1.745 459 324 169 999 826 377 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 377(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
1.745 459 324 169 999 826 377(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: 0.745 459 324 169 999 826 377.

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) 0.745 459 324 169 999 826 377 × 2 = 1 + 0.490 918 648 339 999 652 754;
  • 2) 0.490 918 648 339 999 652 754 × 2 = 0 + 0.981 837 296 679 999 305 508;
  • 3) 0.981 837 296 679 999 305 508 × 2 = 1 + 0.963 674 593 359 998 611 016;
  • 4) 0.963 674 593 359 998 611 016 × 2 = 1 + 0.927 349 186 719 997 222 032;
  • 5) 0.927 349 186 719 997 222 032 × 2 = 1 + 0.854 698 373 439 994 444 064;
  • 6) 0.854 698 373 439 994 444 064 × 2 = 1 + 0.709 396 746 879 988 888 128;
  • 7) 0.709 396 746 879 988 888 128 × 2 = 1 + 0.418 793 493 759 977 776 256;
  • 8) 0.418 793 493 759 977 776 256 × 2 = 0 + 0.837 586 987 519 955 552 512;
  • 9) 0.837 586 987 519 955 552 512 × 2 = 1 + 0.675 173 975 039 911 105 024;
  • 10) 0.675 173 975 039 911 105 024 × 2 = 1 + 0.350 347 950 079 822 210 048;
  • 11) 0.350 347 950 079 822 210 048 × 2 = 0 + 0.700 695 900 159 644 420 096;
  • 12) 0.700 695 900 159 644 420 096 × 2 = 1 + 0.401 391 800 319 288 840 192;
  • 13) 0.401 391 800 319 288 840 192 × 2 = 0 + 0.802 783 600 638 577 680 384;
  • 14) 0.802 783 600 638 577 680 384 × 2 = 1 + 0.605 567 201 277 155 360 768;
  • 15) 0.605 567 201 277 155 360 768 × 2 = 1 + 0.211 134 402 554 310 721 536;
  • 16) 0.211 134 402 554 310 721 536 × 2 = 0 + 0.422 268 805 108 621 443 072;
  • 17) 0.422 268 805 108 621 443 072 × 2 = 0 + 0.844 537 610 217 242 886 144;
  • 18) 0.844 537 610 217 242 886 144 × 2 = 1 + 0.689 075 220 434 485 772 288;
  • 19) 0.689 075 220 434 485 772 288 × 2 = 1 + 0.378 150 440 868 971 544 576;
  • 20) 0.378 150 440 868 971 544 576 × 2 = 0 + 0.756 300 881 737 943 089 152;
  • 21) 0.756 300 881 737 943 089 152 × 2 = 1 + 0.512 601 763 475 886 178 304;
  • 22) 0.512 601 763 475 886 178 304 × 2 = 1 + 0.025 203 526 951 772 356 608;
  • 23) 0.025 203 526 951 772 356 608 × 2 = 0 + 0.050 407 053 903 544 713 216;
  • 24) 0.050 407 053 903 544 713 216 × 2 = 0 + 0.100 814 107 807 089 426 432;
  • 25) 0.100 814 107 807 089 426 432 × 2 = 0 + 0.201 628 215 614 178 852 864;
  • 26) 0.201 628 215 614 178 852 864 × 2 = 0 + 0.403 256 431 228 357 705 728;
  • 27) 0.403 256 431 228 357 705 728 × 2 = 0 + 0.806 512 862 456 715 411 456;
  • 28) 0.806 512 862 456 715 411 456 × 2 = 1 + 0.613 025 724 913 430 822 912;
  • 29) 0.613 025 724 913 430 822 912 × 2 = 1 + 0.226 051 449 826 861 645 824;
  • 30) 0.226 051 449 826 861 645 824 × 2 = 0 + 0.452 102 899 653 723 291 648;
  • 31) 0.452 102 899 653 723 291 648 × 2 = 0 + 0.904 205 799 307 446 583 296;
  • 32) 0.904 205 799 307 446 583 296 × 2 = 1 + 0.808 411 598 614 893 166 592;
  • 33) 0.808 411 598 614 893 166 592 × 2 = 1 + 0.616 823 197 229 786 333 184;
  • 34) 0.616 823 197 229 786 333 184 × 2 = 1 + 0.233 646 394 459 572 666 368;
  • 35) 0.233 646 394 459 572 666 368 × 2 = 0 + 0.467 292 788 919 145 332 736;
  • 36) 0.467 292 788 919 145 332 736 × 2 = 0 + 0.934 585 577 838 290 665 472;
  • 37) 0.934 585 577 838 290 665 472 × 2 = 1 + 0.869 171 155 676 581 330 944;
  • 38) 0.869 171 155 676 581 330 944 × 2 = 1 + 0.738 342 311 353 162 661 888;
  • 39) 0.738 342 311 353 162 661 888 × 2 = 1 + 0.476 684 622 706 325 323 776;
  • 40) 0.476 684 622 706 325 323 776 × 2 = 0 + 0.953 369 245 412 650 647 552;
  • 41) 0.953 369 245 412 650 647 552 × 2 = 1 + 0.906 738 490 825 301 295 104;
  • 42) 0.906 738 490 825 301 295 104 × 2 = 1 + 0.813 476 981 650 602 590 208;
  • 43) 0.813 476 981 650 602 590 208 × 2 = 1 + 0.626 953 963 301 205 180 416;
  • 44) 0.626 953 963 301 205 180 416 × 2 = 1 + 0.253 907 926 602 410 360 832;
  • 45) 0.253 907 926 602 410 360 832 × 2 = 0 + 0.507 815 853 204 820 721 664;
  • 46) 0.507 815 853 204 820 721 664 × 2 = 1 + 0.015 631 706 409 641 443 328;
  • 47) 0.015 631 706 409 641 443 328 × 2 = 0 + 0.031 263 412 819 282 886 656;
  • 48) 0.031 263 412 819 282 886 656 × 2 = 0 + 0.062 526 825 638 565 773 312;
  • 49) 0.062 526 825 638 565 773 312 × 2 = 0 + 0.125 053 651 277 131 546 624;
  • 50) 0.125 053 651 277 131 546 624 × 2 = 0 + 0.250 107 302 554 263 093 248;
  • 51) 0.250 107 302 554 263 093 248 × 2 = 0 + 0.500 214 605 108 526 186 496;
  • 52) 0.500 214 605 108 526 186 496 × 2 = 1 + 0.000 429 210 217 052 372 992;
  • 53) 0.000 429 210 217 052 372 992 × 2 = 0 + 0.000 858 420 434 104 745 984;

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:


0.745 459 324 169 999 826 377(10) =

0.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(2)

5. Positive number before normalization:

1.745 459 324 169 999 826 377(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(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:


1.745 459 324 169 999 826 377(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(2) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(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 0001 0


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 0001 0 =

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


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 0001


Decimal number 1.745 459 324 169 999 826 377 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 0001

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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