64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: -0.004 409 437 274 976 586 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number -0.004 409 437 274 976 586(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Start with the positive version of the number:

|-0.004 409 437 274 976 586| = 0.004 409 437 274 976 586

2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.004 409 437 274 976 586.

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) 0.004 409 437 274 976 586 × 2 = 0 + 0.008 818 874 549 953 172;
  • 2) 0.008 818 874 549 953 172 × 2 = 0 + 0.017 637 749 099 906 344;
  • 3) 0.017 637 749 099 906 344 × 2 = 0 + 0.035 275 498 199 812 688;
  • 4) 0.035 275 498 199 812 688 × 2 = 0 + 0.070 550 996 399 625 376;
  • 5) 0.070 550 996 399 625 376 × 2 = 0 + 0.141 101 992 799 250 752;
  • 6) 0.141 101 992 799 250 752 × 2 = 0 + 0.282 203 985 598 501 504;
  • 7) 0.282 203 985 598 501 504 × 2 = 0 + 0.564 407 971 197 003 008;
  • 8) 0.564 407 971 197 003 008 × 2 = 1 + 0.128 815 942 394 006 016;
  • 9) 0.128 815 942 394 006 016 × 2 = 0 + 0.257 631 884 788 012 032;
  • 10) 0.257 631 884 788 012 032 × 2 = 0 + 0.515 263 769 576 024 064;
  • 11) 0.515 263 769 576 024 064 × 2 = 1 + 0.030 527 539 152 048 128;
  • 12) 0.030 527 539 152 048 128 × 2 = 0 + 0.061 055 078 304 096 256;
  • 13) 0.061 055 078 304 096 256 × 2 = 0 + 0.122 110 156 608 192 512;
  • 14) 0.122 110 156 608 192 512 × 2 = 0 + 0.244 220 313 216 385 024;
  • 15) 0.244 220 313 216 385 024 × 2 = 0 + 0.488 440 626 432 770 048;
  • 16) 0.488 440 626 432 770 048 × 2 = 0 + 0.976 881 252 865 540 096;
  • 17) 0.976 881 252 865 540 096 × 2 = 1 + 0.953 762 505 731 080 192;
  • 18) 0.953 762 505 731 080 192 × 2 = 1 + 0.907 525 011 462 160 384;
  • 19) 0.907 525 011 462 160 384 × 2 = 1 + 0.815 050 022 924 320 768;
  • 20) 0.815 050 022 924 320 768 × 2 = 1 + 0.630 100 045 848 641 536;
  • 21) 0.630 100 045 848 641 536 × 2 = 1 + 0.260 200 091 697 283 072;
  • 22) 0.260 200 091 697 283 072 × 2 = 0 + 0.520 400 183 394 566 144;
  • 23) 0.520 400 183 394 566 144 × 2 = 1 + 0.040 800 366 789 132 288;
  • 24) 0.040 800 366 789 132 288 × 2 = 0 + 0.081 600 733 578 264 576;
  • 25) 0.081 600 733 578 264 576 × 2 = 0 + 0.163 201 467 156 529 152;
  • 26) 0.163 201 467 156 529 152 × 2 = 0 + 0.326 402 934 313 058 304;
  • 27) 0.326 402 934 313 058 304 × 2 = 0 + 0.652 805 868 626 116 608;
  • 28) 0.652 805 868 626 116 608 × 2 = 1 + 0.305 611 737 252 233 216;
  • 29) 0.305 611 737 252 233 216 × 2 = 0 + 0.611 223 474 504 466 432;
  • 30) 0.611 223 474 504 466 432 × 2 = 1 + 0.222 446 949 008 932 864;
  • 31) 0.222 446 949 008 932 864 × 2 = 0 + 0.444 893 898 017 865 728;
  • 32) 0.444 893 898 017 865 728 × 2 = 0 + 0.889 787 796 035 731 456;
  • 33) 0.889 787 796 035 731 456 × 2 = 1 + 0.779 575 592 071 462 912;
  • 34) 0.779 575 592 071 462 912 × 2 = 1 + 0.559 151 184 142 925 824;
  • 35) 0.559 151 184 142 925 824 × 2 = 1 + 0.118 302 368 285 851 648;
  • 36) 0.118 302 368 285 851 648 × 2 = 0 + 0.236 604 736 571 703 296;
  • 37) 0.236 604 736 571 703 296 × 2 = 0 + 0.473 209 473 143 406 592;
  • 38) 0.473 209 473 143 406 592 × 2 = 0 + 0.946 418 946 286 813 184;
  • 39) 0.946 418 946 286 813 184 × 2 = 1 + 0.892 837 892 573 626 368;
  • 40) 0.892 837 892 573 626 368 × 2 = 1 + 0.785 675 785 147 252 736;
  • 41) 0.785 675 785 147 252 736 × 2 = 1 + 0.571 351 570 294 505 472;
  • 42) 0.571 351 570 294 505 472 × 2 = 1 + 0.142 703 140 589 010 944;
  • 43) 0.142 703 140 589 010 944 × 2 = 0 + 0.285 406 281 178 021 888;
  • 44) 0.285 406 281 178 021 888 × 2 = 0 + 0.570 812 562 356 043 776;
  • 45) 0.570 812 562 356 043 776 × 2 = 1 + 0.141 625 124 712 087 552;
  • 46) 0.141 625 124 712 087 552 × 2 = 0 + 0.283 250 249 424 175 104;
  • 47) 0.283 250 249 424 175 104 × 2 = 0 + 0.566 500 498 848 350 208;
  • 48) 0.566 500 498 848 350 208 × 2 = 1 + 0.133 000 997 696 700 416;
  • 49) 0.133 000 997 696 700 416 × 2 = 0 + 0.266 001 995 393 400 832;
  • 50) 0.266 001 995 393 400 832 × 2 = 0 + 0.532 003 990 786 801 664;
  • 51) 0.532 003 990 786 801 664 × 2 = 1 + 0.064 007 981 573 603 328;
  • 52) 0.064 007 981 573 603 328 × 2 = 0 + 0.128 015 963 147 206 656;
  • 53) 0.128 015 963 147 206 656 × 2 = 0 + 0.256 031 926 294 413 312;
  • 54) 0.256 031 926 294 413 312 × 2 = 0 + 0.512 063 852 588 826 624;
  • 55) 0.512 063 852 588 826 624 × 2 = 1 + 0.024 127 705 177 653 248;
  • 56) 0.024 127 705 177 653 248 × 2 = 0 + 0.048 255 410 355 306 496;
  • 57) 0.048 255 410 355 306 496 × 2 = 0 + 0.096 510 820 710 612 992;
  • 58) 0.096 510 820 710 612 992 × 2 = 0 + 0.193 021 641 421 225 984;
  • 59) 0.193 021 641 421 225 984 × 2 = 0 + 0.386 043 282 842 451 968;
  • 60) 0.386 043 282 842 451 968 × 2 = 0 + 0.772 086 565 684 903 936;

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...)


5. 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:


0.004 409 437 274 976 586(10) =


0.0000 0001 0010 0000 1111 1010 0001 0100 1110 0011 1100 1001 0010 0010 0000(2)


6. Positive number before normalization:

0.004 409 437 274 976 586(10) =


0.0000 0001 0010 0000 1111 1010 0001 0100 1110 0011 1100 1001 0010 0010 0000(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 8 positions to the right, so that only one non zero digit remains to the left of it:


0.004 409 437 274 976 586(10) =


0.0000 0001 0010 0000 1111 1010 0001 0100 1110 0011 1100 1001 0010 0010 0000(2) =


0.0000 0001 0010 0000 1111 1010 0001 0100 1110 0011 1100 1001 0010 0010 0000(2) × 20 =


1.0010 0000 1111 1010 0001 0100 1110 0011 1100 1001 0010 0010 0000(2) × 2-8


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): -8


Mantissa (not normalized):
1.0010 0000 1111 1010 0001 0100 1110 0011 1100 1001 0010 0010 0000


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


-8 + 2(11-1) - 1 =


(-8 + 1 023)(10) =


1 015(10)


10. 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;
  • 1 015 ÷ 2 = 507 + 1;
  • 507 ÷ 2 = 253 + 1;
  • 253 ÷ 2 = 126 + 1;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


1015(10) =


011 1111 0111(2)


12. 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 0000 1111 1010 0001 0100 1110 0011 1100 1001 0010 0010 0000 =


0010 0000 1111 1010 0001 0100 1110 0011 1100 1001 0010 0010 0000


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)


Exponent (11 bits) =
011 1111 0111


Mantissa (52 bits) =
0010 0000 1111 1010 0001 0100 1110 0011 1100 1001 0010 0010 0000


The base ten decimal number -0.004 409 437 274 976 586 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
1 - 011 1111 0111 - 0010 0000 1111 1010 0001 0100 1110 0011 1100 1001 0010 0010 0000

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