1.745 459 324 169 999 826 281 696 004 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 696 004(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
1.745 459 324 169 999 826 281 696 004(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: 0.745 459 324 169 999 826 281 696 004.

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) 0.745 459 324 169 999 826 281 696 004 × 2 = 1 + 0.490 918 648 339 999 652 563 392 008;
  • 2) 0.490 918 648 339 999 652 563 392 008 × 2 = 0 + 0.981 837 296 679 999 305 126 784 016;
  • 3) 0.981 837 296 679 999 305 126 784 016 × 2 = 1 + 0.963 674 593 359 998 610 253 568 032;
  • 4) 0.963 674 593 359 998 610 253 568 032 × 2 = 1 + 0.927 349 186 719 997 220 507 136 064;
  • 5) 0.927 349 186 719 997 220 507 136 064 × 2 = 1 + 0.854 698 373 439 994 441 014 272 128;
  • 6) 0.854 698 373 439 994 441 014 272 128 × 2 = 1 + 0.709 396 746 879 988 882 028 544 256;
  • 7) 0.709 396 746 879 988 882 028 544 256 × 2 = 1 + 0.418 793 493 759 977 764 057 088 512;
  • 8) 0.418 793 493 759 977 764 057 088 512 × 2 = 0 + 0.837 586 987 519 955 528 114 177 024;
  • 9) 0.837 586 987 519 955 528 114 177 024 × 2 = 1 + 0.675 173 975 039 911 056 228 354 048;
  • 10) 0.675 173 975 039 911 056 228 354 048 × 2 = 1 + 0.350 347 950 079 822 112 456 708 096;
  • 11) 0.350 347 950 079 822 112 456 708 096 × 2 = 0 + 0.700 695 900 159 644 224 913 416 192;
  • 12) 0.700 695 900 159 644 224 913 416 192 × 2 = 1 + 0.401 391 800 319 288 449 826 832 384;
  • 13) 0.401 391 800 319 288 449 826 832 384 × 2 = 0 + 0.802 783 600 638 576 899 653 664 768;
  • 14) 0.802 783 600 638 576 899 653 664 768 × 2 = 1 + 0.605 567 201 277 153 799 307 329 536;
  • 15) 0.605 567 201 277 153 799 307 329 536 × 2 = 1 + 0.211 134 402 554 307 598 614 659 072;
  • 16) 0.211 134 402 554 307 598 614 659 072 × 2 = 0 + 0.422 268 805 108 615 197 229 318 144;
  • 17) 0.422 268 805 108 615 197 229 318 144 × 2 = 0 + 0.844 537 610 217 230 394 458 636 288;
  • 18) 0.844 537 610 217 230 394 458 636 288 × 2 = 1 + 0.689 075 220 434 460 788 917 272 576;
  • 19) 0.689 075 220 434 460 788 917 272 576 × 2 = 1 + 0.378 150 440 868 921 577 834 545 152;
  • 20) 0.378 150 440 868 921 577 834 545 152 × 2 = 0 + 0.756 300 881 737 843 155 669 090 304;
  • 21) 0.756 300 881 737 843 155 669 090 304 × 2 = 1 + 0.512 601 763 475 686 311 338 180 608;
  • 22) 0.512 601 763 475 686 311 338 180 608 × 2 = 1 + 0.025 203 526 951 372 622 676 361 216;
  • 23) 0.025 203 526 951 372 622 676 361 216 × 2 = 0 + 0.050 407 053 902 745 245 352 722 432;
  • 24) 0.050 407 053 902 745 245 352 722 432 × 2 = 0 + 0.100 814 107 805 490 490 705 444 864;
  • 25) 0.100 814 107 805 490 490 705 444 864 × 2 = 0 + 0.201 628 215 610 980 981 410 889 728;
  • 26) 0.201 628 215 610 980 981 410 889 728 × 2 = 0 + 0.403 256 431 221 961 962 821 779 456;
  • 27) 0.403 256 431 221 961 962 821 779 456 × 2 = 0 + 0.806 512 862 443 923 925 643 558 912;
  • 28) 0.806 512 862 443 923 925 643 558 912 × 2 = 1 + 0.613 025 724 887 847 851 287 117 824;
  • 29) 0.613 025 724 887 847 851 287 117 824 × 2 = 1 + 0.226 051 449 775 695 702 574 235 648;
  • 30) 0.226 051 449 775 695 702 574 235 648 × 2 = 0 + 0.452 102 899 551 391 405 148 471 296;
  • 31) 0.452 102 899 551 391 405 148 471 296 × 2 = 0 + 0.904 205 799 102 782 810 296 942 592;
  • 32) 0.904 205 799 102 782 810 296 942 592 × 2 = 1 + 0.808 411 598 205 565 620 593 885 184;
  • 33) 0.808 411 598 205 565 620 593 885 184 × 2 = 1 + 0.616 823 196 411 131 241 187 770 368;
  • 34) 0.616 823 196 411 131 241 187 770 368 × 2 = 1 + 0.233 646 392 822 262 482 375 540 736;
  • 35) 0.233 646 392 822 262 482 375 540 736 × 2 = 0 + 0.467 292 785 644 524 964 751 081 472;
  • 36) 0.467 292 785 644 524 964 751 081 472 × 2 = 0 + 0.934 585 571 289 049 929 502 162 944;
  • 37) 0.934 585 571 289 049 929 502 162 944 × 2 = 1 + 0.869 171 142 578 099 859 004 325 888;
  • 38) 0.869 171 142 578 099 859 004 325 888 × 2 = 1 + 0.738 342 285 156 199 718 008 651 776;
  • 39) 0.738 342 285 156 199 718 008 651 776 × 2 = 1 + 0.476 684 570 312 399 436 017 303 552;
  • 40) 0.476 684 570 312 399 436 017 303 552 × 2 = 0 + 0.953 369 140 624 798 872 034 607 104;
  • 41) 0.953 369 140 624 798 872 034 607 104 × 2 = 1 + 0.906 738 281 249 597 744 069 214 208;
  • 42) 0.906 738 281 249 597 744 069 214 208 × 2 = 1 + 0.813 476 562 499 195 488 138 428 416;
  • 43) 0.813 476 562 499 195 488 138 428 416 × 2 = 1 + 0.626 953 124 998 390 976 276 856 832;
  • 44) 0.626 953 124 998 390 976 276 856 832 × 2 = 1 + 0.253 906 249 996 781 952 553 713 664;
  • 45) 0.253 906 249 996 781 952 553 713 664 × 2 = 0 + 0.507 812 499 993 563 905 107 427 328;
  • 46) 0.507 812 499 993 563 905 107 427 328 × 2 = 1 + 0.015 624 999 987 127 810 214 854 656;
  • 47) 0.015 624 999 987 127 810 214 854 656 × 2 = 0 + 0.031 249 999 974 255 620 429 709 312;
  • 48) 0.031 249 999 974 255 620 429 709 312 × 2 = 0 + 0.062 499 999 948 511 240 859 418 624;
  • 49) 0.062 499 999 948 511 240 859 418 624 × 2 = 0 + 0.124 999 999 897 022 481 718 837 248;
  • 50) 0.124 999 999 897 022 481 718 837 248 × 2 = 0 + 0.249 999 999 794 044 963 437 674 496;
  • 51) 0.249 999 999 794 044 963 437 674 496 × 2 = 0 + 0.499 999 999 588 089 926 875 348 992;
  • 52) 0.499 999 999 588 089 926 875 348 992 × 2 = 0 + 0.999 999 999 176 179 853 750 697 984;
  • 53) 0.999 999 999 176 179 853 750 697 984 × 2 = 1 + 0.999 999 998 352 359 707 501 395 968;

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:


0.745 459 324 169 999 826 281 696 004(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 281 696 004(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:


1.745 459 324 169 999 826 281 696 004(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 1.745 459 324 169 999 826 281 696 004 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