How to convert the base ten signed integer number 1 234 567 890 123 330 to base two:
- 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.
To convert a base ten signed number (written as an integer in decimal system) to base two, written as a signed binary, follow the steps below.
- Divide the number repeatedly by 2: keep track of each remainder.
- Stop when you get a quotient that is equal to zero.
- Construct the base 2 representation of the positive number: take all the remainders starting from the bottom of the list constructed above.
- Determine the signed binary number bit length.
- Get the binary computer representation: if needed, add extra 0s in front (to the left) of the base 2 number, up to the required length and change the first bit (the leftmost), from 0 to 1, if the number is negative.
- Below you can see the conversion process to a signed binary and the related calculations.
1. Divide the number repeatedly by 2:
Keep track of each remainder.
Stop when you get a quotient that is equal to zero.
- division = quotient + remainder;
- 1 234 567 890 123 330 ÷ 2 = 617 283 945 061 665 + 0;
- 617 283 945 061 665 ÷ 2 = 308 641 972 530 832 + 1;
- 308 641 972 530 832 ÷ 2 = 154 320 986 265 416 + 0;
- 154 320 986 265 416 ÷ 2 = 77 160 493 132 708 + 0;
- 77 160 493 132 708 ÷ 2 = 38 580 246 566 354 + 0;
- 38 580 246 566 354 ÷ 2 = 19 290 123 283 177 + 0;
- 19 290 123 283 177 ÷ 2 = 9 645 061 641 588 + 1;
- 9 645 061 641 588 ÷ 2 = 4 822 530 820 794 + 0;
- 4 822 530 820 794 ÷ 2 = 2 411 265 410 397 + 0;
- 2 411 265 410 397 ÷ 2 = 1 205 632 705 198 + 1;
- 1 205 632 705 198 ÷ 2 = 602 816 352 599 + 0;
- 602 816 352 599 ÷ 2 = 301 408 176 299 + 1;
- 301 408 176 299 ÷ 2 = 150 704 088 149 + 1;
- 150 704 088 149 ÷ 2 = 75 352 044 074 + 1;
- 75 352 044 074 ÷ 2 = 37 676 022 037 + 0;
- 37 676 022 037 ÷ 2 = 18 838 011 018 + 1;
- 18 838 011 018 ÷ 2 = 9 419 005 509 + 0;
- 9 419 005 509 ÷ 2 = 4 709 502 754 + 1;
- 4 709 502 754 ÷ 2 = 2 354 751 377 + 0;
- 2 354 751 377 ÷ 2 = 1 177 375 688 + 1;
- 1 177 375 688 ÷ 2 = 588 687 844 + 0;
- 588 687 844 ÷ 2 = 294 343 922 + 0;
- 294 343 922 ÷ 2 = 147 171 961 + 0;
- 147 171 961 ÷ 2 = 73 585 980 + 1;
- 73 585 980 ÷ 2 = 36 792 990 + 0;
- 36 792 990 ÷ 2 = 18 396 495 + 0;
- 18 396 495 ÷ 2 = 9 198 247 + 1;
- 9 198 247 ÷ 2 = 4 599 123 + 1;
- 4 599 123 ÷ 2 = 2 299 561 + 1;
- 2 299 561 ÷ 2 = 1 149 780 + 1;
- 1 149 780 ÷ 2 = 574 890 + 0;
- 574 890 ÷ 2 = 287 445 + 0;
- 287 445 ÷ 2 = 143 722 + 1;
- 143 722 ÷ 2 = 71 861 + 0;
- 71 861 ÷ 2 = 35 930 + 1;
- 35 930 ÷ 2 = 17 965 + 0;
- 17 965 ÷ 2 = 8 982 + 1;
- 8 982 ÷ 2 = 4 491 + 0;
- 4 491 ÷ 2 = 2 245 + 1;
- 2 245 ÷ 2 = 1 122 + 1;
- 1 122 ÷ 2 = 561 + 0;
- 561 ÷ 2 = 280 + 1;
- 280 ÷ 2 = 140 + 0;
- 140 ÷ 2 = 70 + 0;
- 70 ÷ 2 = 35 + 0;
- 35 ÷ 2 = 17 + 1;
- 17 ÷ 2 = 8 + 1;
- 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.
1 234 567 890 123 330(10) = 100 0110 0010 1101 0101 0011 1100 1000 1010 1011 1010 0100 0010(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 51.
- 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) is reserved for 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, 51,
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:
Number 1 234 567 890 123 330(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
1 234 567 890 123 330(10) = 0000 0000 0000 0100 0110 0010 1101 0101 0011 1100 1000 1010 1011 1010 0100 0010
Spaces were used to group digits: for binary, by 4, for decimal, by 3.