0.000 000 000 000 000 000 000 000 006 134 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 000 000 006 134(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
0.000 000 000 000 000 000 000 000 006 134(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: 0.000 000 000 000 000 000 000 000 006 134.

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) 0.000 000 000 000 000 000 000 000 006 134 × 2 = 0 + 0.000 000 000 000 000 000 000 000 012 268;
  • 2) 0.000 000 000 000 000 000 000 000 012 268 × 2 = 0 + 0.000 000 000 000 000 000 000 000 024 536;
  • 3) 0.000 000 000 000 000 000 000 000 024 536 × 2 = 0 + 0.000 000 000 000 000 000 000 000 049 072;
  • 4) 0.000 000 000 000 000 000 000 000 049 072 × 2 = 0 + 0.000 000 000 000 000 000 000 000 098 144;
  • 5) 0.000 000 000 000 000 000 000 000 098 144 × 2 = 0 + 0.000 000 000 000 000 000 000 000 196 288;
  • 6) 0.000 000 000 000 000 000 000 000 196 288 × 2 = 0 + 0.000 000 000 000 000 000 000 000 392 576;
  • 7) 0.000 000 000 000 000 000 000 000 392 576 × 2 = 0 + 0.000 000 000 000 000 000 000 000 785 152;
  • 8) 0.000 000 000 000 000 000 000 000 785 152 × 2 = 0 + 0.000 000 000 000 000 000 000 001 570 304;
  • 9) 0.000 000 000 000 000 000 000 001 570 304 × 2 = 0 + 0.000 000 000 000 000 000 000 003 140 608;
  • 10) 0.000 000 000 000 000 000 000 003 140 608 × 2 = 0 + 0.000 000 000 000 000 000 000 006 281 216;
  • 11) 0.000 000 000 000 000 000 000 006 281 216 × 2 = 0 + 0.000 000 000 000 000 000 000 012 562 432;
  • 12) 0.000 000 000 000 000 000 000 012 562 432 × 2 = 0 + 0.000 000 000 000 000 000 000 025 124 864;
  • 13) 0.000 000 000 000 000 000 000 025 124 864 × 2 = 0 + 0.000 000 000 000 000 000 000 050 249 728;
  • 14) 0.000 000 000 000 000 000 000 050 249 728 × 2 = 0 + 0.000 000 000 000 000 000 000 100 499 456;
  • 15) 0.000 000 000 000 000 000 000 100 499 456 × 2 = 0 + 0.000 000 000 000 000 000 000 200 998 912;
  • 16) 0.000 000 000 000 000 000 000 200 998 912 × 2 = 0 + 0.000 000 000 000 000 000 000 401 997 824;
  • 17) 0.000 000 000 000 000 000 000 401 997 824 × 2 = 0 + 0.000 000 000 000 000 000 000 803 995 648;
  • 18) 0.000 000 000 000 000 000 000 803 995 648 × 2 = 0 + 0.000 000 000 000 000 000 001 607 991 296;
  • 19) 0.000 000 000 000 000 000 001 607 991 296 × 2 = 0 + 0.000 000 000 000 000 000 003 215 982 592;
  • 20) 0.000 000 000 000 000 000 003 215 982 592 × 2 = 0 + 0.000 000 000 000 000 000 006 431 965 184;
  • 21) 0.000 000 000 000 000 000 006 431 965 184 × 2 = 0 + 0.000 000 000 000 000 000 012 863 930 368;
  • 22) 0.000 000 000 000 000 000 012 863 930 368 × 2 = 0 + 0.000 000 000 000 000 000 025 727 860 736;
  • 23) 0.000 000 000 000 000 000 025 727 860 736 × 2 = 0 + 0.000 000 000 000 000 000 051 455 721 472;
  • 24) 0.000 000 000 000 000 000 051 455 721 472 × 2 = 0 + 0.000 000 000 000 000 000 102 911 442 944;
  • 25) 0.000 000 000 000 000 000 102 911 442 944 × 2 = 0 + 0.000 000 000 000 000 000 205 822 885 888;
  • 26) 0.000 000 000 000 000 000 205 822 885 888 × 2 = 0 + 0.000 000 000 000 000 000 411 645 771 776;
  • 27) 0.000 000 000 000 000 000 411 645 771 776 × 2 = 0 + 0.000 000 000 000 000 000 823 291 543 552;
  • 28) 0.000 000 000 000 000 000 823 291 543 552 × 2 = 0 + 0.000 000 000 000 000 001 646 583 087 104;
  • 29) 0.000 000 000 000 000 001 646 583 087 104 × 2 = 0 + 0.000 000 000 000 000 003 293 166 174 208;
  • 30) 0.000 000 000 000 000 003 293 166 174 208 × 2 = 0 + 0.000 000 000 000 000 006 586 332 348 416;
  • 31) 0.000 000 000 000 000 006 586 332 348 416 × 2 = 0 + 0.000 000 000 000 000 013 172 664 696 832;
  • 32) 0.000 000 000 000 000 013 172 664 696 832 × 2 = 0 + 0.000 000 000 000 000 026 345 329 393 664;
  • 33) 0.000 000 000 000 000 026 345 329 393 664 × 2 = 0 + 0.000 000 000 000 000 052 690 658 787 328;
  • 34) 0.000 000 000 000 000 052 690 658 787 328 × 2 = 0 + 0.000 000 000 000 000 105 381 317 574 656;
  • 35) 0.000 000 000 000 000 105 381 317 574 656 × 2 = 0 + 0.000 000 000 000 000 210 762 635 149 312;
  • 36) 0.000 000 000 000 000 210 762 635 149 312 × 2 = 0 + 0.000 000 000 000 000 421 525 270 298 624;
  • 37) 0.000 000 000 000 000 421 525 270 298 624 × 2 = 0 + 0.000 000 000 000 000 843 050 540 597 248;
  • 38) 0.000 000 000 000 000 843 050 540 597 248 × 2 = 0 + 0.000 000 000 000 001 686 101 081 194 496;
  • 39) 0.000 000 000 000 001 686 101 081 194 496 × 2 = 0 + 0.000 000 000 000 003 372 202 162 388 992;
  • 40) 0.000 000 000 000 003 372 202 162 388 992 × 2 = 0 + 0.000 000 000 000 006 744 404 324 777 984;
  • 41) 0.000 000 000 000 006 744 404 324 777 984 × 2 = 0 + 0.000 000 000 000 013 488 808 649 555 968;
  • 42) 0.000 000 000 000 013 488 808 649 555 968 × 2 = 0 + 0.000 000 000 000 026 977 617 299 111 936;
  • 43) 0.000 000 000 000 026 977 617 299 111 936 × 2 = 0 + 0.000 000 000 000 053 955 234 598 223 872;
  • 44) 0.000 000 000 000 053 955 234 598 223 872 × 2 = 0 + 0.000 000 000 000 107 910 469 196 447 744;
  • 45) 0.000 000 000 000 107 910 469 196 447 744 × 2 = 0 + 0.000 000 000 000 215 820 938 392 895 488;
  • 46) 0.000 000 000 000 215 820 938 392 895 488 × 2 = 0 + 0.000 000 000 000 431 641 876 785 790 976;
  • 47) 0.000 000 000 000 431 641 876 785 790 976 × 2 = 0 + 0.000 000 000 000 863 283 753 571 581 952;
  • 48) 0.000 000 000 000 863 283 753 571 581 952 × 2 = 0 + 0.000 000 000 001 726 567 507 143 163 904;
  • 49) 0.000 000 000 001 726 567 507 143 163 904 × 2 = 0 + 0.000 000 000 003 453 135 014 286 327 808;
  • 50) 0.000 000 000 003 453 135 014 286 327 808 × 2 = 0 + 0.000 000 000 006 906 270 028 572 655 616;
  • 51) 0.000 000 000 006 906 270 028 572 655 616 × 2 = 0 + 0.000 000 000 013 812 540 057 145 311 232;
  • 52) 0.000 000 000 013 812 540 057 145 311 232 × 2 = 0 + 0.000 000 000 027 625 080 114 290 622 464;
  • 53) 0.000 000 000 027 625 080 114 290 622 464 × 2 = 0 + 0.000 000 000 055 250 160 228 581 244 928;
  • 54) 0.000 000 000 055 250 160 228 581 244 928 × 2 = 0 + 0.000 000 000 110 500 320 457 162 489 856;
  • 55) 0.000 000 000 110 500 320 457 162 489 856 × 2 = 0 + 0.000 000 000 221 000 640 914 324 979 712;
  • 56) 0.000 000 000 221 000 640 914 324 979 712 × 2 = 0 + 0.000 000 000 442 001 281 828 649 959 424;
  • 57) 0.000 000 000 442 001 281 828 649 959 424 × 2 = 0 + 0.000 000 000 884 002 563 657 299 918 848;
  • 58) 0.000 000 000 884 002 563 657 299 918 848 × 2 = 0 + 0.000 000 001 768 005 127 314 599 837 696;
  • 59) 0.000 000 001 768 005 127 314 599 837 696 × 2 = 0 + 0.000 000 003 536 010 254 629 199 675 392;
  • 60) 0.000 000 003 536 010 254 629 199 675 392 × 2 = 0 + 0.000 000 007 072 020 509 258 399 350 784;
  • 61) 0.000 000 007 072 020 509 258 399 350 784 × 2 = 0 + 0.000 000 014 144 041 018 516 798 701 568;
  • 62) 0.000 000 014 144 041 018 516 798 701 568 × 2 = 0 + 0.000 000 028 288 082 037 033 597 403 136;
  • 63) 0.000 000 028 288 082 037 033 597 403 136 × 2 = 0 + 0.000 000 056 576 164 074 067 194 806 272;
  • 64) 0.000 000 056 576 164 074 067 194 806 272 × 2 = 0 + 0.000 000 113 152 328 148 134 389 612 544;
  • 65) 0.000 000 113 152 328 148 134 389 612 544 × 2 = 0 + 0.000 000 226 304 656 296 268 779 225 088;
  • 66) 0.000 000 226 304 656 296 268 779 225 088 × 2 = 0 + 0.000 000 452 609 312 592 537 558 450 176;
  • 67) 0.000 000 452 609 312 592 537 558 450 176 × 2 = 0 + 0.000 000 905 218 625 185 075 116 900 352;
  • 68) 0.000 000 905 218 625 185 075 116 900 352 × 2 = 0 + 0.000 001 810 437 250 370 150 233 800 704;
  • 69) 0.000 001 810 437 250 370 150 233 800 704 × 2 = 0 + 0.000 003 620 874 500 740 300 467 601 408;
  • 70) 0.000 003 620 874 500 740 300 467 601 408 × 2 = 0 + 0.000 007 241 749 001 480 600 935 202 816;
  • 71) 0.000 007 241 749 001 480 600 935 202 816 × 2 = 0 + 0.000 014 483 498 002 961 201 870 405 632;
  • 72) 0.000 014 483 498 002 961 201 870 405 632 × 2 = 0 + 0.000 028 966 996 005 922 403 740 811 264;
  • 73) 0.000 028 966 996 005 922 403 740 811 264 × 2 = 0 + 0.000 057 933 992 011 844 807 481 622 528;
  • 74) 0.000 057 933 992 011 844 807 481 622 528 × 2 = 0 + 0.000 115 867 984 023 689 614 963 245 056;
  • 75) 0.000 115 867 984 023 689 614 963 245 056 × 2 = 0 + 0.000 231 735 968 047 379 229 926 490 112;
  • 76) 0.000 231 735 968 047 379 229 926 490 112 × 2 = 0 + 0.000 463 471 936 094 758 459 852 980 224;
  • 77) 0.000 463 471 936 094 758 459 852 980 224 × 2 = 0 + 0.000 926 943 872 189 516 919 705 960 448;
  • 78) 0.000 926 943 872 189 516 919 705 960 448 × 2 = 0 + 0.001 853 887 744 379 033 839 411 920 896;
  • 79) 0.001 853 887 744 379 033 839 411 920 896 × 2 = 0 + 0.003 707 775 488 758 067 678 823 841 792;
  • 80) 0.003 707 775 488 758 067 678 823 841 792 × 2 = 0 + 0.007 415 550 977 516 135 357 647 683 584;
  • 81) 0.007 415 550 977 516 135 357 647 683 584 × 2 = 0 + 0.014 831 101 955 032 270 715 295 367 168;
  • 82) 0.014 831 101 955 032 270 715 295 367 168 × 2 = 0 + 0.029 662 203 910 064 541 430 590 734 336;
  • 83) 0.029 662 203 910 064 541 430 590 734 336 × 2 = 0 + 0.059 324 407 820 129 082 861 181 468 672;
  • 84) 0.059 324 407 820 129 082 861 181 468 672 × 2 = 0 + 0.118 648 815 640 258 165 722 362 937 344;
  • 85) 0.118 648 815 640 258 165 722 362 937 344 × 2 = 0 + 0.237 297 631 280 516 331 444 725 874 688;
  • 86) 0.237 297 631 280 516 331 444 725 874 688 × 2 = 0 + 0.474 595 262 561 032 662 889 451 749 376;
  • 87) 0.474 595 262 561 032 662 889 451 749 376 × 2 = 0 + 0.949 190 525 122 065 325 778 903 498 752;
  • 88) 0.949 190 525 122 065 325 778 903 498 752 × 2 = 1 + 0.898 381 050 244 130 651 557 806 997 504;
  • 89) 0.898 381 050 244 130 651 557 806 997 504 × 2 = 1 + 0.796 762 100 488 261 303 115 613 995 008;
  • 90) 0.796 762 100 488 261 303 115 613 995 008 × 2 = 1 + 0.593 524 200 976 522 606 231 227 990 016;
  • 91) 0.593 524 200 976 522 606 231 227 990 016 × 2 = 1 + 0.187 048 401 953 045 212 462 455 980 032;
  • 92) 0.187 048 401 953 045 212 462 455 980 032 × 2 = 0 + 0.374 096 803 906 090 424 924 911 960 064;
  • 93) 0.374 096 803 906 090 424 924 911 960 064 × 2 = 0 + 0.748 193 607 812 180 849 849 823 920 128;
  • 94) 0.748 193 607 812 180 849 849 823 920 128 × 2 = 1 + 0.496 387 215 624 361 699 699 647 840 256;
  • 95) 0.496 387 215 624 361 699 699 647 840 256 × 2 = 0 + 0.992 774 431 248 723 399 399 295 680 512;
  • 96) 0.992 774 431 248 723 399 399 295 680 512 × 2 = 1 + 0.985 548 862 497 446 798 798 591 361 024;
  • 97) 0.985 548 862 497 446 798 798 591 361 024 × 2 = 1 + 0.971 097 724 994 893 597 597 182 722 048;
  • 98) 0.971 097 724 994 893 597 597 182 722 048 × 2 = 1 + 0.942 195 449 989 787 195 194 365 444 096;
  • 99) 0.942 195 449 989 787 195 194 365 444 096 × 2 = 1 + 0.884 390 899 979 574 390 388 730 888 192;
  • 100) 0.884 390 899 979 574 390 388 730 888 192 × 2 = 1 + 0.768 781 799 959 148 780 777 461 776 384;
  • 101) 0.768 781 799 959 148 780 777 461 776 384 × 2 = 1 + 0.537 563 599 918 297 561 554 923 552 768;
  • 102) 0.537 563 599 918 297 561 554 923 552 768 × 2 = 1 + 0.075 127 199 836 595 123 109 847 105 536;
  • 103) 0.075 127 199 836 595 123 109 847 105 536 × 2 = 0 + 0.150 254 399 673 190 246 219 694 211 072;
  • 104) 0.150 254 399 673 190 246 219 694 211 072 × 2 = 0 + 0.300 508 799 346 380 492 439 388 422 144;
  • 105) 0.300 508 799 346 380 492 439 388 422 144 × 2 = 0 + 0.601 017 598 692 760 984 878 776 844 288;
  • 106) 0.601 017 598 692 760 984 878 776 844 288 × 2 = 1 + 0.202 035 197 385 521 969 757 553 688 576;
  • 107) 0.202 035 197 385 521 969 757 553 688 576 × 2 = 0 + 0.404 070 394 771 043 939 515 107 377 152;
  • 108) 0.404 070 394 771 043 939 515 107 377 152 × 2 = 0 + 0.808 140 789 542 087 879 030 214 754 304;
  • 109) 0.808 140 789 542 087 879 030 214 754 304 × 2 = 1 + 0.616 281 579 084 175 758 060 429 508 608;
  • 110) 0.616 281 579 084 175 758 060 429 508 608 × 2 = 1 + 0.232 563 158 168 351 516 120 859 017 216;
  • 111) 0.232 563 158 168 351 516 120 859 017 216 × 2 = 0 + 0.465 126 316 336 703 032 241 718 034 432;

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:


0.000 000 000 000 000 000 000 000 006 134(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1110 0101 1111 1100 0100 110(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 000 000 006 134(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1110 0101 1111 1100 0100 110(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:


0.000 000 000 000 000 000 000 000 006 134(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1110 0101 1111 1100 0100 110(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1110 0101 1111 1100 0100 110(2) × 20 =

1.1110 0101 1111 1100 0100 110(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):
1.1110 0101 1111 1100 0100 110


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. 111 0010 1111 1110 0010 0110 =

111 0010 1111 1110 0010 0110


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) =
111 0010 1111 1110 0010 0110


Decimal number 0.000 000 000 000 000 000 000 000 006 134 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0010 0111 - 111 0010 1111 1110 0010 0110

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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