101(2) = -4 637 506 650 014 505 717(10)

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


More operations of this kind:

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0100 = ?

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110 = ?


Convert signed binary numbers to integers in decimal system (base 10)

First bit (the leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value.

Entered binary number length must be: 2, 4, 8, 16, 32, or 64 - otherwise extra bits on 0 will be added in front (to the left).

How to convert a signed binary number to an integer in base ten:

1) Construct the unsigned binary number: exclude the first bit (the leftmost); this bit is reserved for the sign, 1 = negative, 0 = positive and does not count when calculating the absolute value (without sign).

2) Multiply each bit of the binary number by its corresponding power of 2 that its place value represents.

3) Add all the terms up to get the positive integer number in base ten.

4) Adjust the sign of the integer number by the first bit of the initial binary number.

Latest signed binary numbers converted to signed integers in decimal system (base ten)

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101 = -4,637,506,650,014,505,717 Jul 24 10:48 UTC (GMT)
0010 0011 0001 0000 = 8,976 Jul 24 10:48 UTC (GMT)
0000 0000 0000 0000 0000 0000 0000 1111 0000 0000 0000 0000 0000 0000 0000 0000 = 64,424,509,440 Jul 24 10:48 UTC (GMT)
0000 0000 0000 1100 1110 0010 0000 0010 0110 0101 0111 0011 1000 0110 0011 1101 = 3,626,199,640,409,661 Jul 24 10:47 UTC (GMT)
1111 1100 1110 0011 = -31,971 Jul 24 10:47 UTC (GMT)
0010 0100 0010 0100 0000 0001 0000 0100 0000 0001 0010 0100 1010 1010 1100 1101 = 2,604,207,601,237,666,509 Jul 24 10:47 UTC (GMT)
0000 0000 0001 0110 0000 0100 0001 0011 = 1,442,835 Jul 24 10:47 UTC (GMT)
0000 0000 0000 0000 0000 0000 0000 0000 0001 0101 1111 1111 1110 1011 0001 0010 = 369,093,394 Jul 24 10:47 UTC (GMT)
0011 0101 0000 1001 = 13,577 Jul 24 10:47 UTC (GMT)
0000 0001 1101 1010 = 474 Jul 24 10:47 UTC (GMT)
0000 0000 0100 1111 1111 1111 1111 1101 = 5,242,877 Jul 24 10:47 UTC (GMT)
1010 1100 1111 1111 1111 1111 1111 1011 = -754,974,715 Jul 24 10:47 UTC (GMT)
1111 1111 1111 1111 1111 1111 1111 1111 1011 1111 1111 1111 1111 1111 1111 1000 = -9,223,372,035,781,033,976 Jul 24 10:46 UTC (GMT)
All the converted signed binary numbers to integers 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