Signed: Binary -> Integer: 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1010 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1010(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

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.

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1010 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:

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1010 = 111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1010


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

      1

    • 258

      1

    • 257

      1

    • 256

      1

    • 255

      1

    • 254

      1

    • 253

      1

    • 252

      1

    • 251

      1

    • 250

      1

    • 249

      1

    • 248

      1

    • 247

      1

    • 246

      1

    • 245

      1

    • 244

      1

    • 243

      1

    • 242

      1

    • 241

      1

    • 240

      1

    • 239

      1

    • 238

      1

    • 237

      1

    • 236

      1

    • 235

      1

    • 234

      1

    • 233

      1

    • 232

      1

    • 231

      1

    • 230

      1

    • 229

      1

    • 228

      1

    • 227

      1

    • 226

      1

    • 225

      1

    • 224

      1

    • 223

      1

    • 222

      1

    • 221

      1

    • 220

      1

    • 219

      1

    • 218

      1

    • 217

      1

    • 216

      1

    • 215

      1

    • 214

      0

    • 213

      0

    • 212

      1

    • 211

      1

    • 210

      1

    • 29

      0

    • 28

      0

    • 27

      0

    • 26

      1

    • 25

      1

    • 24

      0

    • 23

      1

    • 22

      0

    • 21

      1

    • 20

      0

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

111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1010(2) =

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

(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 0 + 0 + 4 096 + 2 048 + 1 024 + 0 + 0 + 0 + 64 + 32 + 0 + 8 + 0 + 2 + 0)(10) =

(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 4 096 + 2 048 + 1 024 + 64 + 32 + 8 + 2)(10) =

9 223 372 036 854 750 314(10)

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

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1010(2) = -9 223 372 036 854 750 314(10)

The number 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1010(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1010(2) = -9 223 372 036 854 750 314(10)

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

The signed binary number 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1001 (in base two), converted and written as an integer in decimal system (in base ten) = ?

The signed binary number 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1011 (in base two), converted and written as an integer in decimal system (in base ten) = ?

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

The first bit (the leftmost) is reserved for the sign (1 = negative, 0 = positive) and it doesn't count when calculating the absolute value.

Binary number's length must be: 2, 4, 8, 16, 32, 64 - or else extra bits on 0 are 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.

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

Convert the signed binary number 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1100 0110 1010, write it as a decimal system integer number (written in base ten) Nov 28 10:49 UTC (GMT)
Convert the signed binary number 0000 0000 0001 1000 1110 0000 0110 1000, write it as a decimal system integer number (written in base ten) Nov 28 10:49 UTC (GMT)
Convert the signed binary number 0000 0000 0100 0000 0001 0101 1000 0010, write it as a decimal system integer number (written in base ten) Nov 28 10:49 UTC (GMT)
Convert the signed binary number 0100 0100 1010 0100 0010 1010 0000 1001, write it as a decimal system integer number (written in base ten) Nov 28 10:49 UTC (GMT)
Convert the signed binary number 1101 1100 1111 0010 0010 0000 0010 1001, write it as a decimal system integer number (written in base ten) Nov 28 10:49 UTC (GMT)
Convert the signed binary number 0000 0000 0000 0000 0000 0000 0000 0011 0010 0000 1000 0000 0000 1000 0000 1001, write it as a decimal system integer number (written in base ten) Nov 28 10:49 UTC (GMT)
Convert the signed binary number 0100 0001 1001 0000 0000 0000 0000 0110, write it as a decimal system integer number (written in base ten) Nov 28 10:49 UTC (GMT)
Convert the signed binary number 0000 0000 1101 1001 1001 1001 1010 0100, write it as a decimal system integer number (written in base ten) Nov 28 10:49 UTC (GMT)
Convert the signed binary number 1111 1111 1111 1111 0000 1000 1001 0101, write it as a decimal system integer number (written in base ten) Nov 28 10:49 UTC (GMT)
Convert the signed binary number 0000 0000 1010 1001 1101 1010 1101 0110, write it as a decimal system integer number (written in base ten) Nov 28 10:49 UTC (GMT)
All the signed binary numbers converted to integers in decimal system (written 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