Convert the Signed Integer Number -908 to a Signed Binary in One's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two). Detailed Explanations
Signed integer number -908(10) converted and written as a signed binary in one's complement representation (base 2) = ?
The first steps we'll go through to make the conversion:
1. Start with the positive version of the number
2. Divide the number repeatedly by 2
3. Construct the base 2 representation of the positive number
1. Start with the positive version of the number:
|-908| = 908
2. Divide the number repeatedly by 2:
Keep track of each remainder.
We stop when we get a quotient that is equal to zero.
- division = quotient + remainder;
- 908 ÷ 2 = 454 + 0;
- 454 ÷ 2 = 227 + 0;
- 227 ÷ 2 = 113 + 1;
- 113 ÷ 2 = 56 + 1;
- 56 ÷ 2 = 28 + 0;
- 28 ÷ 2 = 14 + 0;
- 14 ÷ 2 = 7 + 0;
- 7 ÷ 2 = 3 + 1;
- 3 ÷ 2 = 1 + 1;
- 1 ÷ 2 = 0 + 1;
3. Construct the base 2 representation of the positive number:
Take all the remainders starting from the bottom of the list constructed above.
908(10) = 11 1000 1100(2)
The last steps we'll go through to make the conversion:
4. Determine the signed binary number bit length
5. Get the positive binary computer representation on 16 bits (2 Bytes)
6. Get the negative integer number representation
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 10.
A signed binary's bit length must be equal to a power of 2, as of:
21 = 2; 22 = 4; 23 = 8; 24 = 16; 25 = 32; 26 = 64; ...
The first bit (the leftmost) indicates the sign:
0 = positive integer number, 1 = negative integer number
The least number that is:
1) a power of 2
2) and is larger than the actual length, 10,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 16.
5. Get the positive binary computer representation on 16 bits (2 Bytes):
If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 16.
908(10) = 0000 0011 1000 1100
6. Get the negative integer number representation:
To write the negative integer number on 16 bits (2 Bytes),
as a signed binary in one's complement representation,
... replace all the bits on 0 with 1s and all the bits set on 1 with 0s.
Reverse the digits, flip the digits:
Replace the bits set on 0 with 1s and the bits set on 1 with 0s.
-908(10) = !(0000 0011 1000 1100)
Number -908(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in one's complement representation:
-908(10) = 1111 1100 0111 0011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
Convert signed integer numbers from the decimal system (base ten) to signed binary in one's complement representation
How to convert a base 10 signed integer number to signed binary in one's complement representation:
1) Divide the positive version of the number repeatedly by 2, keeping track of each remainder, till getting a quotient that is 0.
2) Construct the base 2 representation by taking the previously calculated remainders starting from the last remainder up to the first one, in that order.
3) Construct the positive binary computer representation so that the first bit is 0.
4) Only if the initial number is negative, switch all the bits from 0 to 1 and from 1 to 0 (reversing the digits).
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How to convert signed integers from the decimal system to signed binary in one's complement representation
Follow the steps below to convert a signed base 10 integer number to signed binary in one's complement representation:
- 1. If the number to be converted is negative, start with the positive version of the number.
- 2. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, keeping track of each remainder, until we get a quotient that is equal to ZERO.
- 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
- 4. Binary numbers represented in computer language must have 4, 8, 16, 32, 64, ... bit length (a power of 2) - if needed, fill in '0' bits in front (to the left) of the base 2 number calculated above, up to the right length; this way the first bit (leftmost) will always be '0', correctly representing a positive number.
- 5. To get the negative integer number representation in signed binary one's complement, replace all '0' bits with '1's and all '1' bits with '0's.
Example: convert the negative number -49 from the decimal system (base ten) to signed binary one's complement:
- 1. Start with the positive version of the number: |-49| = 49
- 2. Divide repeatedly 49 by 2, keeping track of each remainder:
- division = quotient + remainder
- 49 ÷ 2 = 24 + 1
- 24 ÷ 2 = 12 + 0
- 12 ÷ 2 = 6 + 0
- 6 ÷ 2 = 3 + 0
- 3 ÷ 2 = 1 + 1
- 1 ÷ 2 = 0 + 1
- 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above:
49(10) = 11 0001(2) - 4. The actual bit length of base 2 representation is 6, so the positive binary computer representation of a signed binary will take in this case 8 bits (the least power of 2 that is larger than 6) - add '0's in front of the base 2 number, up to the required length:
49(10) = 0011 0001(2) - 5. To get the negative integer number representation in signed binary one's complement, replace all '0' bits with '1's and all '1' bits with '0's:
-49(10) = 1100 1110 Number -49(10), signed integer, converted from the decimal system (base 10) to signed binary in one's complement representation = 1100 1110
Available Base Conversions Between Decimal and Binary Systems
Conversions Between Decimal System Numbers (Written in Base Ten) and Binary System Numbers (Base Two and Computer Representation):
1. Integer -> Binary
2. Decimal -> Binary
3. Binary -> Integer
4. Binary -> Decimal