1.745 459 324 169 999 826 281 696 186 924 818 903 156 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 696 186 924 818 903 156(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
1.745 459 324 169 999 826 281 696 186 924 818 903 156(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: 0.745 459 324 169 999 826 281 696 186 924 818 903 156.

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) 0.745 459 324 169 999 826 281 696 186 924 818 903 156 × 2 = 1 + 0.490 918 648 339 999 652 563 392 373 849 637 806 312;
  • 2) 0.490 918 648 339 999 652 563 392 373 849 637 806 312 × 2 = 0 + 0.981 837 296 679 999 305 126 784 747 699 275 612 624;
  • 3) 0.981 837 296 679 999 305 126 784 747 699 275 612 624 × 2 = 1 + 0.963 674 593 359 998 610 253 569 495 398 551 225 248;
  • 4) 0.963 674 593 359 998 610 253 569 495 398 551 225 248 × 2 = 1 + 0.927 349 186 719 997 220 507 138 990 797 102 450 496;
  • 5) 0.927 349 186 719 997 220 507 138 990 797 102 450 496 × 2 = 1 + 0.854 698 373 439 994 441 014 277 981 594 204 900 992;
  • 6) 0.854 698 373 439 994 441 014 277 981 594 204 900 992 × 2 = 1 + 0.709 396 746 879 988 882 028 555 963 188 409 801 984;
  • 7) 0.709 396 746 879 988 882 028 555 963 188 409 801 984 × 2 = 1 + 0.418 793 493 759 977 764 057 111 926 376 819 603 968;
  • 8) 0.418 793 493 759 977 764 057 111 926 376 819 603 968 × 2 = 0 + 0.837 586 987 519 955 528 114 223 852 753 639 207 936;
  • 9) 0.837 586 987 519 955 528 114 223 852 753 639 207 936 × 2 = 1 + 0.675 173 975 039 911 056 228 447 705 507 278 415 872;
  • 10) 0.675 173 975 039 911 056 228 447 705 507 278 415 872 × 2 = 1 + 0.350 347 950 079 822 112 456 895 411 014 556 831 744;
  • 11) 0.350 347 950 079 822 112 456 895 411 014 556 831 744 × 2 = 0 + 0.700 695 900 159 644 224 913 790 822 029 113 663 488;
  • 12) 0.700 695 900 159 644 224 913 790 822 029 113 663 488 × 2 = 1 + 0.401 391 800 319 288 449 827 581 644 058 227 326 976;
  • 13) 0.401 391 800 319 288 449 827 581 644 058 227 326 976 × 2 = 0 + 0.802 783 600 638 576 899 655 163 288 116 454 653 952;
  • 14) 0.802 783 600 638 576 899 655 163 288 116 454 653 952 × 2 = 1 + 0.605 567 201 277 153 799 310 326 576 232 909 307 904;
  • 15) 0.605 567 201 277 153 799 310 326 576 232 909 307 904 × 2 = 1 + 0.211 134 402 554 307 598 620 653 152 465 818 615 808;
  • 16) 0.211 134 402 554 307 598 620 653 152 465 818 615 808 × 2 = 0 + 0.422 268 805 108 615 197 241 306 304 931 637 231 616;
  • 17) 0.422 268 805 108 615 197 241 306 304 931 637 231 616 × 2 = 0 + 0.844 537 610 217 230 394 482 612 609 863 274 463 232;
  • 18) 0.844 537 610 217 230 394 482 612 609 863 274 463 232 × 2 = 1 + 0.689 075 220 434 460 788 965 225 219 726 548 926 464;
  • 19) 0.689 075 220 434 460 788 965 225 219 726 548 926 464 × 2 = 1 + 0.378 150 440 868 921 577 930 450 439 453 097 852 928;
  • 20) 0.378 150 440 868 921 577 930 450 439 453 097 852 928 × 2 = 0 + 0.756 300 881 737 843 155 860 900 878 906 195 705 856;
  • 21) 0.756 300 881 737 843 155 860 900 878 906 195 705 856 × 2 = 1 + 0.512 601 763 475 686 311 721 801 757 812 391 411 712;
  • 22) 0.512 601 763 475 686 311 721 801 757 812 391 411 712 × 2 = 1 + 0.025 203 526 951 372 623 443 603 515 624 782 823 424;
  • 23) 0.025 203 526 951 372 623 443 603 515 624 782 823 424 × 2 = 0 + 0.050 407 053 902 745 246 887 207 031 249 565 646 848;
  • 24) 0.050 407 053 902 745 246 887 207 031 249 565 646 848 × 2 = 0 + 0.100 814 107 805 490 493 774 414 062 499 131 293 696;
  • 25) 0.100 814 107 805 490 493 774 414 062 499 131 293 696 × 2 = 0 + 0.201 628 215 610 980 987 548 828 124 998 262 587 392;
  • 26) 0.201 628 215 610 980 987 548 828 124 998 262 587 392 × 2 = 0 + 0.403 256 431 221 961 975 097 656 249 996 525 174 784;
  • 27) 0.403 256 431 221 961 975 097 656 249 996 525 174 784 × 2 = 0 + 0.806 512 862 443 923 950 195 312 499 993 050 349 568;
  • 28) 0.806 512 862 443 923 950 195 312 499 993 050 349 568 × 2 = 1 + 0.613 025 724 887 847 900 390 624 999 986 100 699 136;
  • 29) 0.613 025 724 887 847 900 390 624 999 986 100 699 136 × 2 = 1 + 0.226 051 449 775 695 800 781 249 999 972 201 398 272;
  • 30) 0.226 051 449 775 695 800 781 249 999 972 201 398 272 × 2 = 0 + 0.452 102 899 551 391 601 562 499 999 944 402 796 544;
  • 31) 0.452 102 899 551 391 601 562 499 999 944 402 796 544 × 2 = 0 + 0.904 205 799 102 783 203 124 999 999 888 805 593 088;
  • 32) 0.904 205 799 102 783 203 124 999 999 888 805 593 088 × 2 = 1 + 0.808 411 598 205 566 406 249 999 999 777 611 186 176;
  • 33) 0.808 411 598 205 566 406 249 999 999 777 611 186 176 × 2 = 1 + 0.616 823 196 411 132 812 499 999 999 555 222 372 352;
  • 34) 0.616 823 196 411 132 812 499 999 999 555 222 372 352 × 2 = 1 + 0.233 646 392 822 265 624 999 999 999 110 444 744 704;
  • 35) 0.233 646 392 822 265 624 999 999 999 110 444 744 704 × 2 = 0 + 0.467 292 785 644 531 249 999 999 998 220 889 489 408;
  • 36) 0.467 292 785 644 531 249 999 999 998 220 889 489 408 × 2 = 0 + 0.934 585 571 289 062 499 999 999 996 441 778 978 816;
  • 37) 0.934 585 571 289 062 499 999 999 996 441 778 978 816 × 2 = 1 + 0.869 171 142 578 124 999 999 999 992 883 557 957 632;
  • 38) 0.869 171 142 578 124 999 999 999 992 883 557 957 632 × 2 = 1 + 0.738 342 285 156 249 999 999 999 985 767 115 915 264;
  • 39) 0.738 342 285 156 249 999 999 999 985 767 115 915 264 × 2 = 1 + 0.476 684 570 312 499 999 999 999 971 534 231 830 528;
  • 40) 0.476 684 570 312 499 999 999 999 971 534 231 830 528 × 2 = 0 + 0.953 369 140 624 999 999 999 999 943 068 463 661 056;
  • 41) 0.953 369 140 624 999 999 999 999 943 068 463 661 056 × 2 = 1 + 0.906 738 281 249 999 999 999 999 886 136 927 322 112;
  • 42) 0.906 738 281 249 999 999 999 999 886 136 927 322 112 × 2 = 1 + 0.813 476 562 499 999 999 999 999 772 273 854 644 224;
  • 43) 0.813 476 562 499 999 999 999 999 772 273 854 644 224 × 2 = 1 + 0.626 953 124 999 999 999 999 999 544 547 709 288 448;
  • 44) 0.626 953 124 999 999 999 999 999 544 547 709 288 448 × 2 = 1 + 0.253 906 249 999 999 999 999 999 089 095 418 576 896;
  • 45) 0.253 906 249 999 999 999 999 999 089 095 418 576 896 × 2 = 0 + 0.507 812 499 999 999 999 999 998 178 190 837 153 792;
  • 46) 0.507 812 499 999 999 999 999 998 178 190 837 153 792 × 2 = 1 + 0.015 624 999 999 999 999 999 996 356 381 674 307 584;
  • 47) 0.015 624 999 999 999 999 999 996 356 381 674 307 584 × 2 = 0 + 0.031 249 999 999 999 999 999 992 712 763 348 615 168;
  • 48) 0.031 249 999 999 999 999 999 992 712 763 348 615 168 × 2 = 0 + 0.062 499 999 999 999 999 999 985 425 526 697 230 336;
  • 49) 0.062 499 999 999 999 999 999 985 425 526 697 230 336 × 2 = 0 + 0.124 999 999 999 999 999 999 970 851 053 394 460 672;
  • 50) 0.124 999 999 999 999 999 999 970 851 053 394 460 672 × 2 = 0 + 0.249 999 999 999 999 999 999 941 702 106 788 921 344;
  • 51) 0.249 999 999 999 999 999 999 941 702 106 788 921 344 × 2 = 0 + 0.499 999 999 999 999 999 999 883 404 213 577 842 688;
  • 52) 0.499 999 999 999 999 999 999 883 404 213 577 842 688 × 2 = 0 + 0.999 999 999 999 999 999 999 766 808 427 155 685 376;
  • 53) 0.999 999 999 999 999 999 999 766 808 427 155 685 376 × 2 = 1 + 0.999 999 999 999 999 999 999 533 616 854 311 370 752;

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:


0.745 459 324 169 999 826 281 696 186 924 818 903 156(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 281 696 186 924 818 903 156(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:


1.745 459 324 169 999 826 281 696 186 924 818 903 156(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 1.745 459 324 169 999 826 281 696 186 924 818 903 156 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

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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