0010 0101 0101 0111 Signed Binary Number in One's Complement Representation, Converted and Written as a Decimal System Integer Number (in Base Ten). Steps Explained in Detail

The signed binary in one's complement representation 0010 0101 0101 0110 converted and written as an integer number in decimal system (in base ten) = ?

The signed binary in one's complement representation 0010 0101 0101 1000 converted and written as an integer number in decimal system (in base ten) = ?

How to convert a signed binary number in one's complement representation to an integer in base ten:

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

2) 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.

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

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

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

To understand how to convert a signed binary number in one's complement representation from binary system to decimal (base ten), the easiest way is to do it through an example - convert binary, 1001 1101, to base ten:

• In a signed binary one's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive. The first bit is 1, so our number is negative.
• 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:
  !(1001 1101) = 0110 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 by increasing each corresonding power of 2 by exactly one unit:

• powers of 2:	7	6	5	4	3	2	1	0
  digits:	0	1	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:
  

  0110 0010₍₂₎ =

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

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  (0 + 64 + 32 + 0 + 0 + 0 + 2 + 0)₍₁₀₎ =

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  (64 + 32 + 2)₍₁₀₎ =

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  98₍₁₀₎

• Signed binary number in one's complement representation, 1001 1110 = -98₍₁₀₎, a signed negative integer in base 10

Conversions Between Decimal System Numbers (Written in Base Ten) and Binary System Numbers (Base Two and Computer Representation):

1. Integer -> Binary

• 1.1. Unsigned integer -> Unsigned binary

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  1.2. Signed integer -> Signed binary

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  1.3. Signed integer -> Signed binary one's complement

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  1.4. Signed integer -> Signed binary two's complement

2. Decimal -> Binary

• 2.1. Decimal -> 32bit single precision IEEE 754 binary floating point

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  2.2. Decimal -> 64bit double precision IEEE 754 binary floating point

3. Binary -> Integer

• 3.1. Unsigned binary -> Unsigned integer

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  3.2. Signed binary -> Signed integer

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  3.3. Signed binary one's complement -> Signed integer

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  3.4. Signed binary two's complement -> Signed integer

4. Binary -> Decimal

• 4.1. 32bit single precision IEEE 754 binary floating point -> Float

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  4.2. 64bit double precision IEEE 754 binary floating point -> Double