Convert 144 680 345 658 434 397 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 144 680 345 658 434 397(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
144 680 345 658 434 397 to a signed binary in two's (2's) complement representation?

  • A signed integer, written in base ten, or a decimal system number, is a number written using the digits 0 through 9 and the sign, which can be positive (+) or negative (-). If positive, the sign is usually not written. A number written in base two, or binary, is a number written using only the digits 0 and 1.

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;
  • 144 680 345 658 434 397 ÷ 2 = 72 340 172 829 217 198 + 1;
  • 72 340 172 829 217 198 ÷ 2 = 36 170 086 414 608 599 + 0;
  • 36 170 086 414 608 599 ÷ 2 = 18 085 043 207 304 299 + 1;
  • 18 085 043 207 304 299 ÷ 2 = 9 042 521 603 652 149 + 1;
  • 9 042 521 603 652 149 ÷ 2 = 4 521 260 801 826 074 + 1;
  • 4 521 260 801 826 074 ÷ 2 = 2 260 630 400 913 037 + 0;
  • 2 260 630 400 913 037 ÷ 2 = 1 130 315 200 456 518 + 1;
  • 1 130 315 200 456 518 ÷ 2 = 565 157 600 228 259 + 0;
  • 565 157 600 228 259 ÷ 2 = 282 578 800 114 129 + 1;
  • 282 578 800 114 129 ÷ 2 = 141 289 400 057 064 + 1;
  • 141 289 400 057 064 ÷ 2 = 70 644 700 028 532 + 0;
  • 70 644 700 028 532 ÷ 2 = 35 322 350 014 266 + 0;
  • 35 322 350 014 266 ÷ 2 = 17 661 175 007 133 + 0;
  • 17 661 175 007 133 ÷ 2 = 8 830 587 503 566 + 1;
  • 8 830 587 503 566 ÷ 2 = 4 415 293 751 783 + 0;
  • 4 415 293 751 783 ÷ 2 = 2 207 646 875 891 + 1;
  • 2 207 646 875 891 ÷ 2 = 1 103 823 437 945 + 1;
  • 1 103 823 437 945 ÷ 2 = 551 911 718 972 + 1;
  • 551 911 718 972 ÷ 2 = 275 955 859 486 + 0;
  • 275 955 859 486 ÷ 2 = 137 977 929 743 + 0;
  • 137 977 929 743 ÷ 2 = 68 988 964 871 + 1;
  • 68 988 964 871 ÷ 2 = 34 494 482 435 + 1;
  • 34 494 482 435 ÷ 2 = 17 247 241 217 + 1;
  • 17 247 241 217 ÷ 2 = 8 623 620 608 + 1;
  • 8 623 620 608 ÷ 2 = 4 311 810 304 + 0;
  • 4 311 810 304 ÷ 2 = 2 155 905 152 + 0;
  • 2 155 905 152 ÷ 2 = 1 077 952 576 + 0;
  • 1 077 952 576 ÷ 2 = 538 976 288 + 0;
  • 538 976 288 ÷ 2 = 269 488 144 + 0;
  • 269 488 144 ÷ 2 = 134 744 072 + 0;
  • 134 744 072 ÷ 2 = 67 372 036 + 0;
  • 67 372 036 ÷ 2 = 33 686 018 + 0;
  • 33 686 018 ÷ 2 = 16 843 009 + 0;
  • 16 843 009 ÷ 2 = 8 421 504 + 1;
  • 8 421 504 ÷ 2 = 4 210 752 + 0;
  • 4 210 752 ÷ 2 = 2 105 376 + 0;
  • 2 105 376 ÷ 2 = 1 052 688 + 0;
  • 1 052 688 ÷ 2 = 526 344 + 0;
  • 526 344 ÷ 2 = 263 172 + 0;
  • 263 172 ÷ 2 = 131 586 + 0;
  • 131 586 ÷ 2 = 65 793 + 0;
  • 65 793 ÷ 2 = 32 896 + 1;
  • 32 896 ÷ 2 = 16 448 + 0;
  • 16 448 ÷ 2 = 8 224 + 0;
  • 8 224 ÷ 2 = 4 112 + 0;
  • 4 112 ÷ 2 = 2 056 + 0;
  • 2 056 ÷ 2 = 1 028 + 0;
  • 1 028 ÷ 2 = 514 + 0;
  • 514 ÷ 2 = 257 + 0;
  • 257 ÷ 2 = 128 + 1;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 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.

144 680 345 658 434 397(10) = 10 0000 0010 0000 0010 0000 0010 0000 0000 1111 0011 1010 0011 0101 1101(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 58.

  • 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, 58,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


4. Get the 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, 64.


Decimal Number 144 680 345 658 434 397(10) converted to signed binary in two's complement representation:

144 680 345 658 434 397(10) = 0000 0010 0000 0010 0000 0010 0000 0010 0000 0000 1111 0011 1010 0011 0101 1101

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

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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