0.000 000 000 000 000 000 000 000 000 98 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 000 000 000 98(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
0.000 000 000 000 000 000 000 000 000 98(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: 0.000 000 000 000 000 000 000 000 000 98.

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) 0.000 000 000 000 000 000 000 000 000 98 × 2 = 0 + 0.000 000 000 000 000 000 000 000 001 96;
  • 2) 0.000 000 000 000 000 000 000 000 001 96 × 2 = 0 + 0.000 000 000 000 000 000 000 000 003 92;
  • 3) 0.000 000 000 000 000 000 000 000 003 92 × 2 = 0 + 0.000 000 000 000 000 000 000 000 007 84;
  • 4) 0.000 000 000 000 000 000 000 000 007 84 × 2 = 0 + 0.000 000 000 000 000 000 000 000 015 68;
  • 5) 0.000 000 000 000 000 000 000 000 015 68 × 2 = 0 + 0.000 000 000 000 000 000 000 000 031 36;
  • 6) 0.000 000 000 000 000 000 000 000 031 36 × 2 = 0 + 0.000 000 000 000 000 000 000 000 062 72;
  • 7) 0.000 000 000 000 000 000 000 000 062 72 × 2 = 0 + 0.000 000 000 000 000 000 000 000 125 44;
  • 8) 0.000 000 000 000 000 000 000 000 125 44 × 2 = 0 + 0.000 000 000 000 000 000 000 000 250 88;
  • 9) 0.000 000 000 000 000 000 000 000 250 88 × 2 = 0 + 0.000 000 000 000 000 000 000 000 501 76;
  • 10) 0.000 000 000 000 000 000 000 000 501 76 × 2 = 0 + 0.000 000 000 000 000 000 000 001 003 52;
  • 11) 0.000 000 000 000 000 000 000 001 003 52 × 2 = 0 + 0.000 000 000 000 000 000 000 002 007 04;
  • 12) 0.000 000 000 000 000 000 000 002 007 04 × 2 = 0 + 0.000 000 000 000 000 000 000 004 014 08;
  • 13) 0.000 000 000 000 000 000 000 004 014 08 × 2 = 0 + 0.000 000 000 000 000 000 000 008 028 16;
  • 14) 0.000 000 000 000 000 000 000 008 028 16 × 2 = 0 + 0.000 000 000 000 000 000 000 016 056 32;
  • 15) 0.000 000 000 000 000 000 000 016 056 32 × 2 = 0 + 0.000 000 000 000 000 000 000 032 112 64;
  • 16) 0.000 000 000 000 000 000 000 032 112 64 × 2 = 0 + 0.000 000 000 000 000 000 000 064 225 28;
  • 17) 0.000 000 000 000 000 000 000 064 225 28 × 2 = 0 + 0.000 000 000 000 000 000 000 128 450 56;
  • 18) 0.000 000 000 000 000 000 000 128 450 56 × 2 = 0 + 0.000 000 000 000 000 000 000 256 901 12;
  • 19) 0.000 000 000 000 000 000 000 256 901 12 × 2 = 0 + 0.000 000 000 000 000 000 000 513 802 24;
  • 20) 0.000 000 000 000 000 000 000 513 802 24 × 2 = 0 + 0.000 000 000 000 000 000 001 027 604 48;
  • 21) 0.000 000 000 000 000 000 001 027 604 48 × 2 = 0 + 0.000 000 000 000 000 000 002 055 208 96;
  • 22) 0.000 000 000 000 000 000 002 055 208 96 × 2 = 0 + 0.000 000 000 000 000 000 004 110 417 92;
  • 23) 0.000 000 000 000 000 000 004 110 417 92 × 2 = 0 + 0.000 000 000 000 000 000 008 220 835 84;
  • 24) 0.000 000 000 000 000 000 008 220 835 84 × 2 = 0 + 0.000 000 000 000 000 000 016 441 671 68;
  • 25) 0.000 000 000 000 000 000 016 441 671 68 × 2 = 0 + 0.000 000 000 000 000 000 032 883 343 36;
  • 26) 0.000 000 000 000 000 000 032 883 343 36 × 2 = 0 + 0.000 000 000 000 000 000 065 766 686 72;
  • 27) 0.000 000 000 000 000 000 065 766 686 72 × 2 = 0 + 0.000 000 000 000 000 000 131 533 373 44;
  • 28) 0.000 000 000 000 000 000 131 533 373 44 × 2 = 0 + 0.000 000 000 000 000 000 263 066 746 88;
  • 29) 0.000 000 000 000 000 000 263 066 746 88 × 2 = 0 + 0.000 000 000 000 000 000 526 133 493 76;
  • 30) 0.000 000 000 000 000 000 526 133 493 76 × 2 = 0 + 0.000 000 000 000 000 001 052 266 987 52;
  • 31) 0.000 000 000 000 000 001 052 266 987 52 × 2 = 0 + 0.000 000 000 000 000 002 104 533 975 04;
  • 32) 0.000 000 000 000 000 002 104 533 975 04 × 2 = 0 + 0.000 000 000 000 000 004 209 067 950 08;
  • 33) 0.000 000 000 000 000 004 209 067 950 08 × 2 = 0 + 0.000 000 000 000 000 008 418 135 900 16;
  • 34) 0.000 000 000 000 000 008 418 135 900 16 × 2 = 0 + 0.000 000 000 000 000 016 836 271 800 32;
  • 35) 0.000 000 000 000 000 016 836 271 800 32 × 2 = 0 + 0.000 000 000 000 000 033 672 543 600 64;
  • 36) 0.000 000 000 000 000 033 672 543 600 64 × 2 = 0 + 0.000 000 000 000 000 067 345 087 201 28;
  • 37) 0.000 000 000 000 000 067 345 087 201 28 × 2 = 0 + 0.000 000 000 000 000 134 690 174 402 56;
  • 38) 0.000 000 000 000 000 134 690 174 402 56 × 2 = 0 + 0.000 000 000 000 000 269 380 348 805 12;
  • 39) 0.000 000 000 000 000 269 380 348 805 12 × 2 = 0 + 0.000 000 000 000 000 538 760 697 610 24;
  • 40) 0.000 000 000 000 000 538 760 697 610 24 × 2 = 0 + 0.000 000 000 000 001 077 521 395 220 48;
  • 41) 0.000 000 000 000 001 077 521 395 220 48 × 2 = 0 + 0.000 000 000 000 002 155 042 790 440 96;
  • 42) 0.000 000 000 000 002 155 042 790 440 96 × 2 = 0 + 0.000 000 000 000 004 310 085 580 881 92;
  • 43) 0.000 000 000 000 004 310 085 580 881 92 × 2 = 0 + 0.000 000 000 000 008 620 171 161 763 84;
  • 44) 0.000 000 000 000 008 620 171 161 763 84 × 2 = 0 + 0.000 000 000 000 017 240 342 323 527 68;
  • 45) 0.000 000 000 000 017 240 342 323 527 68 × 2 = 0 + 0.000 000 000 000 034 480 684 647 055 36;
  • 46) 0.000 000 000 000 034 480 684 647 055 36 × 2 = 0 + 0.000 000 000 000 068 961 369 294 110 72;
  • 47) 0.000 000 000 000 068 961 369 294 110 72 × 2 = 0 + 0.000 000 000 000 137 922 738 588 221 44;
  • 48) 0.000 000 000 000 137 922 738 588 221 44 × 2 = 0 + 0.000 000 000 000 275 845 477 176 442 88;
  • 49) 0.000 000 000 000 275 845 477 176 442 88 × 2 = 0 + 0.000 000 000 000 551 690 954 352 885 76;
  • 50) 0.000 000 000 000 551 690 954 352 885 76 × 2 = 0 + 0.000 000 000 001 103 381 908 705 771 52;
  • 51) 0.000 000 000 001 103 381 908 705 771 52 × 2 = 0 + 0.000 000 000 002 206 763 817 411 543 04;
  • 52) 0.000 000 000 002 206 763 817 411 543 04 × 2 = 0 + 0.000 000 000 004 413 527 634 823 086 08;
  • 53) 0.000 000 000 004 413 527 634 823 086 08 × 2 = 0 + 0.000 000 000 008 827 055 269 646 172 16;
  • 54) 0.000 000 000 008 827 055 269 646 172 16 × 2 = 0 + 0.000 000 000 017 654 110 539 292 344 32;
  • 55) 0.000 000 000 017 654 110 539 292 344 32 × 2 = 0 + 0.000 000 000 035 308 221 078 584 688 64;
  • 56) 0.000 000 000 035 308 221 078 584 688 64 × 2 = 0 + 0.000 000 000 070 616 442 157 169 377 28;
  • 57) 0.000 000 000 070 616 442 157 169 377 28 × 2 = 0 + 0.000 000 000 141 232 884 314 338 754 56;
  • 58) 0.000 000 000 141 232 884 314 338 754 56 × 2 = 0 + 0.000 000 000 282 465 768 628 677 509 12;
  • 59) 0.000 000 000 282 465 768 628 677 509 12 × 2 = 0 + 0.000 000 000 564 931 537 257 355 018 24;
  • 60) 0.000 000 000 564 931 537 257 355 018 24 × 2 = 0 + 0.000 000 001 129 863 074 514 710 036 48;
  • 61) 0.000 000 001 129 863 074 514 710 036 48 × 2 = 0 + 0.000 000 002 259 726 149 029 420 072 96;
  • 62) 0.000 000 002 259 726 149 029 420 072 96 × 2 = 0 + 0.000 000 004 519 452 298 058 840 145 92;
  • 63) 0.000 000 004 519 452 298 058 840 145 92 × 2 = 0 + 0.000 000 009 038 904 596 117 680 291 84;
  • 64) 0.000 000 009 038 904 596 117 680 291 84 × 2 = 0 + 0.000 000 018 077 809 192 235 360 583 68;
  • 65) 0.000 000 018 077 809 192 235 360 583 68 × 2 = 0 + 0.000 000 036 155 618 384 470 721 167 36;
  • 66) 0.000 000 036 155 618 384 470 721 167 36 × 2 = 0 + 0.000 000 072 311 236 768 941 442 334 72;
  • 67) 0.000 000 072 311 236 768 941 442 334 72 × 2 = 0 + 0.000 000 144 622 473 537 882 884 669 44;
  • 68) 0.000 000 144 622 473 537 882 884 669 44 × 2 = 0 + 0.000 000 289 244 947 075 765 769 338 88;
  • 69) 0.000 000 289 244 947 075 765 769 338 88 × 2 = 0 + 0.000 000 578 489 894 151 531 538 677 76;
  • 70) 0.000 000 578 489 894 151 531 538 677 76 × 2 = 0 + 0.000 001 156 979 788 303 063 077 355 52;
  • 71) 0.000 001 156 979 788 303 063 077 355 52 × 2 = 0 + 0.000 002 313 959 576 606 126 154 711 04;
  • 72) 0.000 002 313 959 576 606 126 154 711 04 × 2 = 0 + 0.000 004 627 919 153 212 252 309 422 08;
  • 73) 0.000 004 627 919 153 212 252 309 422 08 × 2 = 0 + 0.000 009 255 838 306 424 504 618 844 16;
  • 74) 0.000 009 255 838 306 424 504 618 844 16 × 2 = 0 + 0.000 018 511 676 612 849 009 237 688 32;
  • 75) 0.000 018 511 676 612 849 009 237 688 32 × 2 = 0 + 0.000 037 023 353 225 698 018 475 376 64;
  • 76) 0.000 037 023 353 225 698 018 475 376 64 × 2 = 0 + 0.000 074 046 706 451 396 036 950 753 28;
  • 77) 0.000 074 046 706 451 396 036 950 753 28 × 2 = 0 + 0.000 148 093 412 902 792 073 901 506 56;
  • 78) 0.000 148 093 412 902 792 073 901 506 56 × 2 = 0 + 0.000 296 186 825 805 584 147 803 013 12;
  • 79) 0.000 296 186 825 805 584 147 803 013 12 × 2 = 0 + 0.000 592 373 651 611 168 295 606 026 24;
  • 80) 0.000 592 373 651 611 168 295 606 026 24 × 2 = 0 + 0.001 184 747 303 222 336 591 212 052 48;
  • 81) 0.001 184 747 303 222 336 591 212 052 48 × 2 = 0 + 0.002 369 494 606 444 673 182 424 104 96;
  • 82) 0.002 369 494 606 444 673 182 424 104 96 × 2 = 0 + 0.004 738 989 212 889 346 364 848 209 92;
  • 83) 0.004 738 989 212 889 346 364 848 209 92 × 2 = 0 + 0.009 477 978 425 778 692 729 696 419 84;
  • 84) 0.009 477 978 425 778 692 729 696 419 84 × 2 = 0 + 0.018 955 956 851 557 385 459 392 839 68;
  • 85) 0.018 955 956 851 557 385 459 392 839 68 × 2 = 0 + 0.037 911 913 703 114 770 918 785 679 36;
  • 86) 0.037 911 913 703 114 770 918 785 679 36 × 2 = 0 + 0.075 823 827 406 229 541 837 571 358 72;
  • 87) 0.075 823 827 406 229 541 837 571 358 72 × 2 = 0 + 0.151 647 654 812 459 083 675 142 717 44;
  • 88) 0.151 647 654 812 459 083 675 142 717 44 × 2 = 0 + 0.303 295 309 624 918 167 350 285 434 88;
  • 89) 0.303 295 309 624 918 167 350 285 434 88 × 2 = 0 + 0.606 590 619 249 836 334 700 570 869 76;
  • 90) 0.606 590 619 249 836 334 700 570 869 76 × 2 = 1 + 0.213 181 238 499 672 669 401 141 739 52;
  • 91) 0.213 181 238 499 672 669 401 141 739 52 × 2 = 0 + 0.426 362 476 999 345 338 802 283 479 04;
  • 92) 0.426 362 476 999 345 338 802 283 479 04 × 2 = 0 + 0.852 724 953 998 690 677 604 566 958 08;
  • 93) 0.852 724 953 998 690 677 604 566 958 08 × 2 = 1 + 0.705 449 907 997 381 355 209 133 916 16;
  • 94) 0.705 449 907 997 381 355 209 133 916 16 × 2 = 1 + 0.410 899 815 994 762 710 418 267 832 32;
  • 95) 0.410 899 815 994 762 710 418 267 832 32 × 2 = 0 + 0.821 799 631 989 525 420 836 535 664 64;
  • 96) 0.821 799 631 989 525 420 836 535 664 64 × 2 = 1 + 0.643 599 263 979 050 841 673 071 329 28;
  • 97) 0.643 599 263 979 050 841 673 071 329 28 × 2 = 1 + 0.287 198 527 958 101 683 346 142 658 56;
  • 98) 0.287 198 527 958 101 683 346 142 658 56 × 2 = 0 + 0.574 397 055 916 203 366 692 285 317 12;
  • 99) 0.574 397 055 916 203 366 692 285 317 12 × 2 = 1 + 0.148 794 111 832 406 733 384 570 634 24;
  • 100) 0.148 794 111 832 406 733 384 570 634 24 × 2 = 0 + 0.297 588 223 664 813 466 769 141 268 48;
  • 101) 0.297 588 223 664 813 466 769 141 268 48 × 2 = 0 + 0.595 176 447 329 626 933 538 282 536 96;
  • 102) 0.595 176 447 329 626 933 538 282 536 96 × 2 = 1 + 0.190 352 894 659 253 867 076 565 073 92;
  • 103) 0.190 352 894 659 253 867 076 565 073 92 × 2 = 0 + 0.380 705 789 318 507 734 153 130 147 84;
  • 104) 0.380 705 789 318 507 734 153 130 147 84 × 2 = 0 + 0.761 411 578 637 015 468 306 260 295 68;
  • 105) 0.761 411 578 637 015 468 306 260 295 68 × 2 = 1 + 0.522 823 157 274 030 936 612 520 591 36;
  • 106) 0.522 823 157 274 030 936 612 520 591 36 × 2 = 1 + 0.045 646 314 548 061 873 225 041 182 72;
  • 107) 0.045 646 314 548 061 873 225 041 182 72 × 2 = 0 + 0.091 292 629 096 123 746 450 082 365 44;
  • 108) 0.091 292 629 096 123 746 450 082 365 44 × 2 = 0 + 0.182 585 258 192 247 492 900 164 730 88;
  • 109) 0.182 585 258 192 247 492 900 164 730 88 × 2 = 0 + 0.365 170 516 384 494 985 800 329 461 76;
  • 110) 0.365 170 516 384 494 985 800 329 461 76 × 2 = 0 + 0.730 341 032 768 989 971 600 658 923 52;
  • 111) 0.730 341 032 768 989 971 600 658 923 52 × 2 = 1 + 0.460 682 065 537 979 943 201 317 847 04;
  • 112) 0.460 682 065 537 979 943 201 317 847 04 × 2 = 0 + 0.921 364 131 075 959 886 402 635 694 08;
  • 113) 0.921 364 131 075 959 886 402 635 694 08 × 2 = 1 + 0.842 728 262 151 919 772 805 271 388 16;

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:


0.000 000 000 000 000 000 000 000 000 98(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 1101 1010 0100 1100 0010 1(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 000 000 000 98(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 1101 1010 0100 1100 0010 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 90 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 000 000 000 98(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 1101 1010 0100 1100 0010 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 1101 1010 0100 1100 0010 1(2) × 20 =

1.0011 0110 1001 0011 0000 101(2) × 2-90


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): -90

Mantissa (not normalized):
1.0011 0110 1001 0011 0000 101


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-90 + 2(8-1) - 1 =

(-90 + 127)(10) =

37(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;
  • 37 ÷ 2 = 18 + 1;
  • 18 ÷ 2 = 9 + 0;
  • 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) =

37(10) =

0010 0101(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. 001 1011 0100 1001 1000 0101 =

001 1011 0100 1001 1000 0101


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 0101

Mantissa (23 bits) =
001 1011 0100 1001 1000 0101


Decimal number 0.000 000 000 000 000 000 000 000 000 98 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0010 0101 - 001 1011 0100 1001 1000 0101

Share

» Convert decimal 0.000 000 000 000 000 000 000 000 001 52 to 32 bit single precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 09, 2026 [September]: Decimal Numbers Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: September - by our visitors

» Improve your social skills: My Manager Only Says “We” When Referring to “My Work”. Gemini Notebook YouTube Video of 11.13 minutes

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