Signed binary two's complement number 1100 0101 converted to decimal system (base ten) signed integer

Signed binary two's complement 1100 0101(2) to an integer in decimal system (in base 10) = ?

1. Is this a positive or a negative number?

In a signed binary two's complement,

The first bit (the leftmost) indicates the sign,

1 = negative, 0 = positive.

1100 0101 is the binary representation of a negative integer, on 8 bits.


2. Get the binary representation in one's complement:

* Run this step only if the number is negative *

Note: 11 - 1 = 10; 10 - 1 = 1; 1 - 0 = 1; 1 - 1 = 0.

Subtract 1 from the initial binary number.

1100 0101 - 1 = 1100 0100


3. Get the binary representation of the positive (unsigned) number:

* Run this step only if the number is negative *

Flip all the bits in the signed binary one's complement representation (reverse the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:

!(1100 0100) = 0011 1011


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

    • 27

      0

    • 26

      0

    • 25

      1

    • 24

      1

    • 23

      1

    • 22

      0

    • 21

      1

    • 20

      1

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

0011 1011(2) =

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

(0 + 0 + 32 + 16 + 8 + 0 + 2 + 1)(10) =

(32 + 16 + 8 + 2 + 1)(10) =

59(10)

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

1100 0101(2) = -59(10)

Number 1100 0101(2) converted from signed binary two's complement representation to an integer in decimal system (in base 10):
1100 0101(2) = -59(10)

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

More operations of this kind:

1100 0100 = ?

1100 0110 = ?


Convert signed binary two's complement numbers to decimal system (base ten) integers

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 in two's complement representation to an integer in base ten:

1) In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive.

2) Get the signed binary representation in one's complement, subtract 1 from the initial number.

3) Construct the unsigned binary number: flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s.

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

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

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

Latest binary numbers in two's complement representation converted to signed integers in decimal system (base ten)

1100 0101 = -59 Mar 24 09:54 UTC (GMT)
1101 1111 1001 0000 0000 0000 0000 0001 = -544,210,943 Mar 24 09:53 UTC (GMT)
1111 1101 0011 0000 = -720 Mar 24 09:53 UTC (GMT)
0101 1001 1011 0001 = 22,961 Mar 24 09:53 UTC (GMT)
1111 1011 0011 0000 = -1,232 Mar 24 09:53 UTC (GMT)
0111 1111 1111 1111 1111 1001 1101 0000 = 2,147,482,064 Mar 24 09:53 UTC (GMT)
10 = -2 Mar 24 09:52 UTC (GMT)
0110 0110 1000 0011 = 26,243 Mar 24 09:52 UTC (GMT)
1110 0111 1111 1010 = -6,150 Mar 24 09:52 UTC (GMT)
0000 0000 0000 0000 0000 0000 0000 0001 0101 0001 1101 0101 1111 1111 1110 1110 = 5,667,946,478 Mar 24 09:52 UTC (GMT)
0100 0100 1101 0000 0000 0000 0000 0001 = 1,154,482,177 Mar 24 09:52 UTC (GMT)
1100 0000 1011 0000 0000 0000 0001 0000 = -1,062,207,472 Mar 24 09:51 UTC (GMT)
0100 0000 1100 1100 0000 0000 0001 0101 = 1,087,111,189 Mar 24 09:50 UTC (GMT)
All the converted signed binary two's complement numbers

How to convert signed binary numbers in two's complement representation from binary system to decimal

To understand how to convert a signed binary number in two's complement representation from the binary system to decimal (base ten), the easiest way is to do it by an example - convert binary, 1101 1110, to base ten:

  • In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive. The first bit is 1, so our number is negative.
  • Get the signed binary representation in one's complement, subtract 1 from the initial number:
    1101 1110 - 1 = 1101 1101
  • Get the binary representation of the positive number, flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:
    !(1101 1101) = 0010 0010
  • Write bellow the positive binary number representation 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, increasing each corresonding power of 2 by exactly one unit:
  • powers of 2: 7 6 5 4 3 2 1 0
    digits: 0 0 1 0 0 0 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:

    0010 0010(2) =

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

    (0 + 0 + 32 + 0 + 0 + 0 + 2 + 0)(10) =

    (32 + 2)(10) =

    34(10)

  • Signed binary number in two's complement representation, 1101 1110 = -34(10), a signed negative integer in base 10