0.000 000 000 000 000 000 000 000 672 084 213 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 000 000 672 084 213(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
0.000 000 000 000 000 000 000 000 672 084 213(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: 0.000 000 000 000 000 000 000 000 672 084 213.

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) 0.000 000 000 000 000 000 000 000 672 084 213 × 2 = 0 + 0.000 000 000 000 000 000 000 001 344 168 426;
  • 2) 0.000 000 000 000 000 000 000 001 344 168 426 × 2 = 0 + 0.000 000 000 000 000 000 000 002 688 336 852;
  • 3) 0.000 000 000 000 000 000 000 002 688 336 852 × 2 = 0 + 0.000 000 000 000 000 000 000 005 376 673 704;
  • 4) 0.000 000 000 000 000 000 000 005 376 673 704 × 2 = 0 + 0.000 000 000 000 000 000 000 010 753 347 408;
  • 5) 0.000 000 000 000 000 000 000 010 753 347 408 × 2 = 0 + 0.000 000 000 000 000 000 000 021 506 694 816;
  • 6) 0.000 000 000 000 000 000 000 021 506 694 816 × 2 = 0 + 0.000 000 000 000 000 000 000 043 013 389 632;
  • 7) 0.000 000 000 000 000 000 000 043 013 389 632 × 2 = 0 + 0.000 000 000 000 000 000 000 086 026 779 264;
  • 8) 0.000 000 000 000 000 000 000 086 026 779 264 × 2 = 0 + 0.000 000 000 000 000 000 000 172 053 558 528;
  • 9) 0.000 000 000 000 000 000 000 172 053 558 528 × 2 = 0 + 0.000 000 000 000 000 000 000 344 107 117 056;
  • 10) 0.000 000 000 000 000 000 000 344 107 117 056 × 2 = 0 + 0.000 000 000 000 000 000 000 688 214 234 112;
  • 11) 0.000 000 000 000 000 000 000 688 214 234 112 × 2 = 0 + 0.000 000 000 000 000 000 001 376 428 468 224;
  • 12) 0.000 000 000 000 000 000 001 376 428 468 224 × 2 = 0 + 0.000 000 000 000 000 000 002 752 856 936 448;
  • 13) 0.000 000 000 000 000 000 002 752 856 936 448 × 2 = 0 + 0.000 000 000 000 000 000 005 505 713 872 896;
  • 14) 0.000 000 000 000 000 000 005 505 713 872 896 × 2 = 0 + 0.000 000 000 000 000 000 011 011 427 745 792;
  • 15) 0.000 000 000 000 000 000 011 011 427 745 792 × 2 = 0 + 0.000 000 000 000 000 000 022 022 855 491 584;
  • 16) 0.000 000 000 000 000 000 022 022 855 491 584 × 2 = 0 + 0.000 000 000 000 000 000 044 045 710 983 168;
  • 17) 0.000 000 000 000 000 000 044 045 710 983 168 × 2 = 0 + 0.000 000 000 000 000 000 088 091 421 966 336;
  • 18) 0.000 000 000 000 000 000 088 091 421 966 336 × 2 = 0 + 0.000 000 000 000 000 000 176 182 843 932 672;
  • 19) 0.000 000 000 000 000 000 176 182 843 932 672 × 2 = 0 + 0.000 000 000 000 000 000 352 365 687 865 344;
  • 20) 0.000 000 000 000 000 000 352 365 687 865 344 × 2 = 0 + 0.000 000 000 000 000 000 704 731 375 730 688;
  • 21) 0.000 000 000 000 000 000 704 731 375 730 688 × 2 = 0 + 0.000 000 000 000 000 001 409 462 751 461 376;
  • 22) 0.000 000 000 000 000 001 409 462 751 461 376 × 2 = 0 + 0.000 000 000 000 000 002 818 925 502 922 752;
  • 23) 0.000 000 000 000 000 002 818 925 502 922 752 × 2 = 0 + 0.000 000 000 000 000 005 637 851 005 845 504;
  • 24) 0.000 000 000 000 000 005 637 851 005 845 504 × 2 = 0 + 0.000 000 000 000 000 011 275 702 011 691 008;
  • 25) 0.000 000 000 000 000 011 275 702 011 691 008 × 2 = 0 + 0.000 000 000 000 000 022 551 404 023 382 016;
  • 26) 0.000 000 000 000 000 022 551 404 023 382 016 × 2 = 0 + 0.000 000 000 000 000 045 102 808 046 764 032;
  • 27) 0.000 000 000 000 000 045 102 808 046 764 032 × 2 = 0 + 0.000 000 000 000 000 090 205 616 093 528 064;
  • 28) 0.000 000 000 000 000 090 205 616 093 528 064 × 2 = 0 + 0.000 000 000 000 000 180 411 232 187 056 128;
  • 29) 0.000 000 000 000 000 180 411 232 187 056 128 × 2 = 0 + 0.000 000 000 000 000 360 822 464 374 112 256;
  • 30) 0.000 000 000 000 000 360 822 464 374 112 256 × 2 = 0 + 0.000 000 000 000 000 721 644 928 748 224 512;
  • 31) 0.000 000 000 000 000 721 644 928 748 224 512 × 2 = 0 + 0.000 000 000 000 001 443 289 857 496 449 024;
  • 32) 0.000 000 000 000 001 443 289 857 496 449 024 × 2 = 0 + 0.000 000 000 000 002 886 579 714 992 898 048;
  • 33) 0.000 000 000 000 002 886 579 714 992 898 048 × 2 = 0 + 0.000 000 000 000 005 773 159 429 985 796 096;
  • 34) 0.000 000 000 000 005 773 159 429 985 796 096 × 2 = 0 + 0.000 000 000 000 011 546 318 859 971 592 192;
  • 35) 0.000 000 000 000 011 546 318 859 971 592 192 × 2 = 0 + 0.000 000 000 000 023 092 637 719 943 184 384;
  • 36) 0.000 000 000 000 023 092 637 719 943 184 384 × 2 = 0 + 0.000 000 000 000 046 185 275 439 886 368 768;
  • 37) 0.000 000 000 000 046 185 275 439 886 368 768 × 2 = 0 + 0.000 000 000 000 092 370 550 879 772 737 536;
  • 38) 0.000 000 000 000 092 370 550 879 772 737 536 × 2 = 0 + 0.000 000 000 000 184 741 101 759 545 475 072;
  • 39) 0.000 000 000 000 184 741 101 759 545 475 072 × 2 = 0 + 0.000 000 000 000 369 482 203 519 090 950 144;
  • 40) 0.000 000 000 000 369 482 203 519 090 950 144 × 2 = 0 + 0.000 000 000 000 738 964 407 038 181 900 288;
  • 41) 0.000 000 000 000 738 964 407 038 181 900 288 × 2 = 0 + 0.000 000 000 001 477 928 814 076 363 800 576;
  • 42) 0.000 000 000 001 477 928 814 076 363 800 576 × 2 = 0 + 0.000 000 000 002 955 857 628 152 727 601 152;
  • 43) 0.000 000 000 002 955 857 628 152 727 601 152 × 2 = 0 + 0.000 000 000 005 911 715 256 305 455 202 304;
  • 44) 0.000 000 000 005 911 715 256 305 455 202 304 × 2 = 0 + 0.000 000 000 011 823 430 512 610 910 404 608;
  • 45) 0.000 000 000 011 823 430 512 610 910 404 608 × 2 = 0 + 0.000 000 000 023 646 861 025 221 820 809 216;
  • 46) 0.000 000 000 023 646 861 025 221 820 809 216 × 2 = 0 + 0.000 000 000 047 293 722 050 443 641 618 432;
  • 47) 0.000 000 000 047 293 722 050 443 641 618 432 × 2 = 0 + 0.000 000 000 094 587 444 100 887 283 236 864;
  • 48) 0.000 000 000 094 587 444 100 887 283 236 864 × 2 = 0 + 0.000 000 000 189 174 888 201 774 566 473 728;
  • 49) 0.000 000 000 189 174 888 201 774 566 473 728 × 2 = 0 + 0.000 000 000 378 349 776 403 549 132 947 456;
  • 50) 0.000 000 000 378 349 776 403 549 132 947 456 × 2 = 0 + 0.000 000 000 756 699 552 807 098 265 894 912;
  • 51) 0.000 000 000 756 699 552 807 098 265 894 912 × 2 = 0 + 0.000 000 001 513 399 105 614 196 531 789 824;
  • 52) 0.000 000 001 513 399 105 614 196 531 789 824 × 2 = 0 + 0.000 000 003 026 798 211 228 393 063 579 648;
  • 53) 0.000 000 003 026 798 211 228 393 063 579 648 × 2 = 0 + 0.000 000 006 053 596 422 456 786 127 159 296;
  • 54) 0.000 000 006 053 596 422 456 786 127 159 296 × 2 = 0 + 0.000 000 012 107 192 844 913 572 254 318 592;
  • 55) 0.000 000 012 107 192 844 913 572 254 318 592 × 2 = 0 + 0.000 000 024 214 385 689 827 144 508 637 184;
  • 56) 0.000 000 024 214 385 689 827 144 508 637 184 × 2 = 0 + 0.000 000 048 428 771 379 654 289 017 274 368;
  • 57) 0.000 000 048 428 771 379 654 289 017 274 368 × 2 = 0 + 0.000 000 096 857 542 759 308 578 034 548 736;
  • 58) 0.000 000 096 857 542 759 308 578 034 548 736 × 2 = 0 + 0.000 000 193 715 085 518 617 156 069 097 472;
  • 59) 0.000 000 193 715 085 518 617 156 069 097 472 × 2 = 0 + 0.000 000 387 430 171 037 234 312 138 194 944;
  • 60) 0.000 000 387 430 171 037 234 312 138 194 944 × 2 = 0 + 0.000 000 774 860 342 074 468 624 276 389 888;
  • 61) 0.000 000 774 860 342 074 468 624 276 389 888 × 2 = 0 + 0.000 001 549 720 684 148 937 248 552 779 776;
  • 62) 0.000 001 549 720 684 148 937 248 552 779 776 × 2 = 0 + 0.000 003 099 441 368 297 874 497 105 559 552;
  • 63) 0.000 003 099 441 368 297 874 497 105 559 552 × 2 = 0 + 0.000 006 198 882 736 595 748 994 211 119 104;
  • 64) 0.000 006 198 882 736 595 748 994 211 119 104 × 2 = 0 + 0.000 012 397 765 473 191 497 988 422 238 208;
  • 65) 0.000 012 397 765 473 191 497 988 422 238 208 × 2 = 0 + 0.000 024 795 530 946 382 995 976 844 476 416;
  • 66) 0.000 024 795 530 946 382 995 976 844 476 416 × 2 = 0 + 0.000 049 591 061 892 765 991 953 688 952 832;
  • 67) 0.000 049 591 061 892 765 991 953 688 952 832 × 2 = 0 + 0.000 099 182 123 785 531 983 907 377 905 664;
  • 68) 0.000 099 182 123 785 531 983 907 377 905 664 × 2 = 0 + 0.000 198 364 247 571 063 967 814 755 811 328;
  • 69) 0.000 198 364 247 571 063 967 814 755 811 328 × 2 = 0 + 0.000 396 728 495 142 127 935 629 511 622 656;
  • 70) 0.000 396 728 495 142 127 935 629 511 622 656 × 2 = 0 + 0.000 793 456 990 284 255 871 259 023 245 312;
  • 71) 0.000 793 456 990 284 255 871 259 023 245 312 × 2 = 0 + 0.001 586 913 980 568 511 742 518 046 490 624;
  • 72) 0.001 586 913 980 568 511 742 518 046 490 624 × 2 = 0 + 0.003 173 827 961 137 023 485 036 092 981 248;
  • 73) 0.003 173 827 961 137 023 485 036 092 981 248 × 2 = 0 + 0.006 347 655 922 274 046 970 072 185 962 496;
  • 74) 0.006 347 655 922 274 046 970 072 185 962 496 × 2 = 0 + 0.012 695 311 844 548 093 940 144 371 924 992;
  • 75) 0.012 695 311 844 548 093 940 144 371 924 992 × 2 = 0 + 0.025 390 623 689 096 187 880 288 743 849 984;
  • 76) 0.025 390 623 689 096 187 880 288 743 849 984 × 2 = 0 + 0.050 781 247 378 192 375 760 577 487 699 968;
  • 77) 0.050 781 247 378 192 375 760 577 487 699 968 × 2 = 0 + 0.101 562 494 756 384 751 521 154 975 399 936;
  • 78) 0.101 562 494 756 384 751 521 154 975 399 936 × 2 = 0 + 0.203 124 989 512 769 503 042 309 950 799 872;
  • 79) 0.203 124 989 512 769 503 042 309 950 799 872 × 2 = 0 + 0.406 249 979 025 539 006 084 619 901 599 744;
  • 80) 0.406 249 979 025 539 006 084 619 901 599 744 × 2 = 0 + 0.812 499 958 051 078 012 169 239 803 199 488;
  • 81) 0.812 499 958 051 078 012 169 239 803 199 488 × 2 = 1 + 0.624 999 916 102 156 024 338 479 606 398 976;
  • 82) 0.624 999 916 102 156 024 338 479 606 398 976 × 2 = 1 + 0.249 999 832 204 312 048 676 959 212 797 952;
  • 83) 0.249 999 832 204 312 048 676 959 212 797 952 × 2 = 0 + 0.499 999 664 408 624 097 353 918 425 595 904;
  • 84) 0.499 999 664 408 624 097 353 918 425 595 904 × 2 = 0 + 0.999 999 328 817 248 194 707 836 851 191 808;
  • 85) 0.999 999 328 817 248 194 707 836 851 191 808 × 2 = 1 + 0.999 998 657 634 496 389 415 673 702 383 616;
  • 86) 0.999 998 657 634 496 389 415 673 702 383 616 × 2 = 1 + 0.999 997 315 268 992 778 831 347 404 767 232;
  • 87) 0.999 997 315 268 992 778 831 347 404 767 232 × 2 = 1 + 0.999 994 630 537 985 557 662 694 809 534 464;
  • 88) 0.999 994 630 537 985 557 662 694 809 534 464 × 2 = 1 + 0.999 989 261 075 971 115 325 389 619 068 928;
  • 89) 0.999 989 261 075 971 115 325 389 619 068 928 × 2 = 1 + 0.999 978 522 151 942 230 650 779 238 137 856;
  • 90) 0.999 978 522 151 942 230 650 779 238 137 856 × 2 = 1 + 0.999 957 044 303 884 461 301 558 476 275 712;
  • 91) 0.999 957 044 303 884 461 301 558 476 275 712 × 2 = 1 + 0.999 914 088 607 768 922 603 116 952 551 424;
  • 92) 0.999 914 088 607 768 922 603 116 952 551 424 × 2 = 1 + 0.999 828 177 215 537 845 206 233 905 102 848;
  • 93) 0.999 828 177 215 537 845 206 233 905 102 848 × 2 = 1 + 0.999 656 354 431 075 690 412 467 810 205 696;
  • 94) 0.999 656 354 431 075 690 412 467 810 205 696 × 2 = 1 + 0.999 312 708 862 151 380 824 935 620 411 392;
  • 95) 0.999 312 708 862 151 380 824 935 620 411 392 × 2 = 1 + 0.998 625 417 724 302 761 649 871 240 822 784;
  • 96) 0.998 625 417 724 302 761 649 871 240 822 784 × 2 = 1 + 0.997 250 835 448 605 523 299 742 481 645 568;
  • 97) 0.997 250 835 448 605 523 299 742 481 645 568 × 2 = 1 + 0.994 501 670 897 211 046 599 484 963 291 136;
  • 98) 0.994 501 670 897 211 046 599 484 963 291 136 × 2 = 1 + 0.989 003 341 794 422 093 198 969 926 582 272;
  • 99) 0.989 003 341 794 422 093 198 969 926 582 272 × 2 = 1 + 0.978 006 683 588 844 186 397 939 853 164 544;
  • 100) 0.978 006 683 588 844 186 397 939 853 164 544 × 2 = 1 + 0.956 013 367 177 688 372 795 879 706 329 088;
  • 101) 0.956 013 367 177 688 372 795 879 706 329 088 × 2 = 1 + 0.912 026 734 355 376 745 591 759 412 658 176;
  • 102) 0.912 026 734 355 376 745 591 759 412 658 176 × 2 = 1 + 0.824 053 468 710 753 491 183 518 825 316 352;
  • 103) 0.824 053 468 710 753 491 183 518 825 316 352 × 2 = 1 + 0.648 106 937 421 506 982 367 037 650 632 704;
  • 104) 0.648 106 937 421 506 982 367 037 650 632 704 × 2 = 1 + 0.296 213 874 843 013 964 734 075 301 265 408;

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:


0.000 000 000 000 000 000 000 000 672 084 213(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 1111 1111 1111 1111 1111(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 000 000 672 084 213(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 1111 1111 1111 1111 1111(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 81 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 000 000 672 084 213(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 1111 1111 1111 1111 1111(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 1111 1111 1111 1111 1111(2) × 20 =

1.1001 1111 1111 1111 1111 111(2) × 2-81


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): -81

Mantissa (not normalized):
1.1001 1111 1111 1111 1111 111


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-81 + 2(8-1) - 1 =

(-81 + 127)(10) =

46(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;
  • 46 ÷ 2 = 23 + 0;
  • 23 ÷ 2 = 11 + 1;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 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) =

46(10) =

0010 1110(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. 100 1111 1111 1111 1111 1111 =

100 1111 1111 1111 1111 1111


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 1110

Mantissa (23 bits) =
100 1111 1111 1111 1111 1111


Decimal number 0.000 000 000 000 000 000 000 000 672 084 213 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0010 1110 - 100 1111 1111 1111 1111 1111

» Convert decimal 0.000 000 000 000 000 000 000 000 672 084 19 to 32 bit single precision IEEE 754 binary floating point representation standard

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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