0.000 000 000 000 000 000 000 000 006 179 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 000 000 006 179(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
0.000 000 000 000 000 000 000 000 006 179(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: 0.000 000 000 000 000 000 000 000 006 179.

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) 0.000 000 000 000 000 000 000 000 006 179 × 2 = 0 + 0.000 000 000 000 000 000 000 000 012 358;
  • 2) 0.000 000 000 000 000 000 000 000 012 358 × 2 = 0 + 0.000 000 000 000 000 000 000 000 024 716;
  • 3) 0.000 000 000 000 000 000 000 000 024 716 × 2 = 0 + 0.000 000 000 000 000 000 000 000 049 432;
  • 4) 0.000 000 000 000 000 000 000 000 049 432 × 2 = 0 + 0.000 000 000 000 000 000 000 000 098 864;
  • 5) 0.000 000 000 000 000 000 000 000 098 864 × 2 = 0 + 0.000 000 000 000 000 000 000 000 197 728;
  • 6) 0.000 000 000 000 000 000 000 000 197 728 × 2 = 0 + 0.000 000 000 000 000 000 000 000 395 456;
  • 7) 0.000 000 000 000 000 000 000 000 395 456 × 2 = 0 + 0.000 000 000 000 000 000 000 000 790 912;
  • 8) 0.000 000 000 000 000 000 000 000 790 912 × 2 = 0 + 0.000 000 000 000 000 000 000 001 581 824;
  • 9) 0.000 000 000 000 000 000 000 001 581 824 × 2 = 0 + 0.000 000 000 000 000 000 000 003 163 648;
  • 10) 0.000 000 000 000 000 000 000 003 163 648 × 2 = 0 + 0.000 000 000 000 000 000 000 006 327 296;
  • 11) 0.000 000 000 000 000 000 000 006 327 296 × 2 = 0 + 0.000 000 000 000 000 000 000 012 654 592;
  • 12) 0.000 000 000 000 000 000 000 012 654 592 × 2 = 0 + 0.000 000 000 000 000 000 000 025 309 184;
  • 13) 0.000 000 000 000 000 000 000 025 309 184 × 2 = 0 + 0.000 000 000 000 000 000 000 050 618 368;
  • 14) 0.000 000 000 000 000 000 000 050 618 368 × 2 = 0 + 0.000 000 000 000 000 000 000 101 236 736;
  • 15) 0.000 000 000 000 000 000 000 101 236 736 × 2 = 0 + 0.000 000 000 000 000 000 000 202 473 472;
  • 16) 0.000 000 000 000 000 000 000 202 473 472 × 2 = 0 + 0.000 000 000 000 000 000 000 404 946 944;
  • 17) 0.000 000 000 000 000 000 000 404 946 944 × 2 = 0 + 0.000 000 000 000 000 000 000 809 893 888;
  • 18) 0.000 000 000 000 000 000 000 809 893 888 × 2 = 0 + 0.000 000 000 000 000 000 001 619 787 776;
  • 19) 0.000 000 000 000 000 000 001 619 787 776 × 2 = 0 + 0.000 000 000 000 000 000 003 239 575 552;
  • 20) 0.000 000 000 000 000 000 003 239 575 552 × 2 = 0 + 0.000 000 000 000 000 000 006 479 151 104;
  • 21) 0.000 000 000 000 000 000 006 479 151 104 × 2 = 0 + 0.000 000 000 000 000 000 012 958 302 208;
  • 22) 0.000 000 000 000 000 000 012 958 302 208 × 2 = 0 + 0.000 000 000 000 000 000 025 916 604 416;
  • 23) 0.000 000 000 000 000 000 025 916 604 416 × 2 = 0 + 0.000 000 000 000 000 000 051 833 208 832;
  • 24) 0.000 000 000 000 000 000 051 833 208 832 × 2 = 0 + 0.000 000 000 000 000 000 103 666 417 664;
  • 25) 0.000 000 000 000 000 000 103 666 417 664 × 2 = 0 + 0.000 000 000 000 000 000 207 332 835 328;
  • 26) 0.000 000 000 000 000 000 207 332 835 328 × 2 = 0 + 0.000 000 000 000 000 000 414 665 670 656;
  • 27) 0.000 000 000 000 000 000 414 665 670 656 × 2 = 0 + 0.000 000 000 000 000 000 829 331 341 312;
  • 28) 0.000 000 000 000 000 000 829 331 341 312 × 2 = 0 + 0.000 000 000 000 000 001 658 662 682 624;
  • 29) 0.000 000 000 000 000 001 658 662 682 624 × 2 = 0 + 0.000 000 000 000 000 003 317 325 365 248;
  • 30) 0.000 000 000 000 000 003 317 325 365 248 × 2 = 0 + 0.000 000 000 000 000 006 634 650 730 496;
  • 31) 0.000 000 000 000 000 006 634 650 730 496 × 2 = 0 + 0.000 000 000 000 000 013 269 301 460 992;
  • 32) 0.000 000 000 000 000 013 269 301 460 992 × 2 = 0 + 0.000 000 000 000 000 026 538 602 921 984;
  • 33) 0.000 000 000 000 000 026 538 602 921 984 × 2 = 0 + 0.000 000 000 000 000 053 077 205 843 968;
  • 34) 0.000 000 000 000 000 053 077 205 843 968 × 2 = 0 + 0.000 000 000 000 000 106 154 411 687 936;
  • 35) 0.000 000 000 000 000 106 154 411 687 936 × 2 = 0 + 0.000 000 000 000 000 212 308 823 375 872;
  • 36) 0.000 000 000 000 000 212 308 823 375 872 × 2 = 0 + 0.000 000 000 000 000 424 617 646 751 744;
  • 37) 0.000 000 000 000 000 424 617 646 751 744 × 2 = 0 + 0.000 000 000 000 000 849 235 293 503 488;
  • 38) 0.000 000 000 000 000 849 235 293 503 488 × 2 = 0 + 0.000 000 000 000 001 698 470 587 006 976;
  • 39) 0.000 000 000 000 001 698 470 587 006 976 × 2 = 0 + 0.000 000 000 000 003 396 941 174 013 952;
  • 40) 0.000 000 000 000 003 396 941 174 013 952 × 2 = 0 + 0.000 000 000 000 006 793 882 348 027 904;
  • 41) 0.000 000 000 000 006 793 882 348 027 904 × 2 = 0 + 0.000 000 000 000 013 587 764 696 055 808;
  • 42) 0.000 000 000 000 013 587 764 696 055 808 × 2 = 0 + 0.000 000 000 000 027 175 529 392 111 616;
  • 43) 0.000 000 000 000 027 175 529 392 111 616 × 2 = 0 + 0.000 000 000 000 054 351 058 784 223 232;
  • 44) 0.000 000 000 000 054 351 058 784 223 232 × 2 = 0 + 0.000 000 000 000 108 702 117 568 446 464;
  • 45) 0.000 000 000 000 108 702 117 568 446 464 × 2 = 0 + 0.000 000 000 000 217 404 235 136 892 928;
  • 46) 0.000 000 000 000 217 404 235 136 892 928 × 2 = 0 + 0.000 000 000 000 434 808 470 273 785 856;
  • 47) 0.000 000 000 000 434 808 470 273 785 856 × 2 = 0 + 0.000 000 000 000 869 616 940 547 571 712;
  • 48) 0.000 000 000 000 869 616 940 547 571 712 × 2 = 0 + 0.000 000 000 001 739 233 881 095 143 424;
  • 49) 0.000 000 000 001 739 233 881 095 143 424 × 2 = 0 + 0.000 000 000 003 478 467 762 190 286 848;
  • 50) 0.000 000 000 003 478 467 762 190 286 848 × 2 = 0 + 0.000 000 000 006 956 935 524 380 573 696;
  • 51) 0.000 000 000 006 956 935 524 380 573 696 × 2 = 0 + 0.000 000 000 013 913 871 048 761 147 392;
  • 52) 0.000 000 000 013 913 871 048 761 147 392 × 2 = 0 + 0.000 000 000 027 827 742 097 522 294 784;
  • 53) 0.000 000 000 027 827 742 097 522 294 784 × 2 = 0 + 0.000 000 000 055 655 484 195 044 589 568;
  • 54) 0.000 000 000 055 655 484 195 044 589 568 × 2 = 0 + 0.000 000 000 111 310 968 390 089 179 136;
  • 55) 0.000 000 000 111 310 968 390 089 179 136 × 2 = 0 + 0.000 000 000 222 621 936 780 178 358 272;
  • 56) 0.000 000 000 222 621 936 780 178 358 272 × 2 = 0 + 0.000 000 000 445 243 873 560 356 716 544;
  • 57) 0.000 000 000 445 243 873 560 356 716 544 × 2 = 0 + 0.000 000 000 890 487 747 120 713 433 088;
  • 58) 0.000 000 000 890 487 747 120 713 433 088 × 2 = 0 + 0.000 000 001 780 975 494 241 426 866 176;
  • 59) 0.000 000 001 780 975 494 241 426 866 176 × 2 = 0 + 0.000 000 003 561 950 988 482 853 732 352;
  • 60) 0.000 000 003 561 950 988 482 853 732 352 × 2 = 0 + 0.000 000 007 123 901 976 965 707 464 704;
  • 61) 0.000 000 007 123 901 976 965 707 464 704 × 2 = 0 + 0.000 000 014 247 803 953 931 414 929 408;
  • 62) 0.000 000 014 247 803 953 931 414 929 408 × 2 = 0 + 0.000 000 028 495 607 907 862 829 858 816;
  • 63) 0.000 000 028 495 607 907 862 829 858 816 × 2 = 0 + 0.000 000 056 991 215 815 725 659 717 632;
  • 64) 0.000 000 056 991 215 815 725 659 717 632 × 2 = 0 + 0.000 000 113 982 431 631 451 319 435 264;
  • 65) 0.000 000 113 982 431 631 451 319 435 264 × 2 = 0 + 0.000 000 227 964 863 262 902 638 870 528;
  • 66) 0.000 000 227 964 863 262 902 638 870 528 × 2 = 0 + 0.000 000 455 929 726 525 805 277 741 056;
  • 67) 0.000 000 455 929 726 525 805 277 741 056 × 2 = 0 + 0.000 000 911 859 453 051 610 555 482 112;
  • 68) 0.000 000 911 859 453 051 610 555 482 112 × 2 = 0 + 0.000 001 823 718 906 103 221 110 964 224;
  • 69) 0.000 001 823 718 906 103 221 110 964 224 × 2 = 0 + 0.000 003 647 437 812 206 442 221 928 448;
  • 70) 0.000 003 647 437 812 206 442 221 928 448 × 2 = 0 + 0.000 007 294 875 624 412 884 443 856 896;
  • 71) 0.000 007 294 875 624 412 884 443 856 896 × 2 = 0 + 0.000 014 589 751 248 825 768 887 713 792;
  • 72) 0.000 014 589 751 248 825 768 887 713 792 × 2 = 0 + 0.000 029 179 502 497 651 537 775 427 584;
  • 73) 0.000 029 179 502 497 651 537 775 427 584 × 2 = 0 + 0.000 058 359 004 995 303 075 550 855 168;
  • 74) 0.000 058 359 004 995 303 075 550 855 168 × 2 = 0 + 0.000 116 718 009 990 606 151 101 710 336;
  • 75) 0.000 116 718 009 990 606 151 101 710 336 × 2 = 0 + 0.000 233 436 019 981 212 302 203 420 672;
  • 76) 0.000 233 436 019 981 212 302 203 420 672 × 2 = 0 + 0.000 466 872 039 962 424 604 406 841 344;
  • 77) 0.000 466 872 039 962 424 604 406 841 344 × 2 = 0 + 0.000 933 744 079 924 849 208 813 682 688;
  • 78) 0.000 933 744 079 924 849 208 813 682 688 × 2 = 0 + 0.001 867 488 159 849 698 417 627 365 376;
  • 79) 0.001 867 488 159 849 698 417 627 365 376 × 2 = 0 + 0.003 734 976 319 699 396 835 254 730 752;
  • 80) 0.003 734 976 319 699 396 835 254 730 752 × 2 = 0 + 0.007 469 952 639 398 793 670 509 461 504;
  • 81) 0.007 469 952 639 398 793 670 509 461 504 × 2 = 0 + 0.014 939 905 278 797 587 341 018 923 008;
  • 82) 0.014 939 905 278 797 587 341 018 923 008 × 2 = 0 + 0.029 879 810 557 595 174 682 037 846 016;
  • 83) 0.029 879 810 557 595 174 682 037 846 016 × 2 = 0 + 0.059 759 621 115 190 349 364 075 692 032;
  • 84) 0.059 759 621 115 190 349 364 075 692 032 × 2 = 0 + 0.119 519 242 230 380 698 728 151 384 064;
  • 85) 0.119 519 242 230 380 698 728 151 384 064 × 2 = 0 + 0.239 038 484 460 761 397 456 302 768 128;
  • 86) 0.239 038 484 460 761 397 456 302 768 128 × 2 = 0 + 0.478 076 968 921 522 794 912 605 536 256;
  • 87) 0.478 076 968 921 522 794 912 605 536 256 × 2 = 0 + 0.956 153 937 843 045 589 825 211 072 512;
  • 88) 0.956 153 937 843 045 589 825 211 072 512 × 2 = 1 + 0.912 307 875 686 091 179 650 422 145 024;
  • 89) 0.912 307 875 686 091 179 650 422 145 024 × 2 = 1 + 0.824 615 751 372 182 359 300 844 290 048;
  • 90) 0.824 615 751 372 182 359 300 844 290 048 × 2 = 1 + 0.649 231 502 744 364 718 601 688 580 096;
  • 91) 0.649 231 502 744 364 718 601 688 580 096 × 2 = 1 + 0.298 463 005 488 729 437 203 377 160 192;
  • 92) 0.298 463 005 488 729 437 203 377 160 192 × 2 = 0 + 0.596 926 010 977 458 874 406 754 320 384;
  • 93) 0.596 926 010 977 458 874 406 754 320 384 × 2 = 1 + 0.193 852 021 954 917 748 813 508 640 768;
  • 94) 0.193 852 021 954 917 748 813 508 640 768 × 2 = 0 + 0.387 704 043 909 835 497 627 017 281 536;
  • 95) 0.387 704 043 909 835 497 627 017 281 536 × 2 = 0 + 0.775 408 087 819 670 995 254 034 563 072;
  • 96) 0.775 408 087 819 670 995 254 034 563 072 × 2 = 1 + 0.550 816 175 639 341 990 508 069 126 144;
  • 97) 0.550 816 175 639 341 990 508 069 126 144 × 2 = 1 + 0.101 632 351 278 683 981 016 138 252 288;
  • 98) 0.101 632 351 278 683 981 016 138 252 288 × 2 = 0 + 0.203 264 702 557 367 962 032 276 504 576;
  • 99) 0.203 264 702 557 367 962 032 276 504 576 × 2 = 0 + 0.406 529 405 114 735 924 064 553 009 152;
  • 100) 0.406 529 405 114 735 924 064 553 009 152 × 2 = 0 + 0.813 058 810 229 471 848 129 106 018 304;
  • 101) 0.813 058 810 229 471 848 129 106 018 304 × 2 = 1 + 0.626 117 620 458 943 696 258 212 036 608;
  • 102) 0.626 117 620 458 943 696 258 212 036 608 × 2 = 1 + 0.252 235 240 917 887 392 516 424 073 216;
  • 103) 0.252 235 240 917 887 392 516 424 073 216 × 2 = 0 + 0.504 470 481 835 774 785 032 848 146 432;
  • 104) 0.504 470 481 835 774 785 032 848 146 432 × 2 = 1 + 0.008 940 963 671 549 570 065 696 292 864;
  • 105) 0.008 940 963 671 549 570 065 696 292 864 × 2 = 0 + 0.017 881 927 343 099 140 131 392 585 728;
  • 106) 0.017 881 927 343 099 140 131 392 585 728 × 2 = 0 + 0.035 763 854 686 198 280 262 785 171 456;
  • 107) 0.035 763 854 686 198 280 262 785 171 456 × 2 = 0 + 0.071 527 709 372 396 560 525 570 342 912;
  • 108) 0.071 527 709 372 396 560 525 570 342 912 × 2 = 0 + 0.143 055 418 744 793 121 051 140 685 824;
  • 109) 0.143 055 418 744 793 121 051 140 685 824 × 2 = 0 + 0.286 110 837 489 586 242 102 281 371 648;
  • 110) 0.286 110 837 489 586 242 102 281 371 648 × 2 = 0 + 0.572 221 674 979 172 484 204 562 743 296;
  • 111) 0.572 221 674 979 172 484 204 562 743 296 × 2 = 1 + 0.144 443 349 958 344 968 409 125 486 592;

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:


0.000 000 000 000 000 000 000 000 006 179(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 1001 1000 1101 0000 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 000 000 006 179(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 1001 1000 1101 0000 001(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:


0.000 000 000 000 000 000 000 000 006 179(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 1001 1000 1101 0000 001(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 1001 1000 1101 0000 001(2) × 20 =

1.1110 1001 1000 1101 0000 001(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 1001 1000 1101 0000 001


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 0100 1100 0110 1000 0001 =

111 0100 1100 0110 1000 0001


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 0100 1100 0110 1000 0001


Decimal number 0.000 000 000 000 000 000 000 000 006 179 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0010 0111 - 111 0100 1100 0110 1000 0001

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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