Signed binary number 1111 1111 1111 1111 1001 1000 0010 0101 converted to an integer in base ten

Signed binary 1111 1111 1111 1111 1001 1000 0010 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.

1111 1111 1111 1111 1001 1000 0010 0101 is the binary representation of a negative integer, on 32 bits (4 Bytes).


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

1111 1111 1111 1111 1001 1000 0010 0101 = 111 1111 1111 1111 1001 1000 0010 0101

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

    • 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

      0

    • 29

      0

    • 28

      0

    • 27

      0

    • 26

      0

    • 25

      1

    • 24

      0

    • 23

      0

    • 22

      1

    • 21

      0

    • 20

      1

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

111 1111 1111 1111 1001 1000 0010 0101(2) =

(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 + 0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20)(10) =

(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 + 0 + 0 + 0 + 0 + 0 + 32 + 0 + 0 + 4 + 0 + 1)(10) =

(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 + 32 + 4 + 1)(10) =

2 147 457 061(10)

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

1111 1111 1111 1111 1001 1000 0010 0101(2) = -2 147 457 061(10)

Number 1111 1111 1111 1111 1001 1000 0010 0101(2) converted from signed binary to an integer in decimal system (in base 10):
1111 1111 1111 1111 1001 1000 0010 0101(2) = -2 147 457 061(10)

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

More operations of this kind:

1111 1111 1111 1111 1001 1000 0010 0100 = ?

1111 1111 1111 1111 1001 1000 0010 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)

1111 1111 1111 1111 1001 1000 0010 0101 = -2,147,457,061 Mar 08 12:22 UTC (GMT)
0000 0000 0000 0001 0000 0000 0100 1101 = 65,613 Mar 08 12:22 UTC (GMT)
0111 0111 = 119 Mar 08 12:22 UTC (GMT)
0001 1011 0011 1000 = 6,968 Mar 08 12:22 UTC (GMT)
0111 0111 0100 1110 = 30,542 Mar 08 12:21 UTC (GMT)
1011 0101 1101 0011 = -13,779 Mar 08 12:21 UTC (GMT)
0000 0000 0000 0000 0000 1111 1010 1000 = 4,008 Mar 08 12:21 UTC (GMT)
0000 0000 1000 0100 1100 0010 0100 0000 1000 1110 1101 0010 1100 0110 1100 0100 = 37,368,279,455,680,196 Mar 08 12:20 UTC (GMT)
1100 1010 1101 1000 = -19,160 Mar 08 12:20 UTC (GMT)
1011 1101 0100 0000 0000 0000 0000 1110 = -1,027,604,494 Mar 08 12:20 UTC (GMT)
0001 0011 0101 1001 = 4,953 Mar 08 12:20 UTC (GMT)
1100 0000 1100 1111 1111 1111 1111 1100 = -1,087,373,308 Mar 08 12:20 UTC (GMT)
0000 0000 0000 0000 0000 0000 1011 1011 0001 1101 0101 1001 1011 0101 0000 0100 = 803,651,302,660 Mar 08 12:19 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