Base ten decimal system signed integer number 10 111 000 001 converted to signed binary

How to convert the signed integer in decimal system (in base 10):
10 111 000 001(10)
to a signed binary

1. Divide the number repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:

  • division = quotient + remainder;
  • 10 111 000 001 ÷ 2 = 5 055 500 000 + 1;
  • 5 055 500 000 ÷ 2 = 2 527 750 000 + 0;
  • 2 527 750 000 ÷ 2 = 1 263 875 000 + 0;
  • 1 263 875 000 ÷ 2 = 631 937 500 + 0;
  • 631 937 500 ÷ 2 = 315 968 750 + 0;
  • 315 968 750 ÷ 2 = 157 984 375 + 0;
  • 157 984 375 ÷ 2 = 78 992 187 + 1;
  • 78 992 187 ÷ 2 = 39 496 093 + 1;
  • 39 496 093 ÷ 2 = 19 748 046 + 1;
  • 19 748 046 ÷ 2 = 9 874 023 + 0;
  • 9 874 023 ÷ 2 = 4 937 011 + 1;
  • 4 937 011 ÷ 2 = 2 468 505 + 1;
  • 2 468 505 ÷ 2 = 1 234 252 + 1;
  • 1 234 252 ÷ 2 = 617 126 + 0;
  • 617 126 ÷ 2 = 308 563 + 0;
  • 308 563 ÷ 2 = 154 281 + 1;
  • 154 281 ÷ 2 = 77 140 + 1;
  • 77 140 ÷ 2 = 38 570 + 0;
  • 38 570 ÷ 2 = 19 285 + 0;
  • 19 285 ÷ 2 = 9 642 + 1;
  • 9 642 ÷ 2 = 4 821 + 0;
  • 4 821 ÷ 2 = 2 410 + 1;
  • 2 410 ÷ 2 = 1 205 + 0;
  • 1 205 ÷ 2 = 602 + 1;
  • 602 ÷ 2 = 301 + 0;
  • 301 ÷ 2 = 150 + 1;
  • 150 ÷ 2 = 75 + 0;
  • 75 ÷ 2 = 37 + 1;
  • 37 ÷ 2 = 18 + 1;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above:

10 111 000 001(10) = 10 0101 1010 1010 1001 1001 1101 1100 0001(2)

3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 34.

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; ...

First bit (the leftmost) is reserved for the sign:
0 = positive integer number, 1 = negative integer number

The least number that is a power of 2 and is larger than the actual length so that the first bit (leftmost) could be zero is: 64.

4. Positive binary computer representation on 64 bits (8 Bytes) - if needed, add extra 0s in front (to the left) of the base 2 number, up to the required length:

10 111 000 001(10) = 0000 0000 0000 0000 0000 0000 0000 0010 0101 1010 1010 1001 1001 1101 1100 0001

Conclusion:

Number 10 111 000 001, a signed integer, converted from decimal system (base 10) to signed binary:
10 111 000 001(10) = 0000 0000 0000 0000 0000 0000 0000 0010 0101 1010 1010 1001 1001 1101 1100 0001

First bit (the leftmost) is reserved for the sign:
0 = positive integer number, 1 = negative integer number


Spaces used to group numbers digits: for binary, by 4; for decimal, by 3.

Convert signed integer numbers from the decimal system (base ten) to signed binary

How to convert a base ten signed integer number to signed binary:

1) Divide the positive version of number repeatedly by 2, keeping track of each remainder, till we get a quotient that is zero.

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 zero.

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.

Latest signed integers numbers converted from decimal (base ten) to signed binary

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