Convert the Signed Integer Number 99 999 924 to a Signed Binary. Converting and Writing the Base Ten Decimal System Signed Integer Number as a Signed Binary Code (Written in Base Two). Detailed Explanations
Signed integer number 99 999 924(10)
converted and written as a signed binary (base 2) = ?
The first steps we'll go through to make the conversion:
1. Divide the number repeatedly by 2
2. Construct the base 2 representation of the positive number
1. 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;
- 99 999 924 ÷ 2 = 49 999 962 + 0;
- 49 999 962 ÷ 2 = 24 999 981 + 0;
- 24 999 981 ÷ 2 = 12 499 990 + 1;
- 12 499 990 ÷ 2 = 6 249 995 + 0;
- 6 249 995 ÷ 2 = 3 124 997 + 1;
- 3 124 997 ÷ 2 = 1 562 498 + 1;
- 1 562 498 ÷ 2 = 781 249 + 0;
- 781 249 ÷ 2 = 390 624 + 1;
- 390 624 ÷ 2 = 195 312 + 0;
- 195 312 ÷ 2 = 97 656 + 0;
- 97 656 ÷ 2 = 48 828 + 0;
- 48 828 ÷ 2 = 24 414 + 0;
- 24 414 ÷ 2 = 12 207 + 0;
- 12 207 ÷ 2 = 6 103 + 1;
- 6 103 ÷ 2 = 3 051 + 1;
- 3 051 ÷ 2 = 1 525 + 1;
- 1 525 ÷ 2 = 762 + 1;
- 762 ÷ 2 = 381 + 0;
- 381 ÷ 2 = 190 + 1;
- 190 ÷ 2 = 95 + 0;
- 95 ÷ 2 = 47 + 1;
- 47 ÷ 2 = 23 + 1;
- 23 ÷ 2 = 11 + 1;
- 11 ÷ 2 = 5 + 1;
- 5 ÷ 2 = 2 + 1;
- 2 ÷ 2 = 1 + 0;
- 1 ÷ 2 = 0 + 1;
2. Construct the base 2 representation of the positive number:
Take all the remainders starting from the bottom of the list constructed above.
99 999 924(10) = 101 1111 0101 1110 0000 1011 0100(2)
The last steps we'll go through to make the conversion:
3. Determine the signed binary number bit length
4. Get the positive binary computer representation on 32 bits (4 Bytes)
3. 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) is reserved for 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.
4. 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:
Number 99 999 924(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
99 999 924(10) = 0000 0101 1111 0101 1110 0000 1011 0100
The first bit (the leftmost) is reserved for the sign:
0 = positive integer number, 1 = negative integer number
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 (written in base two)
How to convert a base ten signed integer number to signed binary:
1) Divide the positive version of the number repeatedly by 2, keeping track of each remainder. Stop when getting a quotient that is 0.
2) Construct the base two representation by taking the previously calculated remainders starting from the last remainder up to the first one.
3) Construct the positive binary computer representation so that the first bit is 0.
4) Only if the initial number is negative, change the first bit (the leftmost), from 0 to 1. The leftmost bit is reserved for the sign, 1 = negative, 0 = positive.
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How to convert signed integers from decimal system to binary code system
Follow the steps below to convert a signed base ten integer number to signed binary:
- 1. In a signed binary, first bit (the leftmost) is reserved for sign: 0 = positive integer number, 1 = positive integer number. If the number to be converted is negative, start with its positive version.
- 2. Divide repeatedly by 2 the positive integer number keeping track of each remainder. STOP when 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 have a length of 4, 8, 16, 32, 64, ... bits (power of 2) - if needed, fill in extra '0' bits in front of the base 2 number (to the left), up to the right length; this way the first bit (the leftmost one) is always '0', as for a positive representation.
- 5. To get the negative reprezentation of the number, simply switch the first bit (the leftmost one), from '0' to '1'.
Example: convert the negative number -63 from decimal system (base ten) to signed binary code system:
- 1. Start with the positive version of the number: |-63| = 63;
- 2. Divide repeatedly 63 by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
- division = quotient + remainder
- 63 ÷ 2 = 31 + 1
- 31 ÷ 2 = 15 + 1
- 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:
63(10) = 11 1111(2) - 4. The actual length of base 2 representation number is 6, so the positive binary computer representation length of the signed binary will take in this case 8 bits (the least power of 2 higher than 6) - add extra '0's in front (to the left), up to the required length; this way the first bit (the leftmost one) is to be '0', as for a positive number:
63(10) = 0011 1111(2) - 5. To get the negative integer number representation simply change the first bit (the leftmost), from '0' to '1':
-63(10) = 1011 1111 Number -63(10), signed integer, converted from decimal system (base 10) to signed binary = 1011 1111
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