0.000 000 000 000 000 002 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 002(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
0.000 000 000 000 000 002(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: 0.
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;
  • 0 ÷ 2 = 0 + 0;

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.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 002.

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) 0.000 000 000 000 000 002 × 2 = 0 + 0.000 000 000 000 000 004;
  • 2) 0.000 000 000 000 000 004 × 2 = 0 + 0.000 000 000 000 000 008;
  • 3) 0.000 000 000 000 000 008 × 2 = 0 + 0.000 000 000 000 000 016;
  • 4) 0.000 000 000 000 000 016 × 2 = 0 + 0.000 000 000 000 000 032;
  • 5) 0.000 000 000 000 000 032 × 2 = 0 + 0.000 000 000 000 000 064;
  • 6) 0.000 000 000 000 000 064 × 2 = 0 + 0.000 000 000 000 000 128;
  • 7) 0.000 000 000 000 000 128 × 2 = 0 + 0.000 000 000 000 000 256;
  • 8) 0.000 000 000 000 000 256 × 2 = 0 + 0.000 000 000 000 000 512;
  • 9) 0.000 000 000 000 000 512 × 2 = 0 + 0.000 000 000 000 001 024;
  • 10) 0.000 000 000 000 001 024 × 2 = 0 + 0.000 000 000 000 002 048;
  • 11) 0.000 000 000 000 002 048 × 2 = 0 + 0.000 000 000 000 004 096;
  • 12) 0.000 000 000 000 004 096 × 2 = 0 + 0.000 000 000 000 008 192;
  • 13) 0.000 000 000 000 008 192 × 2 = 0 + 0.000 000 000 000 016 384;
  • 14) 0.000 000 000 000 016 384 × 2 = 0 + 0.000 000 000 000 032 768;
  • 15) 0.000 000 000 000 032 768 × 2 = 0 + 0.000 000 000 000 065 536;
  • 16) 0.000 000 000 000 065 536 × 2 = 0 + 0.000 000 000 000 131 072;
  • 17) 0.000 000 000 000 131 072 × 2 = 0 + 0.000 000 000 000 262 144;
  • 18) 0.000 000 000 000 262 144 × 2 = 0 + 0.000 000 000 000 524 288;
  • 19) 0.000 000 000 000 524 288 × 2 = 0 + 0.000 000 000 001 048 576;
  • 20) 0.000 000 000 001 048 576 × 2 = 0 + 0.000 000 000 002 097 152;
  • 21) 0.000 000 000 002 097 152 × 2 = 0 + 0.000 000 000 004 194 304;
  • 22) 0.000 000 000 004 194 304 × 2 = 0 + 0.000 000 000 008 388 608;
  • 23) 0.000 000 000 008 388 608 × 2 = 0 + 0.000 000 000 016 777 216;
  • 24) 0.000 000 000 016 777 216 × 2 = 0 + 0.000 000 000 033 554 432;
  • 25) 0.000 000 000 033 554 432 × 2 = 0 + 0.000 000 000 067 108 864;
  • 26) 0.000 000 000 067 108 864 × 2 = 0 + 0.000 000 000 134 217 728;
  • 27) 0.000 000 000 134 217 728 × 2 = 0 + 0.000 000 000 268 435 456;
  • 28) 0.000 000 000 268 435 456 × 2 = 0 + 0.000 000 000 536 870 912;
  • 29) 0.000 000 000 536 870 912 × 2 = 0 + 0.000 000 001 073 741 824;
  • 30) 0.000 000 001 073 741 824 × 2 = 0 + 0.000 000 002 147 483 648;
  • 31) 0.000 000 002 147 483 648 × 2 = 0 + 0.000 000 004 294 967 296;
  • 32) 0.000 000 004 294 967 296 × 2 = 0 + 0.000 000 008 589 934 592;
  • 33) 0.000 000 008 589 934 592 × 2 = 0 + 0.000 000 017 179 869 184;
  • 34) 0.000 000 017 179 869 184 × 2 = 0 + 0.000 000 034 359 738 368;
  • 35) 0.000 000 034 359 738 368 × 2 = 0 + 0.000 000 068 719 476 736;
  • 36) 0.000 000 068 719 476 736 × 2 = 0 + 0.000 000 137 438 953 472;
  • 37) 0.000 000 137 438 953 472 × 2 = 0 + 0.000 000 274 877 906 944;
  • 38) 0.000 000 274 877 906 944 × 2 = 0 + 0.000 000 549 755 813 888;
  • 39) 0.000 000 549 755 813 888 × 2 = 0 + 0.000 001 099 511 627 776;
  • 40) 0.000 001 099 511 627 776 × 2 = 0 + 0.000 002 199 023 255 552;
  • 41) 0.000 002 199 023 255 552 × 2 = 0 + 0.000 004 398 046 511 104;
  • 42) 0.000 004 398 046 511 104 × 2 = 0 + 0.000 008 796 093 022 208;
  • 43) 0.000 008 796 093 022 208 × 2 = 0 + 0.000 017 592 186 044 416;
  • 44) 0.000 017 592 186 044 416 × 2 = 0 + 0.000 035 184 372 088 832;
  • 45) 0.000 035 184 372 088 832 × 2 = 0 + 0.000 070 368 744 177 664;
  • 46) 0.000 070 368 744 177 664 × 2 = 0 + 0.000 140 737 488 355 328;
  • 47) 0.000 140 737 488 355 328 × 2 = 0 + 0.000 281 474 976 710 656;
  • 48) 0.000 281 474 976 710 656 × 2 = 0 + 0.000 562 949 953 421 312;
  • 49) 0.000 562 949 953 421 312 × 2 = 0 + 0.001 125 899 906 842 624;
  • 50) 0.001 125 899 906 842 624 × 2 = 0 + 0.002 251 799 813 685 248;
  • 51) 0.002 251 799 813 685 248 × 2 = 0 + 0.004 503 599 627 370 496;
  • 52) 0.004 503 599 627 370 496 × 2 = 0 + 0.009 007 199 254 740 992;
  • 53) 0.009 007 199 254 740 992 × 2 = 0 + 0.018 014 398 509 481 984;
  • 54) 0.018 014 398 509 481 984 × 2 = 0 + 0.036 028 797 018 963 968;
  • 55) 0.036 028 797 018 963 968 × 2 = 0 + 0.072 057 594 037 927 936;
  • 56) 0.072 057 594 037 927 936 × 2 = 0 + 0.144 115 188 075 855 872;
  • 57) 0.144 115 188 075 855 872 × 2 = 0 + 0.288 230 376 151 711 744;
  • 58) 0.288 230 376 151 711 744 × 2 = 0 + 0.576 460 752 303 423 488;
  • 59) 0.576 460 752 303 423 488 × 2 = 1 + 0.152 921 504 606 846 976;
  • 60) 0.152 921 504 606 846 976 × 2 = 0 + 0.305 843 009 213 693 952;
  • 61) 0.305 843 009 213 693 952 × 2 = 0 + 0.611 686 018 427 387 904;
  • 62) 0.611 686 018 427 387 904 × 2 = 1 + 0.223 372 036 854 775 808;
  • 63) 0.223 372 036 854 775 808 × 2 = 0 + 0.446 744 073 709 551 616;
  • 64) 0.446 744 073 709 551 616 × 2 = 0 + 0.893 488 147 419 103 232;
  • 65) 0.893 488 147 419 103 232 × 2 = 1 + 0.786 976 294 838 206 464;
  • 66) 0.786 976 294 838 206 464 × 2 = 1 + 0.573 952 589 676 412 928;
  • 67) 0.573 952 589 676 412 928 × 2 = 1 + 0.147 905 179 352 825 856;
  • 68) 0.147 905 179 352 825 856 × 2 = 0 + 0.295 810 358 705 651 712;
  • 69) 0.295 810 358 705 651 712 × 2 = 0 + 0.591 620 717 411 303 424;
  • 70) 0.591 620 717 411 303 424 × 2 = 1 + 0.183 241 434 822 606 848;
  • 71) 0.183 241 434 822 606 848 × 2 = 0 + 0.366 482 869 645 213 696;
  • 72) 0.366 482 869 645 213 696 × 2 = 0 + 0.732 965 739 290 427 392;
  • 73) 0.732 965 739 290 427 392 × 2 = 1 + 0.465 931 478 580 854 784;
  • 74) 0.465 931 478 580 854 784 × 2 = 0 + 0.931 862 957 161 709 568;
  • 75) 0.931 862 957 161 709 568 × 2 = 1 + 0.863 725 914 323 419 136;
  • 76) 0.863 725 914 323 419 136 × 2 = 1 + 0.727 451 828 646 838 272;
  • 77) 0.727 451 828 646 838 272 × 2 = 1 + 0.454 903 657 293 676 544;
  • 78) 0.454 903 657 293 676 544 × 2 = 0 + 0.909 807 314 587 353 088;
  • 79) 0.909 807 314 587 353 088 × 2 = 1 + 0.819 614 629 174 706 176;
  • 80) 0.819 614 629 174 706 176 × 2 = 1 + 0.639 229 258 349 412 352;
  • 81) 0.639 229 258 349 412 352 × 2 = 1 + 0.278 458 516 698 824 704;
  • 82) 0.278 458 516 698 824 704 × 2 = 0 + 0.556 917 033 397 649 408;
  • 83) 0.556 917 033 397 649 408 × 2 = 1 + 0.113 834 066 795 298 816;
  • 84) 0.113 834 066 795 298 816 × 2 = 0 + 0.227 668 133 590 597 632;
  • 85) 0.227 668 133 590 597 632 × 2 = 0 + 0.455 336 267 181 195 264;
  • 86) 0.455 336 267 181 195 264 × 2 = 0 + 0.910 672 534 362 390 528;
  • 87) 0.910 672 534 362 390 528 × 2 = 1 + 0.821 345 068 724 781 056;
  • 88) 0.821 345 068 724 781 056 × 2 = 1 + 0.642 690 137 449 562 112;
  • 89) 0.642 690 137 449 562 112 × 2 = 1 + 0.285 380 274 899 124 224;
  • 90) 0.285 380 274 899 124 224 × 2 = 0 + 0.570 760 549 798 248 448;
  • 91) 0.570 760 549 798 248 448 × 2 = 1 + 0.141 521 099 596 496 896;
  • 92) 0.141 521 099 596 496 896 × 2 = 0 + 0.283 042 199 192 993 792;
  • 93) 0.283 042 199 192 993 792 × 2 = 0 + 0.566 084 398 385 987 584;
  • 94) 0.566 084 398 385 987 584 × 2 = 1 + 0.132 168 796 771 975 168;
  • 95) 0.132 168 796 771 975 168 × 2 = 0 + 0.264 337 593 543 950 336;
  • 96) 0.264 337 593 543 950 336 × 2 = 0 + 0.528 675 187 087 900 672;
  • 97) 0.528 675 187 087 900 672 × 2 = 1 + 0.057 350 374 175 801 344;
  • 98) 0.057 350 374 175 801 344 × 2 = 0 + 0.114 700 748 351 602 688;
  • 99) 0.114 700 748 351 602 688 × 2 = 0 + 0.229 401 496 703 205 376;
  • 100) 0.229 401 496 703 205 376 × 2 = 0 + 0.458 802 993 406 410 752;
  • 101) 0.458 802 993 406 410 752 × 2 = 0 + 0.917 605 986 812 821 504;
  • 102) 0.917 605 986 812 821 504 × 2 = 1 + 0.835 211 973 625 643 008;
  • 103) 0.835 211 973 625 643 008 × 2 = 1 + 0.670 423 947 251 286 016;
  • 104) 0.670 423 947 251 286 016 × 2 = 1 + 0.340 847 894 502 572 032;
  • 105) 0.340 847 894 502 572 032 × 2 = 0 + 0.681 695 789 005 144 064;
  • 106) 0.681 695 789 005 144 064 × 2 = 1 + 0.363 391 578 010 288 128;
  • 107) 0.363 391 578 010 288 128 × 2 = 0 + 0.726 783 156 020 576 256;
  • 108) 0.726 783 156 020 576 256 × 2 = 1 + 0.453 566 312 041 152 512;
  • 109) 0.453 566 312 041 152 512 × 2 = 0 + 0.907 132 624 082 305 024;
  • 110) 0.907 132 624 082 305 024 × 2 = 1 + 0.814 265 248 164 610 048;
  • 111) 0.814 265 248 164 610 048 × 2 = 1 + 0.628 530 496 329 220 096;

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:


0.000 000 000 000 000 002(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0100 1110 0100 1011 1011 1010 0011 1010 0100 1000 0111 0101 011(2)

5. Positive number before normalization:

0.000 000 000 000 000 002(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0100 1110 0100 1011 1011 1010 0011 1010 0100 1000 0111 0101 011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 59 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 002(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0100 1110 0100 1011 1011 1010 0011 1010 0100 1000 0111 0101 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0100 1110 0100 1011 1011 1010 0011 1010 0100 1000 0111 0101 011(2) × 20 =

1.0010 0111 0010 0101 1101 1101 0001 1101 0010 0100 0011 1010 1011(2) × 2-59


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): -59

Mantissa (not normalized):
1.0010 0111 0010 0101 1101 1101 0001 1101 0010 0100 0011 1010 1011


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-59 + 2(11-1) - 1 =

(-59 + 1 023)(10) =

964(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;
  • 964 ÷ 2 = 482 + 0;
  • 482 ÷ 2 = 241 + 0;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

964(10) =

011 1100 0100(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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0010 0111 0010 0101 1101 1101 0001 1101 0010 0100 0011 1010 1011 =

0010 0111 0010 0101 1101 1101 0001 1101 0010 0100 0011 1010 1011


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 1100 0100

Mantissa (52 bits) =
0010 0111 0010 0101 1101 1101 0001 1101 0010 0100 0011 1010 1011


Decimal number 0.000 000 000 000 000 002 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 0100 - 0010 0111 0010 0101 1101 1101 0001 1101 0010 0100 0011 1010 1011

» Convert decimal -0.000 000 000 000 000 046 to 64 bit double precision IEEE 754 binary floating point representation standard

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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