Signed binary number 0111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1101 converted to an integer in base ten

Signed binary 0111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1101(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.

0111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1101 is the binary representation of a positive integer, on 64 bits (8 Bytes).


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

0111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1101 = 111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1101

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

    • 262

      1
    • 261

      1
    • 260

      1
    • 259

      0
    • 258

      0
    • 257

      1
    • 256

      0
    • 255

      1
    • 254

      0
    • 253

      1
    • 252

      0
    • 251

      1
    • 250

      0
    • 249

      1
    • 248

      0
    • 247

      0
    • 246

      1
    • 245

      1
    • 244

      1
    • 243

      0
    • 242

      1
    • 241

      0
    • 240

      1
    • 239

      0
    • 238

      0
    • 237

      1
    • 236

      1
    • 235

      0
    • 234

      1
    • 233

      0
    • 232

      1
    • 231

      0
    • 230

      0
    • 229

      1
    • 228

      1
    • 227

      0
    • 226

      0
    • 225

      1
    • 224

      0
    • 223

      1
    • 222

      1
    • 221

      0
    • 220

      1
    • 219

      0
    • 218

      1
    • 217

      0
    • 216

      0
    • 215

      0
    • 214

      1
    • 213

      1
    • 212

      0
    • 211

      0
    • 210

      0
    • 29

      1
    • 28

      1
    • 27

      0
    • 26

      0
    • 25

      1
    • 24

      1
    • 23

      1
    • 22

      1
    • 21

      0
    • 20

      1

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

111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1101(2) =


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


(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 0 + 0 + 144 115 188 075 855 872 + 0 + 36 028 797 018 963 968 + 0 + 9 007 199 254 740 992 + 0 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 0 + 0 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 0 + 0 + 137 438 953 472 + 68 719 476 736 + 0 + 17 179 869 184 + 0 + 4 294 967 296 + 0 + 0 + 536 870 912 + 268 435 456 + 0 + 0 + 33 554 432 + 0 + 8 388 608 + 4 194 304 + 0 + 1 048 576 + 0 + 262 144 + 0 + 0 + 0 + 16 384 + 8 192 + 0 + 0 + 0 + 512 + 256 + 0 + 0 + 32 + 16 + 8 + 4 + 0 + 1)(10) =


(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 144 115 188 075 855 872 + 36 028 797 018 963 968 + 9 007 199 254 740 992 + 2 251 799 813 685 248 + 562 949 953 421 312 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 4 398 046 511 104 + 1 099 511 627 776 + 137 438 953 472 + 68 719 476 736 + 17 179 869 184 + 4 294 967 296 + 536 870 912 + 268 435 456 + 33 554 432 + 8 388 608 + 4 194 304 + 1 048 576 + 262 144 + 16 384 + 8 192 + 512 + 256 + 32 + 16 + 8 + 4 + 1)(10) =


8 262 545 337 711 092 541(10)

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

0111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1101(2) = 8 262 545 337 711 092 541(10)

Number 0111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1101(2) converted from signed binary to an integer in decimal system (in base 10):
0111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1101(2) = 8 262 545 337 711 092 541(10)

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


More operations of this kind:

0111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1100 = ?

0111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1110 = ?


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)

0111 0010 1010 1010 0111 0101 0011 0101 0011 0010 1101 0100 0110 0011 0011 1101 = 8,262,545,337,711,092,541 Nov 30 09:06 UTC (GMT)
0011 1001 1111 0100 = 14,836 Nov 30 09:06 UTC (GMT)
1111 1111 1111 1111 1101 0010 1000 0000 = -2,147,472,000 Nov 30 09:06 UTC (GMT)
0101 1101 = 93 Nov 30 09:05 UTC (GMT)
0011 0011 0001 0111 = 13,079 Nov 30 09:05 UTC (GMT)
0100 1010 1011 1101 = 19,133 Nov 30 09:04 UTC (GMT)
1111 1111 1111 1111 1111 0110 0100 0100 = -2,147,481,156 Nov 30 09:04 UTC (GMT)
1011 1101 1100 0001 = -15,809 Nov 30 09:04 UTC (GMT)
0000 0000 0000 0000 1110 1100 0111 1110 = 60,542 Nov 30 09:04 UTC (GMT)
0111 1001 1101 0110 = 31,190 Nov 30 09:03 UTC (GMT)
0001 1101 0110 1100 = 7,532 Nov 30 09:03 UTC (GMT)
1110 0000 1001 1101 = -24,733 Nov 30 09:03 UTC (GMT)
1100 0011 1010 1001 1001 1111 1111 1001 = -1,135,190,009 Nov 30 09:02 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