Base ten decimal system signed integer number 1 000 000 010 000 converted to signed binary

How to convert the signed integer in decimal system (in base 10):
1 000 000 010 000(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;
  • 1 000 000 010 000 ÷ 2 = 500 000 005 000 + 0;
  • 500 000 005 000 ÷ 2 = 250 000 002 500 + 0;
  • 250 000 002 500 ÷ 2 = 125 000 001 250 + 0;
  • 125 000 001 250 ÷ 2 = 62 500 000 625 + 0;
  • 62 500 000 625 ÷ 2 = 31 250 000 312 + 1;
  • 31 250 000 312 ÷ 2 = 15 625 000 156 + 0;
  • 15 625 000 156 ÷ 2 = 7 812 500 078 + 0;
  • 7 812 500 078 ÷ 2 = 3 906 250 039 + 0;
  • 3 906 250 039 ÷ 2 = 1 953 125 019 + 1;
  • 1 953 125 019 ÷ 2 = 976 562 509 + 1;
  • 976 562 509 ÷ 2 = 488 281 254 + 1;
  • 488 281 254 ÷ 2 = 244 140 627 + 0;
  • 244 140 627 ÷ 2 = 122 070 313 + 1;
  • 122 070 313 ÷ 2 = 61 035 156 + 1;
  • 61 035 156 ÷ 2 = 30 517 578 + 0;
  • 30 517 578 ÷ 2 = 15 258 789 + 0;
  • 15 258 789 ÷ 2 = 7 629 394 + 1;
  • 7 629 394 ÷ 2 = 3 814 697 + 0;
  • 3 814 697 ÷ 2 = 1 907 348 + 1;
  • 1 907 348 ÷ 2 = 953 674 + 0;
  • 953 674 ÷ 2 = 476 837 + 0;
  • 476 837 ÷ 2 = 238 418 + 1;
  • 238 418 ÷ 2 = 119 209 + 0;
  • 119 209 ÷ 2 = 59 604 + 1;
  • 59 604 ÷ 2 = 29 802 + 0;
  • 29 802 ÷ 2 = 14 901 + 0;
  • 14 901 ÷ 2 = 7 450 + 1;
  • 7 450 ÷ 2 = 3 725 + 0;
  • 3 725 ÷ 2 = 1 862 + 1;
  • 1 862 ÷ 2 = 931 + 0;
  • 931 ÷ 2 = 465 + 1;
  • 465 ÷ 2 = 232 + 1;
  • 232 ÷ 2 = 116 + 0;
  • 116 ÷ 2 = 58 + 0;
  • 58 ÷ 2 = 29 + 0;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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:

1 000 000 010 000(10) = 1110 1000 1101 0100 1010 0101 0011 0111 0001 0000(2)

3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 40.

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:

1 000 000 010 000(10) = 0000 0000 0000 0000 0000 0000 1110 1000 1101 0100 1010 0101 0011 0111 0001 0000

Conclusion:

Number 1 000 000 010 000, a signed integer, converted from decimal system (base 10) to signed binary:
1 000 000 010 000(10) = 0000 0000 0000 0000 0000 0000 1110 1000 1101 0100 1010 0101 0011 0111 0001 0000

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