Signed binary number 0101 0111 0100 0010 1001 0111 0110 1111 0111 1100 1011 1010 1110 1010 1110 1101 converted to an integer in base ten

How to convert a signed binary:
0101 0111 0100 0010 1001 0111 0110 1111 0111 1100 1011 1010 1110 1010 1110 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.

0101 0111 0100 0010 1001 0111 0110 1111 0111 1100 1011 1010 1110 1010 1110 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:

0101 0111 0100 0010 1001 0111 0110 1111 0111 1100 1011 1010 1110 1010 1110 1101 = 101 0111 0100 0010 1001 0111 0110 1111 0111 1100 1011 1010 1110 1010 1110 1101

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

    • 262

      1
    • 261

      0
    • 260

      1
    • 259

      0
    • 258

      1
    • 257

      1
    • 256

      1
    • 255

      0
    • 254

      1
    • 253

      0
    • 252

      0
    • 251

      0
    • 250

      0
    • 249

      1
    • 248

      0
    • 247

      1
    • 246

      0
    • 245

      0
    • 244

      1
    • 243

      0
    • 242

      1
    • 241

      1
    • 240

      1
    • 239

      0
    • 238

      1
    • 237

      1
    • 236

      0
    • 235

      1
    • 234

      1
    • 233

      1
    • 232

      1
    • 231

      0
    • 230

      1
    • 229

      1
    • 228

      1
    • 227

      1
    • 226

      1
    • 225

      0
    • 224

      0
    • 223

      1
    • 222

      0
    • 221

      1
    • 220

      1
    • 219

      1
    • 218

      0
    • 217

      1
    • 216

      0
    • 215

      1
    • 214

      1
    • 213

      1
    • 212

      0
    • 211

      1
    • 210

      0
    • 29

      1
    • 28

      0
    • 27

      1
    • 26

      1
    • 25

      1
    • 24

      0
    • 23

      1
    • 22

      1
    • 21

      0
    • 20

      1

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

101 0111 0100 0010 1001 0111 0110 1111 0111 1100 1011 1010 1110 1010 1110 1101(2) =


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


(4 611 686 018 427 387 904 + 0 + 1 152 921 504 606 846 976 + 0 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 0 + 18 014 398 509 481 984 + 0 + 0 + 0 + 0 + 562 949 953 421 312 + 0 + 140 737 488 355 328 + 0 + 0 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 0 + 274 877 906 944 + 137 438 953 472 + 0 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 0 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 0 + 0 + 8 388 608 + 0 + 2 097 152 + 1 048 576 + 524 288 + 0 + 131 072 + 0 + 32 768 + 16 384 + 8 192 + 0 + 2 048 + 0 + 512 + 0 + 128 + 64 + 32 + 0 + 8 + 4 + 0 + 1)(10) =


(4 611 686 018 427 387 904 + 1 152 921 504 606 846 976 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 18 014 398 509 481 984 + 562 949 953 421 312 + 140 737 488 355 328 + 17 592 186 044 416 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 274 877 906 944 + 137 438 953 472 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 8 388 608 + 2 097 152 + 1 048 576 + 524 288 + 131 072 + 32 768 + 16 384 + 8 192 + 2 048 + 512 + 128 + 64 + 32 + 8 + 4 + 1)(10) =


6 287 754 534 852 422 381(10)

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

0101 0111 0100 0010 1001 0111 0110 1111 0111 1100 1011 1010 1110 1010 1110 1101(2) = 6 287 754 534 852 422 381(10)

Conclusion:

Number 0101 0111 0100 0010 1001 0111 0110 1111 0111 1100 1011 1010 1110 1010 1110 1101(2) converted from signed binary to an integer in decimal system (in base 10):


0101 0111 0100 0010 1001 0111 0110 1111 0111 1100 1011 1010 1110 1010 1110 1101(2) = 6 287 754 534 852 422 381(10)

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

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)

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