1.745 459 324 169 999 826 281 696 186 921 83 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 696 186 921 83(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
1.745 459 324 169 999 826 281 696 186 921 83(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 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;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.745 459 324 169 999 826 281 696 186 921 83.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.745 459 324 169 999 826 281 696 186 921 83 × 2 = 1 + 0.490 918 648 339 999 652 563 392 373 843 66;
  • 2) 0.490 918 648 339 999 652 563 392 373 843 66 × 2 = 0 + 0.981 837 296 679 999 305 126 784 747 687 32;
  • 3) 0.981 837 296 679 999 305 126 784 747 687 32 × 2 = 1 + 0.963 674 593 359 998 610 253 569 495 374 64;
  • 4) 0.963 674 593 359 998 610 253 569 495 374 64 × 2 = 1 + 0.927 349 186 719 997 220 507 138 990 749 28;
  • 5) 0.927 349 186 719 997 220 507 138 990 749 28 × 2 = 1 + 0.854 698 373 439 994 441 014 277 981 498 56;
  • 6) 0.854 698 373 439 994 441 014 277 981 498 56 × 2 = 1 + 0.709 396 746 879 988 882 028 555 962 997 12;
  • 7) 0.709 396 746 879 988 882 028 555 962 997 12 × 2 = 1 + 0.418 793 493 759 977 764 057 111 925 994 24;
  • 8) 0.418 793 493 759 977 764 057 111 925 994 24 × 2 = 0 + 0.837 586 987 519 955 528 114 223 851 988 48;
  • 9) 0.837 586 987 519 955 528 114 223 851 988 48 × 2 = 1 + 0.675 173 975 039 911 056 228 447 703 976 96;
  • 10) 0.675 173 975 039 911 056 228 447 703 976 96 × 2 = 1 + 0.350 347 950 079 822 112 456 895 407 953 92;
  • 11) 0.350 347 950 079 822 112 456 895 407 953 92 × 2 = 0 + 0.700 695 900 159 644 224 913 790 815 907 84;
  • 12) 0.700 695 900 159 644 224 913 790 815 907 84 × 2 = 1 + 0.401 391 800 319 288 449 827 581 631 815 68;
  • 13) 0.401 391 800 319 288 449 827 581 631 815 68 × 2 = 0 + 0.802 783 600 638 576 899 655 163 263 631 36;
  • 14) 0.802 783 600 638 576 899 655 163 263 631 36 × 2 = 1 + 0.605 567 201 277 153 799 310 326 527 262 72;
  • 15) 0.605 567 201 277 153 799 310 326 527 262 72 × 2 = 1 + 0.211 134 402 554 307 598 620 653 054 525 44;
  • 16) 0.211 134 402 554 307 598 620 653 054 525 44 × 2 = 0 + 0.422 268 805 108 615 197 241 306 109 050 88;
  • 17) 0.422 268 805 108 615 197 241 306 109 050 88 × 2 = 0 + 0.844 537 610 217 230 394 482 612 218 101 76;
  • 18) 0.844 537 610 217 230 394 482 612 218 101 76 × 2 = 1 + 0.689 075 220 434 460 788 965 224 436 203 52;
  • 19) 0.689 075 220 434 460 788 965 224 436 203 52 × 2 = 1 + 0.378 150 440 868 921 577 930 448 872 407 04;
  • 20) 0.378 150 440 868 921 577 930 448 872 407 04 × 2 = 0 + 0.756 300 881 737 843 155 860 897 744 814 08;
  • 21) 0.756 300 881 737 843 155 860 897 744 814 08 × 2 = 1 + 0.512 601 763 475 686 311 721 795 489 628 16;
  • 22) 0.512 601 763 475 686 311 721 795 489 628 16 × 2 = 1 + 0.025 203 526 951 372 623 443 590 979 256 32;
  • 23) 0.025 203 526 951 372 623 443 590 979 256 32 × 2 = 0 + 0.050 407 053 902 745 246 887 181 958 512 64;
  • 24) 0.050 407 053 902 745 246 887 181 958 512 64 × 2 = 0 + 0.100 814 107 805 490 493 774 363 917 025 28;
  • 25) 0.100 814 107 805 490 493 774 363 917 025 28 × 2 = 0 + 0.201 628 215 610 980 987 548 727 834 050 56;
  • 26) 0.201 628 215 610 980 987 548 727 834 050 56 × 2 = 0 + 0.403 256 431 221 961 975 097 455 668 101 12;
  • 27) 0.403 256 431 221 961 975 097 455 668 101 12 × 2 = 0 + 0.806 512 862 443 923 950 194 911 336 202 24;
  • 28) 0.806 512 862 443 923 950 194 911 336 202 24 × 2 = 1 + 0.613 025 724 887 847 900 389 822 672 404 48;
  • 29) 0.613 025 724 887 847 900 389 822 672 404 48 × 2 = 1 + 0.226 051 449 775 695 800 779 645 344 808 96;
  • 30) 0.226 051 449 775 695 800 779 645 344 808 96 × 2 = 0 + 0.452 102 899 551 391 601 559 290 689 617 92;
  • 31) 0.452 102 899 551 391 601 559 290 689 617 92 × 2 = 0 + 0.904 205 799 102 783 203 118 581 379 235 84;
  • 32) 0.904 205 799 102 783 203 118 581 379 235 84 × 2 = 1 + 0.808 411 598 205 566 406 237 162 758 471 68;
  • 33) 0.808 411 598 205 566 406 237 162 758 471 68 × 2 = 1 + 0.616 823 196 411 132 812 474 325 516 943 36;
  • 34) 0.616 823 196 411 132 812 474 325 516 943 36 × 2 = 1 + 0.233 646 392 822 265 624 948 651 033 886 72;
  • 35) 0.233 646 392 822 265 624 948 651 033 886 72 × 2 = 0 + 0.467 292 785 644 531 249 897 302 067 773 44;
  • 36) 0.467 292 785 644 531 249 897 302 067 773 44 × 2 = 0 + 0.934 585 571 289 062 499 794 604 135 546 88;
  • 37) 0.934 585 571 289 062 499 794 604 135 546 88 × 2 = 1 + 0.869 171 142 578 124 999 589 208 271 093 76;
  • 38) 0.869 171 142 578 124 999 589 208 271 093 76 × 2 = 1 + 0.738 342 285 156 249 999 178 416 542 187 52;
  • 39) 0.738 342 285 156 249 999 178 416 542 187 52 × 2 = 1 + 0.476 684 570 312 499 998 356 833 084 375 04;
  • 40) 0.476 684 570 312 499 998 356 833 084 375 04 × 2 = 0 + 0.953 369 140 624 999 996 713 666 168 750 08;
  • 41) 0.953 369 140 624 999 996 713 666 168 750 08 × 2 = 1 + 0.906 738 281 249 999 993 427 332 337 500 16;
  • 42) 0.906 738 281 249 999 993 427 332 337 500 16 × 2 = 1 + 0.813 476 562 499 999 986 854 664 675 000 32;
  • 43) 0.813 476 562 499 999 986 854 664 675 000 32 × 2 = 1 + 0.626 953 124 999 999 973 709 329 350 000 64;
  • 44) 0.626 953 124 999 999 973 709 329 350 000 64 × 2 = 1 + 0.253 906 249 999 999 947 418 658 700 001 28;
  • 45) 0.253 906 249 999 999 947 418 658 700 001 28 × 2 = 0 + 0.507 812 499 999 999 894 837 317 400 002 56;
  • 46) 0.507 812 499 999 999 894 837 317 400 002 56 × 2 = 1 + 0.015 624 999 999 999 789 674 634 800 005 12;
  • 47) 0.015 624 999 999 999 789 674 634 800 005 12 × 2 = 0 + 0.031 249 999 999 999 579 349 269 600 010 24;
  • 48) 0.031 249 999 999 999 579 349 269 600 010 24 × 2 = 0 + 0.062 499 999 999 999 158 698 539 200 020 48;
  • 49) 0.062 499 999 999 999 158 698 539 200 020 48 × 2 = 0 + 0.124 999 999 999 998 317 397 078 400 040 96;
  • 50) 0.124 999 999 999 998 317 397 078 400 040 96 × 2 = 0 + 0.249 999 999 999 996 634 794 156 800 081 92;
  • 51) 0.249 999 999 999 996 634 794 156 800 081 92 × 2 = 0 + 0.499 999 999 999 993 269 588 313 600 163 84;
  • 52) 0.499 999 999 999 993 269 588 313 600 163 84 × 2 = 0 + 0.999 999 999 999 986 539 176 627 200 327 68;
  • 53) 0.999 999 999 999 986 539 176 627 200 327 68 × 2 = 1 + 0.999 999 999 999 973 078 353 254 400 655 36;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.745 459 324 169 999 826 281 696 186 921 83(10) =

0.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1(2)

5. Positive number before normalization:

1.745 459 324 169 999 826 281 696 186 921 83(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.745 459 324 169 999 826 281 696 186 921 83(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1(2) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1(2) × 20


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): 0

Mantissa (not normalized):
1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1023(10) =

011 1111 1111(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1 =

1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1111 1111

Mantissa (52 bits) =
1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000


Decimal number 1.745 459 324 169 999 826 281 696 186 921 83 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000

Share

» Convert decimal 1.745 459 324 169 999 826 281 696 186 922 71 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 07, 2026 [July]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: July - by our visitors

» Improve your social skills: My Manager Only Says “We” When Referring to “My Work”. NotebookLM YouTube Video of 11.13 minutes

How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, 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. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100