1.745 459 324 169 999 826 281 696 186 679 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 696 186 679(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
1.745 459 324 169 999 826 281 696 186 679(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: 0.745 459 324 169 999 826 281 696 186 679.

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) 0.745 459 324 169 999 826 281 696 186 679 × 2 = 1 + 0.490 918 648 339 999 652 563 392 373 358;
  • 2) 0.490 918 648 339 999 652 563 392 373 358 × 2 = 0 + 0.981 837 296 679 999 305 126 784 746 716;
  • 3) 0.981 837 296 679 999 305 126 784 746 716 × 2 = 1 + 0.963 674 593 359 998 610 253 569 493 432;
  • 4) 0.963 674 593 359 998 610 253 569 493 432 × 2 = 1 + 0.927 349 186 719 997 220 507 138 986 864;
  • 5) 0.927 349 186 719 997 220 507 138 986 864 × 2 = 1 + 0.854 698 373 439 994 441 014 277 973 728;
  • 6) 0.854 698 373 439 994 441 014 277 973 728 × 2 = 1 + 0.709 396 746 879 988 882 028 555 947 456;
  • 7) 0.709 396 746 879 988 882 028 555 947 456 × 2 = 1 + 0.418 793 493 759 977 764 057 111 894 912;
  • 8) 0.418 793 493 759 977 764 057 111 894 912 × 2 = 0 + 0.837 586 987 519 955 528 114 223 789 824;
  • 9) 0.837 586 987 519 955 528 114 223 789 824 × 2 = 1 + 0.675 173 975 039 911 056 228 447 579 648;
  • 10) 0.675 173 975 039 911 056 228 447 579 648 × 2 = 1 + 0.350 347 950 079 822 112 456 895 159 296;
  • 11) 0.350 347 950 079 822 112 456 895 159 296 × 2 = 0 + 0.700 695 900 159 644 224 913 790 318 592;
  • 12) 0.700 695 900 159 644 224 913 790 318 592 × 2 = 1 + 0.401 391 800 319 288 449 827 580 637 184;
  • 13) 0.401 391 800 319 288 449 827 580 637 184 × 2 = 0 + 0.802 783 600 638 576 899 655 161 274 368;
  • 14) 0.802 783 600 638 576 899 655 161 274 368 × 2 = 1 + 0.605 567 201 277 153 799 310 322 548 736;
  • 15) 0.605 567 201 277 153 799 310 322 548 736 × 2 = 1 + 0.211 134 402 554 307 598 620 645 097 472;
  • 16) 0.211 134 402 554 307 598 620 645 097 472 × 2 = 0 + 0.422 268 805 108 615 197 241 290 194 944;
  • 17) 0.422 268 805 108 615 197 241 290 194 944 × 2 = 0 + 0.844 537 610 217 230 394 482 580 389 888;
  • 18) 0.844 537 610 217 230 394 482 580 389 888 × 2 = 1 + 0.689 075 220 434 460 788 965 160 779 776;
  • 19) 0.689 075 220 434 460 788 965 160 779 776 × 2 = 1 + 0.378 150 440 868 921 577 930 321 559 552;
  • 20) 0.378 150 440 868 921 577 930 321 559 552 × 2 = 0 + 0.756 300 881 737 843 155 860 643 119 104;
  • 21) 0.756 300 881 737 843 155 860 643 119 104 × 2 = 1 + 0.512 601 763 475 686 311 721 286 238 208;
  • 22) 0.512 601 763 475 686 311 721 286 238 208 × 2 = 1 + 0.025 203 526 951 372 623 442 572 476 416;
  • 23) 0.025 203 526 951 372 623 442 572 476 416 × 2 = 0 + 0.050 407 053 902 745 246 885 144 952 832;
  • 24) 0.050 407 053 902 745 246 885 144 952 832 × 2 = 0 + 0.100 814 107 805 490 493 770 289 905 664;
  • 25) 0.100 814 107 805 490 493 770 289 905 664 × 2 = 0 + 0.201 628 215 610 980 987 540 579 811 328;
  • 26) 0.201 628 215 610 980 987 540 579 811 328 × 2 = 0 + 0.403 256 431 221 961 975 081 159 622 656;
  • 27) 0.403 256 431 221 961 975 081 159 622 656 × 2 = 0 + 0.806 512 862 443 923 950 162 319 245 312;
  • 28) 0.806 512 862 443 923 950 162 319 245 312 × 2 = 1 + 0.613 025 724 887 847 900 324 638 490 624;
  • 29) 0.613 025 724 887 847 900 324 638 490 624 × 2 = 1 + 0.226 051 449 775 695 800 649 276 981 248;
  • 30) 0.226 051 449 775 695 800 649 276 981 248 × 2 = 0 + 0.452 102 899 551 391 601 298 553 962 496;
  • 31) 0.452 102 899 551 391 601 298 553 962 496 × 2 = 0 + 0.904 205 799 102 783 202 597 107 924 992;
  • 32) 0.904 205 799 102 783 202 597 107 924 992 × 2 = 1 + 0.808 411 598 205 566 405 194 215 849 984;
  • 33) 0.808 411 598 205 566 405 194 215 849 984 × 2 = 1 + 0.616 823 196 411 132 810 388 431 699 968;
  • 34) 0.616 823 196 411 132 810 388 431 699 968 × 2 = 1 + 0.233 646 392 822 265 620 776 863 399 936;
  • 35) 0.233 646 392 822 265 620 776 863 399 936 × 2 = 0 + 0.467 292 785 644 531 241 553 726 799 872;
  • 36) 0.467 292 785 644 531 241 553 726 799 872 × 2 = 0 + 0.934 585 571 289 062 483 107 453 599 744;
  • 37) 0.934 585 571 289 062 483 107 453 599 744 × 2 = 1 + 0.869 171 142 578 124 966 214 907 199 488;
  • 38) 0.869 171 142 578 124 966 214 907 199 488 × 2 = 1 + 0.738 342 285 156 249 932 429 814 398 976;
  • 39) 0.738 342 285 156 249 932 429 814 398 976 × 2 = 1 + 0.476 684 570 312 499 864 859 628 797 952;
  • 40) 0.476 684 570 312 499 864 859 628 797 952 × 2 = 0 + 0.953 369 140 624 999 729 719 257 595 904;
  • 41) 0.953 369 140 624 999 729 719 257 595 904 × 2 = 1 + 0.906 738 281 249 999 459 438 515 191 808;
  • 42) 0.906 738 281 249 999 459 438 515 191 808 × 2 = 1 + 0.813 476 562 499 998 918 877 030 383 616;
  • 43) 0.813 476 562 499 998 918 877 030 383 616 × 2 = 1 + 0.626 953 124 999 997 837 754 060 767 232;
  • 44) 0.626 953 124 999 997 837 754 060 767 232 × 2 = 1 + 0.253 906 249 999 995 675 508 121 534 464;
  • 45) 0.253 906 249 999 995 675 508 121 534 464 × 2 = 0 + 0.507 812 499 999 991 351 016 243 068 928;
  • 46) 0.507 812 499 999 991 351 016 243 068 928 × 2 = 1 + 0.015 624 999 999 982 702 032 486 137 856;
  • 47) 0.015 624 999 999 982 702 032 486 137 856 × 2 = 0 + 0.031 249 999 999 965 404 064 972 275 712;
  • 48) 0.031 249 999 999 965 404 064 972 275 712 × 2 = 0 + 0.062 499 999 999 930 808 129 944 551 424;
  • 49) 0.062 499 999 999 930 808 129 944 551 424 × 2 = 0 + 0.124 999 999 999 861 616 259 889 102 848;
  • 50) 0.124 999 999 999 861 616 259 889 102 848 × 2 = 0 + 0.249 999 999 999 723 232 519 778 205 696;
  • 51) 0.249 999 999 999 723 232 519 778 205 696 × 2 = 0 + 0.499 999 999 999 446 465 039 556 411 392;
  • 52) 0.499 999 999 999 446 465 039 556 411 392 × 2 = 0 + 0.999 999 999 998 892 930 079 112 822 784;
  • 53) 0.999 999 999 998 892 930 079 112 822 784 × 2 = 1 + 0.999 999 999 997 785 860 158 225 645 568;

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:


0.745 459 324 169 999 826 281 696 186 679(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 281 696 186 679(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:


1.745 459 324 169 999 826 281 696 186 679(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 1.745 459 324 169 999 826 281 696 186 679 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