Convert 148 424 101 473 129 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 148 424 101 473 129(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
148 424 101 473 129 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;
  • 148 424 101 473 129 ÷ 2 = 74 212 050 736 564 + 1;
  • 74 212 050 736 564 ÷ 2 = 37 106 025 368 282 + 0;
  • 37 106 025 368 282 ÷ 2 = 18 553 012 684 141 + 0;
  • 18 553 012 684 141 ÷ 2 = 9 276 506 342 070 + 1;
  • 9 276 506 342 070 ÷ 2 = 4 638 253 171 035 + 0;
  • 4 638 253 171 035 ÷ 2 = 2 319 126 585 517 + 1;
  • 2 319 126 585 517 ÷ 2 = 1 159 563 292 758 + 1;
  • 1 159 563 292 758 ÷ 2 = 579 781 646 379 + 0;
  • 579 781 646 379 ÷ 2 = 289 890 823 189 + 1;
  • 289 890 823 189 ÷ 2 = 144 945 411 594 + 1;
  • 144 945 411 594 ÷ 2 = 72 472 705 797 + 0;
  • 72 472 705 797 ÷ 2 = 36 236 352 898 + 1;
  • 36 236 352 898 ÷ 2 = 18 118 176 449 + 0;
  • 18 118 176 449 ÷ 2 = 9 059 088 224 + 1;
  • 9 059 088 224 ÷ 2 = 4 529 544 112 + 0;
  • 4 529 544 112 ÷ 2 = 2 264 772 056 + 0;
  • 2 264 772 056 ÷ 2 = 1 132 386 028 + 0;
  • 1 132 386 028 ÷ 2 = 566 193 014 + 0;
  • 566 193 014 ÷ 2 = 283 096 507 + 0;
  • 283 096 507 ÷ 2 = 141 548 253 + 1;
  • 141 548 253 ÷ 2 = 70 774 126 + 1;
  • 70 774 126 ÷ 2 = 35 387 063 + 0;
  • 35 387 063 ÷ 2 = 17 693 531 + 1;
  • 17 693 531 ÷ 2 = 8 846 765 + 1;
  • 8 846 765 ÷ 2 = 4 423 382 + 1;
  • 4 423 382 ÷ 2 = 2 211 691 + 0;
  • 2 211 691 ÷ 2 = 1 105 845 + 1;
  • 1 105 845 ÷ 2 = 552 922 + 1;
  • 552 922 ÷ 2 = 276 461 + 0;
  • 276 461 ÷ 2 = 138 230 + 1;
  • 138 230 ÷ 2 = 69 115 + 0;
  • 69 115 ÷ 2 = 34 557 + 1;
  • 34 557 ÷ 2 = 17 278 + 1;
  • 17 278 ÷ 2 = 8 639 + 0;
  • 8 639 ÷ 2 = 4 319 + 1;
  • 4 319 ÷ 2 = 2 159 + 1;
  • 2 159 ÷ 2 = 1 079 + 1;
  • 1 079 ÷ 2 = 539 + 1;
  • 539 ÷ 2 = 269 + 1;
  • 269 ÷ 2 = 134 + 1;
  • 134 ÷ 2 = 67 + 0;
  • 67 ÷ 2 = 33 + 1;
  • 33 ÷ 2 = 16 + 1;
  • 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.

148 424 101 473 129(10) = 1000 0110 1111 1101 1010 1101 1101 1000 0010 1011 0110 1001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 48.

  • 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, 48,

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 148 424 101 473 129(10) converted to signed binary in two's complement representation:

148 424 101 473 129(10) = 0000 0000 0000 0000 1000 0110 1111 1101 1010 1101 1101 1000 0010 1011 0110 1001

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