# Two's Complement: Integer -> Binary: 111 110 000 100 008 Convert the Integer Number to a Signed Binary in Two's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

## Signed integer number 111 110 000 100 008(10) converted and written as a signed binary in two's complement representation (base 2) = ?

### 1. Divide the number repeatedly by 2:

#### We stop when we get a quotient that is equal to zero.

• division = quotient + remainder;
• 111 110 000 100 008 ÷ 2 = 55 555 000 050 004 + 0;
• 55 555 000 050 004 ÷ 2 = 27 777 500 025 002 + 0;
• 27 777 500 025 002 ÷ 2 = 13 888 750 012 501 + 0;
• 13 888 750 012 501 ÷ 2 = 6 944 375 006 250 + 1;
• 6 944 375 006 250 ÷ 2 = 3 472 187 503 125 + 0;
• 3 472 187 503 125 ÷ 2 = 1 736 093 751 562 + 1;
• 1 736 093 751 562 ÷ 2 = 868 046 875 781 + 0;
• 868 046 875 781 ÷ 2 = 434 023 437 890 + 1;
• 434 023 437 890 ÷ 2 = 217 011 718 945 + 0;
• 217 011 718 945 ÷ 2 = 108 505 859 472 + 1;
• 108 505 859 472 ÷ 2 = 54 252 929 736 + 0;
• 54 252 929 736 ÷ 2 = 27 126 464 868 + 0;
• 27 126 464 868 ÷ 2 = 13 563 232 434 + 0;
• 13 563 232 434 ÷ 2 = 6 781 616 217 + 0;
• 6 781 616 217 ÷ 2 = 3 390 808 108 + 1;
• 3 390 808 108 ÷ 2 = 1 695 404 054 + 0;
• 1 695 404 054 ÷ 2 = 847 702 027 + 0;
• 847 702 027 ÷ 2 = 423 851 013 + 1;
• 423 851 013 ÷ 2 = 211 925 506 + 1;
• 211 925 506 ÷ 2 = 105 962 753 + 0;
• 105 962 753 ÷ 2 = 52 981 376 + 1;
• 52 981 376 ÷ 2 = 26 490 688 + 0;
• 26 490 688 ÷ 2 = 13 245 344 + 0;
• 13 245 344 ÷ 2 = 6 622 672 + 0;
• 6 622 672 ÷ 2 = 3 311 336 + 0;
• 3 311 336 ÷ 2 = 1 655 668 + 0;
• 1 655 668 ÷ 2 = 827 834 + 0;
• 827 834 ÷ 2 = 413 917 + 0;
• 413 917 ÷ 2 = 206 958 + 1;
• 206 958 ÷ 2 = 103 479 + 0;
• 103 479 ÷ 2 = 51 739 + 1;
• 51 739 ÷ 2 = 25 869 + 1;
• 25 869 ÷ 2 = 12 934 + 1;
• 12 934 ÷ 2 = 6 467 + 0;
• 6 467 ÷ 2 = 3 233 + 1;
• 3 233 ÷ 2 = 1 616 + 1;
• 1 616 ÷ 2 = 808 + 0;
• 808 ÷ 2 = 404 + 0;
• 404 ÷ 2 = 202 + 0;
• 202 ÷ 2 = 101 + 0;
• 101 ÷ 2 = 50 + 1;
• 50 ÷ 2 = 25 + 0;
• 25 ÷ 2 = 12 + 1;
• 12 ÷ 2 = 6 + 0;
• 6 ÷ 2 = 3 + 0;
• 3 ÷ 2 = 1 + 1;
• 1 ÷ 2 = 0 + 1;

## 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