32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: 1 000 010 000 000 999 999 999 999 999 988 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

Number 1 000 010 000 000 999 999 999 999 999 988(10) converted and written in 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

1. 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;
  • 1 000 010 000 000 999 999 999 999 999 988 ÷ 2 = 500 005 000 000 499 999 999 999 999 994 + 0;
  • 500 005 000 000 499 999 999 999 999 994 ÷ 2 = 250 002 500 000 249 999 999 999 999 997 + 0;
  • 250 002 500 000 249 999 999 999 999 997 ÷ 2 = 125 001 250 000 124 999 999 999 999 998 + 1;
  • 125 001 250 000 124 999 999 999 999 998 ÷ 2 = 62 500 625 000 062 499 999 999 999 999 + 0;
  • 62 500 625 000 062 499 999 999 999 999 ÷ 2 = 31 250 312 500 031 249 999 999 999 999 + 1;
  • 31 250 312 500 031 249 999 999 999 999 ÷ 2 = 15 625 156 250 015 624 999 999 999 999 + 1;
  • 15 625 156 250 015 624 999 999 999 999 ÷ 2 = 7 812 578 125 007 812 499 999 999 999 + 1;
  • 7 812 578 125 007 812 499 999 999 999 ÷ 2 = 3 906 289 062 503 906 249 999 999 999 + 1;
  • 3 906 289 062 503 906 249 999 999 999 ÷ 2 = 1 953 144 531 251 953 124 999 999 999 + 1;
  • 1 953 144 531 251 953 124 999 999 999 ÷ 2 = 976 572 265 625 976 562 499 999 999 + 1;
  • 976 572 265 625 976 562 499 999 999 ÷ 2 = 488 286 132 812 988 281 249 999 999 + 1;
  • 488 286 132 812 988 281 249 999 999 ÷ 2 = 244 143 066 406 494 140 624 999 999 + 1;
  • 244 143 066 406 494 140 624 999 999 ÷ 2 = 122 071 533 203 247 070 312 499 999 + 1;
  • 122 071 533 203 247 070 312 499 999 ÷ 2 = 61 035 766 601 623 535 156 249 999 + 1;
  • 61 035 766 601 623 535 156 249 999 ÷ 2 = 30 517 883 300 811 767 578 124 999 + 1;
  • 30 517 883 300 811 767 578 124 999 ÷ 2 = 15 258 941 650 405 883 789 062 499 + 1;
  • 15 258 941 650 405 883 789 062 499 ÷ 2 = 7 629 470 825 202 941 894 531 249 + 1;
  • 7 629 470 825 202 941 894 531 249 ÷ 2 = 3 814 735 412 601 470 947 265 624 + 1;
  • 3 814 735 412 601 470 947 265 624 ÷ 2 = 1 907 367 706 300 735 473 632 812 + 0;
  • 1 907 367 706 300 735 473 632 812 ÷ 2 = 953 683 853 150 367 736 816 406 + 0;
  • 953 683 853 150 367 736 816 406 ÷ 2 = 476 841 926 575 183 868 408 203 + 0;
  • 476 841 926 575 183 868 408 203 ÷ 2 = 238 420 963 287 591 934 204 101 + 1;
  • 238 420 963 287 591 934 204 101 ÷ 2 = 119 210 481 643 795 967 102 050 + 1;
  • 119 210 481 643 795 967 102 050 ÷ 2 = 59 605 240 821 897 983 551 025 + 0;
  • 59 605 240 821 897 983 551 025 ÷ 2 = 29 802 620 410 948 991 775 512 + 1;
  • 29 802 620 410 948 991 775 512 ÷ 2 = 14 901 310 205 474 495 887 756 + 0;
  • 14 901 310 205 474 495 887 756 ÷ 2 = 7 450 655 102 737 247 943 878 + 0;
  • 7 450 655 102 737 247 943 878 ÷ 2 = 3 725 327 551 368 623 971 939 + 0;
  • 3 725 327 551 368 623 971 939 ÷ 2 = 1 862 663 775 684 311 985 969 + 1;
  • 1 862 663 775 684 311 985 969 ÷ 2 = 931 331 887 842 155 992 984 + 1;
  • 931 331 887 842 155 992 984 ÷ 2 = 465 665 943 921 077 996 492 + 0;
  • 465 665 943 921 077 996 492 ÷ 2 = 232 832 971 960 538 998 246 + 0;
  • 232 832 971 960 538 998 246 ÷ 2 = 116 416 485 980 269 499 123 + 0;
  • 116 416 485 980 269 499 123 ÷ 2 = 58 208 242 990 134 749 561 + 1;
  • 58 208 242 990 134 749 561 ÷ 2 = 29 104 121 495 067 374 780 + 1;
  • 29 104 121 495 067 374 780 ÷ 2 = 14 552 060 747 533 687 390 + 0;
  • 14 552 060 747 533 687 390 ÷ 2 = 7 276 030 373 766 843 695 + 0;
  • 7 276 030 373 766 843 695 ÷ 2 = 3 638 015 186 883 421 847 + 1;
  • 3 638 015 186 883 421 847 ÷ 2 = 1 819 007 593 441 710 923 + 1;
  • 1 819 007 593 441 710 923 ÷ 2 = 909 503 796 720 855 461 + 1;
  • 909 503 796 720 855 461 ÷ 2 = 454 751 898 360 427 730 + 1;
  • 454 751 898 360 427 730 ÷ 2 = 227 375 949 180 213 865 + 0;
  • 227 375 949 180 213 865 ÷ 2 = 113 687 974 590 106 932 + 1;
  • 113 687 974 590 106 932 ÷ 2 = 56 843 987 295 053 466 + 0;
  • 56 843 987 295 053 466 ÷ 2 = 28 421 993 647 526 733 + 0;
  • 28 421 993 647 526 733 ÷ 2 = 14 210 996 823 763 366 + 1;
  • 14 210 996 823 763 366 ÷ 2 = 7 105 498 411 881 683 + 0;
  • 7 105 498 411 881 683 ÷ 2 = 3 552 749 205 940 841 + 1;
  • 3 552 749 205 940 841 ÷ 2 = 1 776 374 602 970 420 + 1;
  • 1 776 374 602 970 420 ÷ 2 = 888 187 301 485 210 + 0;
  • 888 187 301 485 210 ÷ 2 = 444 093 650 742 605 + 0;
  • 444 093 650 742 605 ÷ 2 = 222 046 825 371 302 + 1;
  • 222 046 825 371 302 ÷ 2 = 111 023 412 685 651 + 0;
  • 111 023 412 685 651 ÷ 2 = 55 511 706 342 825 + 1;
  • 55 511 706 342 825 ÷ 2 = 27 755 853 171 412 + 1;
  • 27 755 853 171 412 ÷ 2 = 13 877 926 585 706 + 0;
  • 13 877 926 585 706 ÷ 2 = 6 938 963 292 853 + 0;
  • 6 938 963 292 853 ÷ 2 = 3 469 481 646 426 + 1;
  • 3 469 481 646 426 ÷ 2 = 1 734 740 823 213 + 0;
  • 1 734 740 823 213 ÷ 2 = 867 370 411 606 + 1;
  • 867 370 411 606 ÷ 2 = 433 685 205 803 + 0;
  • 433 685 205 803 ÷ 2 = 216 842 602 901 + 1;
  • 216 842 602 901 ÷ 2 = 108 421 301 450 + 1;
  • 108 421 301 450 ÷ 2 = 54 210 650 725 + 0;
  • 54 210 650 725 ÷ 2 = 27 105 325 362 + 1;
  • 27 105 325 362 ÷ 2 = 13 552 662 681 + 0;
  • 13 552 662 681 ÷ 2 = 6 776 331 340 + 1;
  • 6 776 331 340 ÷ 2 = 3 388 165 670 + 0;
  • 3 388 165 670 ÷ 2 = 1 694 082 835 + 0;
  • 1 694 082 835 ÷ 2 = 847 041 417 + 1;
  • 847 041 417 ÷ 2 = 423 520 708 + 1;
  • 423 520 708 ÷ 2 = 211 760 354 + 0;
  • 211 760 354 ÷ 2 = 105 880 177 + 0;
  • 105 880 177 ÷ 2 = 52 940 088 + 1;
  • 52 940 088 ÷ 2 = 26 470 044 + 0;
  • 26 470 044 ÷ 2 = 13 235 022 + 0;
  • 13 235 022 ÷ 2 = 6 617 511 + 0;
  • 6 617 511 ÷ 2 = 3 308 755 + 1;
  • 3 308 755 ÷ 2 = 1 654 377 + 1;
  • 1 654 377 ÷ 2 = 827 188 + 1;
  • 827 188 ÷ 2 = 413 594 + 0;
  • 413 594 ÷ 2 = 206 797 + 0;
  • 206 797 ÷ 2 = 103 398 + 1;
  • 103 398 ÷ 2 = 51 699 + 0;
  • 51 699 ÷ 2 = 25 849 + 1;
  • 25 849 ÷ 2 = 12 924 + 1;
  • 12 924 ÷ 2 = 6 462 + 0;
  • 6 462 ÷ 2 = 3 231 + 0;
  • 3 231 ÷ 2 = 1 615 + 1;
  • 1 615 ÷ 2 = 807 + 1;
  • 807 ÷ 2 = 403 + 1;
  • 403 ÷ 2 = 201 + 1;
  • 201 ÷ 2 = 100 + 1;
  • 100 ÷ 2 = 50 + 0;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number.

Take all the remainders starting from the bottom of the list constructed above.


1 000 010 000 000 999 999 999 999 999 988(10) =


1100 1001 1111 0011 0100 1110 0010 0110 0101 0110 1010 0110 1001 1010 0101 1110 0110 0011 0001 0110 0011 1111 1111 1111 0100(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 99 positions to the left, so that only one non zero digit remains to the left of it:


1 000 010 000 000 999 999 999 999 999 988(10) =


1100 1001 1111 0011 0100 1110 0010 0110 0101 0110 1010 0110 1001 1010 0101 1110 0110 0011 0001 0110 0011 1111 1111 1111 0100(2) =


1100 1001 1111 0011 0100 1110 0010 0110 0101 0110 1010 0110 1001 1010 0101 1110 0110 0011 0001 0110 0011 1111 1111 1111 0100(2) × 20 =


1.1001 0011 1110 0110 1001 1100 0100 1100 1010 1101 0100 1101 0011 0100 1011 1100 1100 0110 0010 1100 0111 1111 1111 1110 100(2) × 299


4. 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): 99


Mantissa (not normalized):
1.1001 0011 1110 0110 1001 1100 0100 1100 1010 1101 0100 1101 0011 0100 1011 1100 1100 0110 0010 1100 0111 1111 1111 1110 100


5. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =


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


99 + 2(8-1) - 1 =


(99 + 127)(10) =


226(10)


6. 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;
  • 226 ÷ 2 = 113 + 0;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

7. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


226(10) =


1110 0010(2)


8. 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =


1. 100 1001 1111 0011 0100 1110 0010 0110 0101 0110 1010 0110 1001 1010 0101 1110 0110 0011 0001 0110 0011 1111 1111 1111 0100 =


100 1001 1111 0011 0100 1110


9. 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) =
1110 0010


Mantissa (23 bits) =
100 1001 1111 0011 0100 1110


The base ten decimal number 1 000 010 000 000 999 999 999 999 999 988 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 1110 0010 - 100 1001 1111 0011 0100 1110

The latest decimal numbers converted from base ten to 32 bit single precision IEEE 754 floating point binary standard representation