Convert the Signed Integer Number -87 000 005 to a Signed Binary in Two's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two). Detailed Explanations
Signed integer number -87 000 005(10) converted and written as a signed binary in two'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:
|-87 000 005| = 87 000 005
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;
- 87 000 005 ÷ 2 = 43 500 002 + 1;
- 43 500 002 ÷ 2 = 21 750 001 + 0;
- 21 750 001 ÷ 2 = 10 875 000 + 1;
- 10 875 000 ÷ 2 = 5 437 500 + 0;
- 5 437 500 ÷ 2 = 2 718 750 + 0;
- 2 718 750 ÷ 2 = 1 359 375 + 0;
- 1 359 375 ÷ 2 = 679 687 + 1;
- 679 687 ÷ 2 = 339 843 + 1;
- 339 843 ÷ 2 = 169 921 + 1;
- 169 921 ÷ 2 = 84 960 + 1;
- 84 960 ÷ 2 = 42 480 + 0;
- 42 480 ÷ 2 = 21 240 + 0;
- 21 240 ÷ 2 = 10 620 + 0;
- 10 620 ÷ 2 = 5 310 + 0;
- 5 310 ÷ 2 = 2 655 + 0;
- 2 655 ÷ 2 = 1 327 + 1;
- 1 327 ÷ 2 = 663 + 1;
- 663 ÷ 2 = 331 + 1;
- 331 ÷ 2 = 165 + 1;
- 165 ÷ 2 = 82 + 1;
- 82 ÷ 2 = 41 + 0;
- 41 ÷ 2 = 20 + 1;
- 20 ÷ 2 = 10 + 0;
- 10 ÷ 2 = 5 + 0;
- 5 ÷ 2 = 2 + 1;
- 2 ÷ 2 = 1 + 0;
- 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.
87 000 005(10) = 101 0010 1111 1000 0011 1100 0101(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 32 bits (4 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: 27.
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, 27,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 32.
5. Get the positive binary computer representation on 32 bits (4 Bytes):
If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 32.
87 000 005(10) = 0000 0101 0010 1111 1000 0011 1100 0101
6. Get the negative integer number representation. Part 1:
To write the negative integer number on 32 bits (4 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.
!(0000 0101 0010 1111 1000 0011 1100 0101)
= 1111 1010 1101 0000 0111 1100 0011 1010
7. Get the negative integer number representation. Part 2:
To write the negative integer number on 32 bits (4 Bytes),
as a signed binary in two's complement representation,
add 1 to the number calculated above
1111 1010 1101 0000 0111 1100 0011 1010
(to the signed binary in one's complement representation)
Binary addition carries on a value of 2:
0 + 0 = 0
0 + 1 = 1
1 + 1 = 10
1 + 10 = 11
1 + 11 = 100
Add 1 to the number calculated above
(to the signed binary number in one's complement representation):
-87 000 005 =
1111 1010 1101 0000 0111 1100 0011 1010 + 1
Number -87 000 005(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:
-87 000 005(10) = 1111 1010 1101 0000 0111 1100 0011 1011
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 two's complement representation
How to convert a base 10 signed integer number to signed binary in two's complement representation:
1) Divide the positive version of number repeatedly by 2, keeping track of each remainder, till getting a quotient that is equal to 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).
5) Only if the initial number is negative, add 1 to the number at the previous point.
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How to convert signed integers from decimal system to signed binary in two's complement representation
Follow the steps below to convert a signed base 10 integer number to signed binary in two'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, keeping track of each remainder, until we get a quotient that is 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, add extra bits on 0 in front (to the left) of the base 2 number above, up to the required length, so that the first bit (the leftmost) will 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 1s and all 1 bits with 0s (reversing the digits).
- 6. To get the negative integer number, in signed binary two's complement representation, add 1 to the number above.
Example: convert the negative number -60 from the decimal system (base ten) to signed binary in two's complement:
- 1. Start with the positive version of the number: |-60| = 60
- 2. Divide repeatedly 60 by 2, keeping track of each remainder:
- division = quotient + remainder
- 60 ÷ 2 = 30 + 0
- 30 ÷ 2 = 15 + 0
- 15 ÷ 2 = 7 + 1
- 7 ÷ 2 = 3 + 1
- 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:
60(10) = 11 1100(2) - 4. Bit length of base 2 representation number is 6, so the positive binary computer representation of a signed binary will take in this particular case 8 bits (the least power of 2 larger than 6) - add extra 0 digits in front of the base 2 number, up to the required length:
60(10) = 0011 1100(2) - 5. To get the negative integer number representation in signed binary one's complement, replace all the 0 bits with 1s and all 1 bits with 0s (reversing the digits):
!(0011 1100) = 1100 0011 - 6. To get the negative integer number, signed binary in two's complement representation, add 1 to the number above:
-60(10) = 1100 0011 + 1 = 1100 0100 Number -60(10), signed integer, converted from decimal system (base 10) to signed binary two's complement representation = 1100 0100
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