Base ten decimal system signed integer number 100 001 100 100 001 converted to signed binary

How to convert the signed integer in decimal system (in base 10):
100 001 100 100 001(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;
  • 100 001 100 100 001 ÷ 2 = 50 000 550 050 000 + 1;
  • 50 000 550 050 000 ÷ 2 = 25 000 275 025 000 + 0;
  • 25 000 275 025 000 ÷ 2 = 12 500 137 512 500 + 0;
  • 12 500 137 512 500 ÷ 2 = 6 250 068 756 250 + 0;
  • 6 250 068 756 250 ÷ 2 = 3 125 034 378 125 + 0;
  • 3 125 034 378 125 ÷ 2 = 1 562 517 189 062 + 1;
  • 1 562 517 189 062 ÷ 2 = 781 258 594 531 + 0;
  • 781 258 594 531 ÷ 2 = 390 629 297 265 + 1;
  • 390 629 297 265 ÷ 2 = 195 314 648 632 + 1;
  • 195 314 648 632 ÷ 2 = 97 657 324 316 + 0;
  • 97 657 324 316 ÷ 2 = 48 828 662 158 + 0;
  • 48 828 662 158 ÷ 2 = 24 414 331 079 + 0;
  • 24 414 331 079 ÷ 2 = 12 207 165 539 + 1;
  • 12 207 165 539 ÷ 2 = 6 103 582 769 + 1;
  • 6 103 582 769 ÷ 2 = 3 051 791 384 + 1;
  • 3 051 791 384 ÷ 2 = 1 525 895 692 + 0;
  • 1 525 895 692 ÷ 2 = 762 947 846 + 0;
  • 762 947 846 ÷ 2 = 381 473 923 + 0;
  • 381 473 923 ÷ 2 = 190 736 961 + 1;
  • 190 736 961 ÷ 2 = 95 368 480 + 1;
  • 95 368 480 ÷ 2 = 47 684 240 + 0;
  • 47 684 240 ÷ 2 = 23 842 120 + 0;
  • 23 842 120 ÷ 2 = 11 921 060 + 0;
  • 11 921 060 ÷ 2 = 5 960 530 + 0;
  • 5 960 530 ÷ 2 = 2 980 265 + 0;
  • 2 980 265 ÷ 2 = 1 490 132 + 1;
  • 1 490 132 ÷ 2 = 745 066 + 0;
  • 745 066 ÷ 2 = 372 533 + 0;
  • 372 533 ÷ 2 = 186 266 + 1;
  • 186 266 ÷ 2 = 93 133 + 0;
  • 93 133 ÷ 2 = 46 566 + 1;
  • 46 566 ÷ 2 = 23 283 + 0;
  • 23 283 ÷ 2 = 11 641 + 1;
  • 11 641 ÷ 2 = 5 820 + 1;
  • 5 820 ÷ 2 = 2 910 + 0;
  • 2 910 ÷ 2 = 1 455 + 0;
  • 1 455 ÷ 2 = 727 + 1;
  • 727 ÷ 2 = 363 + 1;
  • 363 ÷ 2 = 181 + 1;
  • 181 ÷ 2 = 90 + 1;
  • 90 ÷ 2 = 45 + 0;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 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, by taking all the remainders starting from the bottom of the list constructed above:

100 001 100 100 001(10) = 101 1010 1111 0011 0101 0010 0000 1100 0111 0001 1010 0001(2)

3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 47.

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:

100 001 100 100 001(10) = 0000 0000 0000 0000 0101 1010 1111 0011 0101 0010 0000 1100 0111 0001 1010 0001

Conclusion:

Number 100 001 100 100 001, a signed integer, converted from decimal system (base 10) to signed binary:
100 001 100 100 001(10) = 0000 0000 0000 0000 0101 1010 1111 0011 0101 0010 0000 1100 0111 0001 1010 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