0.000 000 000 000 000 000 000 000 005 63 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 000 000 005 63(10) to 32 bit single precision IEEE 754 binary floating point representation standard (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 000 000 000 000 000 005 63(10) to 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 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: 0.000 000 000 000 000 000 000 000 005 63.

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) 0.000 000 000 000 000 000 000 000 005 63 × 2 = 0 + 0.000 000 000 000 000 000 000 000 011 26;
  • 2) 0.000 000 000 000 000 000 000 000 011 26 × 2 = 0 + 0.000 000 000 000 000 000 000 000 022 52;
  • 3) 0.000 000 000 000 000 000 000 000 022 52 × 2 = 0 + 0.000 000 000 000 000 000 000 000 045 04;
  • 4) 0.000 000 000 000 000 000 000 000 045 04 × 2 = 0 + 0.000 000 000 000 000 000 000 000 090 08;
  • 5) 0.000 000 000 000 000 000 000 000 090 08 × 2 = 0 + 0.000 000 000 000 000 000 000 000 180 16;
  • 6) 0.000 000 000 000 000 000 000 000 180 16 × 2 = 0 + 0.000 000 000 000 000 000 000 000 360 32;
  • 7) 0.000 000 000 000 000 000 000 000 360 32 × 2 = 0 + 0.000 000 000 000 000 000 000 000 720 64;
  • 8) 0.000 000 000 000 000 000 000 000 720 64 × 2 = 0 + 0.000 000 000 000 000 000 000 001 441 28;
  • 9) 0.000 000 000 000 000 000 000 001 441 28 × 2 = 0 + 0.000 000 000 000 000 000 000 002 882 56;
  • 10) 0.000 000 000 000 000 000 000 002 882 56 × 2 = 0 + 0.000 000 000 000 000 000 000 005 765 12;
  • 11) 0.000 000 000 000 000 000 000 005 765 12 × 2 = 0 + 0.000 000 000 000 000 000 000 011 530 24;
  • 12) 0.000 000 000 000 000 000 000 011 530 24 × 2 = 0 + 0.000 000 000 000 000 000 000 023 060 48;
  • 13) 0.000 000 000 000 000 000 000 023 060 48 × 2 = 0 + 0.000 000 000 000 000 000 000 046 120 96;
  • 14) 0.000 000 000 000 000 000 000 046 120 96 × 2 = 0 + 0.000 000 000 000 000 000 000 092 241 92;
  • 15) 0.000 000 000 000 000 000 000 092 241 92 × 2 = 0 + 0.000 000 000 000 000 000 000 184 483 84;
  • 16) 0.000 000 000 000 000 000 000 184 483 84 × 2 = 0 + 0.000 000 000 000 000 000 000 368 967 68;
  • 17) 0.000 000 000 000 000 000 000 368 967 68 × 2 = 0 + 0.000 000 000 000 000 000 000 737 935 36;
  • 18) 0.000 000 000 000 000 000 000 737 935 36 × 2 = 0 + 0.000 000 000 000 000 000 001 475 870 72;
  • 19) 0.000 000 000 000 000 000 001 475 870 72 × 2 = 0 + 0.000 000 000 000 000 000 002 951 741 44;
  • 20) 0.000 000 000 000 000 000 002 951 741 44 × 2 = 0 + 0.000 000 000 000 000 000 005 903 482 88;
  • 21) 0.000 000 000 000 000 000 005 903 482 88 × 2 = 0 + 0.000 000 000 000 000 000 011 806 965 76;
  • 22) 0.000 000 000 000 000 000 011 806 965 76 × 2 = 0 + 0.000 000 000 000 000 000 023 613 931 52;
  • 23) 0.000 000 000 000 000 000 023 613 931 52 × 2 = 0 + 0.000 000 000 000 000 000 047 227 863 04;
  • 24) 0.000 000 000 000 000 000 047 227 863 04 × 2 = 0 + 0.000 000 000 000 000 000 094 455 726 08;
  • 25) 0.000 000 000 000 000 000 094 455 726 08 × 2 = 0 + 0.000 000 000 000 000 000 188 911 452 16;
  • 26) 0.000 000 000 000 000 000 188 911 452 16 × 2 = 0 + 0.000 000 000 000 000 000 377 822 904 32;
  • 27) 0.000 000 000 000 000 000 377 822 904 32 × 2 = 0 + 0.000 000 000 000 000 000 755 645 808 64;
  • 28) 0.000 000 000 000 000 000 755 645 808 64 × 2 = 0 + 0.000 000 000 000 000 001 511 291 617 28;
  • 29) 0.000 000 000 000 000 001 511 291 617 28 × 2 = 0 + 0.000 000 000 000 000 003 022 583 234 56;
  • 30) 0.000 000 000 000 000 003 022 583 234 56 × 2 = 0 + 0.000 000 000 000 000 006 045 166 469 12;
  • 31) 0.000 000 000 000 000 006 045 166 469 12 × 2 = 0 + 0.000 000 000 000 000 012 090 332 938 24;
  • 32) 0.000 000 000 000 000 012 090 332 938 24 × 2 = 0 + 0.000 000 000 000 000 024 180 665 876 48;
  • 33) 0.000 000 000 000 000 024 180 665 876 48 × 2 = 0 + 0.000 000 000 000 000 048 361 331 752 96;
  • 34) 0.000 000 000 000 000 048 361 331 752 96 × 2 = 0 + 0.000 000 000 000 000 096 722 663 505 92;
  • 35) 0.000 000 000 000 000 096 722 663 505 92 × 2 = 0 + 0.000 000 000 000 000 193 445 327 011 84;
  • 36) 0.000 000 000 000 000 193 445 327 011 84 × 2 = 0 + 0.000 000 000 000 000 386 890 654 023 68;
  • 37) 0.000 000 000 000 000 386 890 654 023 68 × 2 = 0 + 0.000 000 000 000 000 773 781 308 047 36;
  • 38) 0.000 000 000 000 000 773 781 308 047 36 × 2 = 0 + 0.000 000 000 000 001 547 562 616 094 72;
  • 39) 0.000 000 000 000 001 547 562 616 094 72 × 2 = 0 + 0.000 000 000 000 003 095 125 232 189 44;
  • 40) 0.000 000 000 000 003 095 125 232 189 44 × 2 = 0 + 0.000 000 000 000 006 190 250 464 378 88;
  • 41) 0.000 000 000 000 006 190 250 464 378 88 × 2 = 0 + 0.000 000 000 000 012 380 500 928 757 76;
  • 42) 0.000 000 000 000 012 380 500 928 757 76 × 2 = 0 + 0.000 000 000 000 024 761 001 857 515 52;
  • 43) 0.000 000 000 000 024 761 001 857 515 52 × 2 = 0 + 0.000 000 000 000 049 522 003 715 031 04;
  • 44) 0.000 000 000 000 049 522 003 715 031 04 × 2 = 0 + 0.000 000 000 000 099 044 007 430 062 08;
  • 45) 0.000 000 000 000 099 044 007 430 062 08 × 2 = 0 + 0.000 000 000 000 198 088 014 860 124 16;
  • 46) 0.000 000 000 000 198 088 014 860 124 16 × 2 = 0 + 0.000 000 000 000 396 176 029 720 248 32;
  • 47) 0.000 000 000 000 396 176 029 720 248 32 × 2 = 0 + 0.000 000 000 000 792 352 059 440 496 64;
  • 48) 0.000 000 000 000 792 352 059 440 496 64 × 2 = 0 + 0.000 000 000 001 584 704 118 880 993 28;
  • 49) 0.000 000 000 001 584 704 118 880 993 28 × 2 = 0 + 0.000 000 000 003 169 408 237 761 986 56;
  • 50) 0.000 000 000 003 169 408 237 761 986 56 × 2 = 0 + 0.000 000 000 006 338 816 475 523 973 12;
  • 51) 0.000 000 000 006 338 816 475 523 973 12 × 2 = 0 + 0.000 000 000 012 677 632 951 047 946 24;
  • 52) 0.000 000 000 012 677 632 951 047 946 24 × 2 = 0 + 0.000 000 000 025 355 265 902 095 892 48;
  • 53) 0.000 000 000 025 355 265 902 095 892 48 × 2 = 0 + 0.000 000 000 050 710 531 804 191 784 96;
  • 54) 0.000 000 000 050 710 531 804 191 784 96 × 2 = 0 + 0.000 000 000 101 421 063 608 383 569 92;
  • 55) 0.000 000 000 101 421 063 608 383 569 92 × 2 = 0 + 0.000 000 000 202 842 127 216 767 139 84;
  • 56) 0.000 000 000 202 842 127 216 767 139 84 × 2 = 0 + 0.000 000 000 405 684 254 433 534 279 68;
  • 57) 0.000 000 000 405 684 254 433 534 279 68 × 2 = 0 + 0.000 000 000 811 368 508 867 068 559 36;
  • 58) 0.000 000 000 811 368 508 867 068 559 36 × 2 = 0 + 0.000 000 001 622 737 017 734 137 118 72;
  • 59) 0.000 000 001 622 737 017 734 137 118 72 × 2 = 0 + 0.000 000 003 245 474 035 468 274 237 44;
  • 60) 0.000 000 003 245 474 035 468 274 237 44 × 2 = 0 + 0.000 000 006 490 948 070 936 548 474 88;
  • 61) 0.000 000 006 490 948 070 936 548 474 88 × 2 = 0 + 0.000 000 012 981 896 141 873 096 949 76;
  • 62) 0.000 000 012 981 896 141 873 096 949 76 × 2 = 0 + 0.000 000 025 963 792 283 746 193 899 52;
  • 63) 0.000 000 025 963 792 283 746 193 899 52 × 2 = 0 + 0.000 000 051 927 584 567 492 387 799 04;
  • 64) 0.000 000 051 927 584 567 492 387 799 04 × 2 = 0 + 0.000 000 103 855 169 134 984 775 598 08;
  • 65) 0.000 000 103 855 169 134 984 775 598 08 × 2 = 0 + 0.000 000 207 710 338 269 969 551 196 16;
  • 66) 0.000 000 207 710 338 269 969 551 196 16 × 2 = 0 + 0.000 000 415 420 676 539 939 102 392 32;
  • 67) 0.000 000 415 420 676 539 939 102 392 32 × 2 = 0 + 0.000 000 830 841 353 079 878 204 784 64;
  • 68) 0.000 000 830 841 353 079 878 204 784 64 × 2 = 0 + 0.000 001 661 682 706 159 756 409 569 28;
  • 69) 0.000 001 661 682 706 159 756 409 569 28 × 2 = 0 + 0.000 003 323 365 412 319 512 819 138 56;
  • 70) 0.000 003 323 365 412 319 512 819 138 56 × 2 = 0 + 0.000 006 646 730 824 639 025 638 277 12;
  • 71) 0.000 006 646 730 824 639 025 638 277 12 × 2 = 0 + 0.000 013 293 461 649 278 051 276 554 24;
  • 72) 0.000 013 293 461 649 278 051 276 554 24 × 2 = 0 + 0.000 026 586 923 298 556 102 553 108 48;
  • 73) 0.000 026 586 923 298 556 102 553 108 48 × 2 = 0 + 0.000 053 173 846 597 112 205 106 216 96;
  • 74) 0.000 053 173 846 597 112 205 106 216 96 × 2 = 0 + 0.000 106 347 693 194 224 410 212 433 92;
  • 75) 0.000 106 347 693 194 224 410 212 433 92 × 2 = 0 + 0.000 212 695 386 388 448 820 424 867 84;
  • 76) 0.000 212 695 386 388 448 820 424 867 84 × 2 = 0 + 0.000 425 390 772 776 897 640 849 735 68;
  • 77) 0.000 425 390 772 776 897 640 849 735 68 × 2 = 0 + 0.000 850 781 545 553 795 281 699 471 36;
  • 78) 0.000 850 781 545 553 795 281 699 471 36 × 2 = 0 + 0.001 701 563 091 107 590 563 398 942 72;
  • 79) 0.001 701 563 091 107 590 563 398 942 72 × 2 = 0 + 0.003 403 126 182 215 181 126 797 885 44;
  • 80) 0.003 403 126 182 215 181 126 797 885 44 × 2 = 0 + 0.006 806 252 364 430 362 253 595 770 88;
  • 81) 0.006 806 252 364 430 362 253 595 770 88 × 2 = 0 + 0.013 612 504 728 860 724 507 191 541 76;
  • 82) 0.013 612 504 728 860 724 507 191 541 76 × 2 = 0 + 0.027 225 009 457 721 449 014 383 083 52;
  • 83) 0.027 225 009 457 721 449 014 383 083 52 × 2 = 0 + 0.054 450 018 915 442 898 028 766 167 04;
  • 84) 0.054 450 018 915 442 898 028 766 167 04 × 2 = 0 + 0.108 900 037 830 885 796 057 532 334 08;
  • 85) 0.108 900 037 830 885 796 057 532 334 08 × 2 = 0 + 0.217 800 075 661 771 592 115 064 668 16;
  • 86) 0.217 800 075 661 771 592 115 064 668 16 × 2 = 0 + 0.435 600 151 323 543 184 230 129 336 32;
  • 87) 0.435 600 151 323 543 184 230 129 336 32 × 2 = 0 + 0.871 200 302 647 086 368 460 258 672 64;
  • 88) 0.871 200 302 647 086 368 460 258 672 64 × 2 = 1 + 0.742 400 605 294 172 736 920 517 345 28;
  • 89) 0.742 400 605 294 172 736 920 517 345 28 × 2 = 1 + 0.484 801 210 588 345 473 841 034 690 56;
  • 90) 0.484 801 210 588 345 473 841 034 690 56 × 2 = 0 + 0.969 602 421 176 690 947 682 069 381 12;
  • 91) 0.969 602 421 176 690 947 682 069 381 12 × 2 = 1 + 0.939 204 842 353 381 895 364 138 762 24;
  • 92) 0.939 204 842 353 381 895 364 138 762 24 × 2 = 1 + 0.878 409 684 706 763 790 728 277 524 48;
  • 93) 0.878 409 684 706 763 790 728 277 524 48 × 2 = 1 + 0.756 819 369 413 527 581 456 555 048 96;
  • 94) 0.756 819 369 413 527 581 456 555 048 96 × 2 = 1 + 0.513 638 738 827 055 162 913 110 097 92;
  • 95) 0.513 638 738 827 055 162 913 110 097 92 × 2 = 1 + 0.027 277 477 654 110 325 826 220 195 84;
  • 96) 0.027 277 477 654 110 325 826 220 195 84 × 2 = 0 + 0.054 554 955 308 220 651 652 440 391 68;
  • 97) 0.054 554 955 308 220 651 652 440 391 68 × 2 = 0 + 0.109 109 910 616 441 303 304 880 783 36;
  • 98) 0.109 109 910 616 441 303 304 880 783 36 × 2 = 0 + 0.218 219 821 232 882 606 609 761 566 72;
  • 99) 0.218 219 821 232 882 606 609 761 566 72 × 2 = 0 + 0.436 439 642 465 765 213 219 523 133 44;
  • 100) 0.436 439 642 465 765 213 219 523 133 44 × 2 = 0 + 0.872 879 284 931 530 426 439 046 266 88;
  • 101) 0.872 879 284 931 530 426 439 046 266 88 × 2 = 1 + 0.745 758 569 863 060 852 878 092 533 76;
  • 102) 0.745 758 569 863 060 852 878 092 533 76 × 2 = 1 + 0.491 517 139 726 121 705 756 185 067 52;
  • 103) 0.491 517 139 726 121 705 756 185 067 52 × 2 = 0 + 0.983 034 279 452 243 411 512 370 135 04;
  • 104) 0.983 034 279 452 243 411 512 370 135 04 × 2 = 1 + 0.966 068 558 904 486 823 024 740 270 08;
  • 105) 0.966 068 558 904 486 823 024 740 270 08 × 2 = 1 + 0.932 137 117 808 973 646 049 480 540 16;
  • 106) 0.932 137 117 808 973 646 049 480 540 16 × 2 = 1 + 0.864 274 235 617 947 292 098 961 080 32;
  • 107) 0.864 274 235 617 947 292 098 961 080 32 × 2 = 1 + 0.728 548 471 235 894 584 197 922 160 64;
  • 108) 0.728 548 471 235 894 584 197 922 160 64 × 2 = 1 + 0.457 096 942 471 789 168 395 844 321 28;
  • 109) 0.457 096 942 471 789 168 395 844 321 28 × 2 = 0 + 0.914 193 884 943 578 336 791 688 642 56;
  • 110) 0.914 193 884 943 578 336 791 688 642 56 × 2 = 1 + 0.828 387 769 887 156 673 583 377 285 12;
  • 111) 0.828 387 769 887 156 673 583 377 285 12 × 2 = 1 + 0.656 775 539 774 313 347 166 754 570 24;

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:


0.000 000 000 000 000 000 000 000 005 63(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1011 1110 0000 1101 1111 011(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 000 000 005 63(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1011 1110 0000 1101 1111 011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 88 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 000 000 005 63(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1011 1110 0000 1101 1111 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1011 1110 0000 1101 1111 011(2) × 20 =

1.1011 1110 0000 1101 1111 011(2) × 2-88


7. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -88

Mantissa (not normalized):
1.1011 1110 0000 1101 1111 011


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(8-1) - 1 =

-88 + 2(8-1) - 1 =

(-88 + 127)(10) =

39(10)


9. Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 39 ÷ 2 = 19 + 1;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

39(10) =

0010 0111(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 23 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 101 1111 0000 0110 1111 1011 =

101 1111 0000 0110 1111 1011


12. The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (8 bits) =
0010 0111

Mantissa (23 bits) =
101 1111 0000 0110 1111 1011


Decimal number 0.000 000 000 000 000 000 000 000 005 63 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0010 0111 - 101 1111 0000 0110 1111 1011

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How to convert decimal numbers from base ten to 32 bit single precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 32 bit single 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 base ten 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 of the previous dividing 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 previous multiplying operations, starting from the top of the constructed list 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, by shifting the decimal point (or if you prefer, the decimal mark) "n" positions either to the left or to the right, so that only one non zero digit remains to the left of the decimal point.
  • 7. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign if the case) and adjust its length to 23 bits, either by removing the excess bits from the right (losing precision...) or by adding extra '0' bits 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 -25.347 from decimal system (base ten) to 32 bit single precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-25.347| = 25.347

  • 2. First convert the integer part, 25. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 25 ÷ 2 = 12 + 1;
    • 12 ÷ 2 = 6 + 0;
    • 6 ÷ 2 = 3 + 0;
    • 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:

    25(10) = 1 1001(2)

  • 4. Then convert the fractional part, 0.347. 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) 0.347 × 2 = 0 + 0.694;
    • 2) 0.694 × 2 = 1 + 0.388;
    • 3) 0.388 × 2 = 0 + 0.776;
    • 4) 0.776 × 2 = 1 + 0.552;
    • 5) 0.552 × 2 = 1 + 0.104;
    • 6) 0.104 × 2 = 0 + 0.208;
    • 7) 0.208 × 2 = 0 + 0.416;
    • 8) 0.416 × 2 = 0 + 0.832;
    • 9) 0.832 × 2 = 1 + 0.664;
    • 10) 0.664 × 2 = 1 + 0.328;
    • 11) 0.328 × 2 = 0 + 0.656;
    • 12) 0.656 × 2 = 1 + 0.312;
    • 13) 0.312 × 2 = 0 + 0.624;
    • 14) 0.624 × 2 = 1 + 0.248;
    • 15) 0.248 × 2 = 0 + 0.496;
    • 16) 0.496 × 2 = 0 + 0.992;
    • 17) 0.992 × 2 = 1 + 0.984;
    • 18) 0.984 × 2 = 1 + 0.968;
    • 19) 0.968 × 2 = 1 + 0.936;
    • 20) 0.936 × 2 = 1 + 0.872;
    • 21) 0.872 × 2 = 1 + 0.744;
    • 22) 0.744 × 2 = 1 + 0.488;
    • 23) 0.488 × 2 = 0 + 0.976;
    • 24) 0.976 × 2 = 1 + 0.952;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 23) 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:

    0.347(10) = 0.0101 1000 1101 0100 1111 1101(2)

  • 6. Summarizing - the positive number before normalization:

    25.347(10) = 1 1001.0101 1000 1101 0100 1111 1101(2)

  • 7. Normalize the binary representation of the number, shifting the decimal point 4 positions to the left so that only one non-zero digit stays to the left of the decimal point:

    25.347(10) =
    1 1001.0101 1000 1101 0100 1111 1101(2) =
    1 1001.0101 1000 1101 0100 1111 1101(2) × 20 =
    1.1001 0101 1000 1101 0100 1111 1101(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

  • 9. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as already demonstrated above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1 = (4 + 127)(10) = 131(10) =
    1000 0011(2)

  • 10. Normalize the mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal point) and adjust its length to 23 bits, by removing the excess bits from the right (losing precision...):

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

    Mantissa (normalized): 100 1010 1100 0110 1010 0111

  • Conclusion:

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

    Exponent (8 bits) = 1000 0011

    Mantissa (23 bits) = 100 1010 1100 0110 1010 0111

  • Number -25.347, converted from the decimal system (base 10) to 32 bit single precision IEEE 754 binary floating point =
    1 - 1000 0011 - 100 1010 1100 0110 1010 0111