Fun with Python’s decimal library

You’ve probably run across a link to this cute math fact in the past few days:

[\frac{1}{998001} = 0.000001002\ldots100101102\ldots900901902\ldots999\ldots]

This relatively simple fraction generates a decimal number that contains, in order, every three-digit sequence from 000 to 999 (except, sadly, 998). Pretty cool.

If it’s true. I don’t want to be cynical or disparage the integrity of the people who’ve been posting this, but folks on the internet don’t always get their math right. Shocking, I know.

So I was going to see if I could generate the decimal and check, but it slipped my mind until Matthew McVickar (Hawaii’s latest musical superstar, successor to Don Ho and Israel Kamakawiwoʻole), reminded me of it on Twitter.

@drdrang Have you seen this ‘mind-melting’ equation? Seems to be up your alley of amateur math naïveté. blog.untitledmagazine.net/post/166434898…
  — Matthew McVickar (@matthewmcvickar) Sat Jan 28 2012

Hmmm… I’m not sure how to take the “naïveté” comment, but I appreciate the diacritical marks.

When you need an answer to 3000 decimal places, you can’t just open up Octave and type in the fraction. Double precision floating point is going to come up about 2983 places short. The easiest way to check the fraction is to enter it in Wolfram Alpha. It’ll start out truncated, but you can keep asking for more digits until you get enough. The problem with Alpha is that the answer isn’t very nicely formatted.

I decided to try Python’s decimal library, a library I’d never used before. It’s main claim is its ability to represent decimals exactly rather than with the rounding and truncation associated with floating point arithmetic, but it can also handle huge numbers of decimal places.

Here’s a quick script to print out the decimal in a form that’s easy to read:

python:
 1:  #!/usr/bin/python
 2:  
 3:  from decimal import *
 4:  from sys import stdout
 5:  
 6:  getcontext().prec = 3000
 7:  num = Decimal(1)
 8:  denom = Decimal(998001)
 9:  dec = str(num/denom)[2:3002]
10:  
11:  for i,c in enumerate(dec):
12:    if i % 30 == 29:
13:      stdout.write(c + "\n")
14:    elif i % 3 == 2:
15:      stdout.write(c + " ")
16:    else:
17:      stdout.write(c)

Line 6 sets the precision to 3000 decimal places. Lines 7 and 8 set the numerator and denominator using the library’s Decimal constructor. Line 9 does the division, converts the result to string form, and extracts the 3000 characters after the decimal point. Lines 11-17 print out the result in lines of 30 characters, with spaces between every set of 3. There’s probably a clever way to get the same output with the format command, but I haven’t used format enough to be comfortable with it.

Here’s the result

000 001 002 003 004 005 006 007 008 009
010 011 012 013 014 015 016 017 018 019
020 021 022 023 024 025 026 027 028 029
030 031 032 033 034 035 036 037 038 039
040 041 042 043 044 045 046 047 048 049
050 051 052 053 054 055 056 057 058 059
060 061 062 063 064 065 066 067 068 069
070 071 072 073 074 075 076 077 078 079
080 081 082 083 084 085 086 087 088 089
090 091 092 093 094 095 096 097 098 099
100 101 102 103 104 105 106 107 108 109
110 111 112 113 114 115 116 117 118 119
120 121 122 123 124 125 126 127 128 129
130 131 132 133 134 135 136 137 138 139
140 141 142 143 144 145 146 147 148 149
150 151 152 153 154 155 156 157 158 159
160 161 162 163 164 165 166 167 168 169
170 171 172 173 174 175 176 177 178 179
180 181 182 183 184 185 186 187 188 189
190 191 192 193 194 195 196 197 198 199
200 201 202 203 204 205 206 207 208 209
210 211 212 213 214 215 216 217 218 219
220 221 222 223 224 225 226 227 228 229
230 231 232 233 234 235 236 237 238 239
240 241 242 243 244 245 246 247 248 249
250 251 252 253 254 255 256 257 258 259
260 261 262 263 264 265 266 267 268 269
270 271 272 273 274 275 276 277 278 279
280 281 282 283 284 285 286 287 288 289
290 291 292 293 294 295 296 297 298 299
300 301 302 303 304 305 306 307 308 309
310 311 312 313 314 315 316 317 318 319
320 321 322 323 324 325 326 327 328 329
330 331 332 333 334 335 336 337 338 339
340 341 342 343 344 345 346 347 348 349
350 351 352 353 354 355 356 357 358 359
360 361 362 363 364 365 366 367 368 369
370 371 372 373 374 375 376 377 378 379
380 381 382 383 384 385 386 387 388 389
390 391 392 393 394 395 396 397 398 399
400 401 402 403 404 405 406 407 408 409
410 411 412 413 414 415 416 417 418 419
420 421 422 423 424 425 426 427 428 429
430 431 432 433 434 435 436 437 438 439
440 441 442 443 444 445 446 447 448 449
450 451 452 453 454 455 456 457 458 459
460 461 462 463 464 465 466 467 468 469
470 471 472 473 474 475 476 477 478 479
480 481 482 483 484 485 486 487 488 489
490 491 492 493 494 495 496 497 498 499
500 501 502 503 504 505 506 507 508 509
510 511 512 513 514 515 516 517 518 519
520 521 522 523 524 525 526 527 528 529
530 531 532 533 534 535 536 537 538 539
540 541 542 543 544 545 546 547 548 549
550 551 552 553 554 555 556 557 558 559
560 561 562 563 564 565 566 567 568 569
570 571 572 573 574 575 576 577 578 579
580 581 582 583 584 585 586 587 588 589
590 591 592 593 594 595 596 597 598 599
600 601 602 603 604 605 606 607 608 609
610 611 612 613 614 615 616 617 618 619
620 621 622 623 624 625 626 627 628 629
630 631 632 633 634 635 636 637 638 639
640 641 642 643 644 645 646 647 648 649
650 651 652 653 654 655 656 657 658 659
660 661 662 663 664 665 666 667 668 669
670 671 672 673 674 675 676 677 678 679
680 681 682 683 684 685 686 687 688 689
690 691 692 693 694 695 696 697 698 699
700 701 702 703 704 705 706 707 708 709
710 711 712 713 714 715 716 717 718 719
720 721 722 723 724 725 726 727 728 729
730 731 732 733 734 735 736 737 738 739
740 741 742 743 744 745 746 747 748 749
750 751 752 753 754 755 756 757 758 759
760 761 762 763 764 765 766 767 768 769
770 771 772 773 774 775 776 777 778 779
780 781 782 783 784 785 786 787 788 789
790 791 792 793 794 795 796 797 798 799
800 801 802 803 804 805 806 807 808 809
810 811 812 813 814 815 816 817 818 819
820 821 822 823 824 825 826 827 828 829
830 831 832 833 834 835 836 837 838 839
840 841 842 843 844 845 846 847 848 849
850 851 852 853 854 855 856 857 858 859
860 861 862 863 864 865 866 867 868 869
870 871 872 873 874 875 876 877 878 879
880 881 882 883 884 885 886 887 888 889
890 891 892 893 894 895 896 897 898 899
900 901 902 903 904 905 906 907 908 909
910 911 912 913 914 915 916 917 918 919
920 921 922 923 924 925 926 927 928 929
930 931 932 933 934 935 936 937 938 939
940 941 942 943 944 945 946 947 948 949
950 951 952 953 954 955 956 957 958 959
960 961 962 963 964 965 966 967 968 969
970 971 972 973 974 975 976 977 978 979
980 981 982 983 984 985 986 987 988 989
990 991 992 993 994 995 996 997 999 000

Formatting the output this way makes it easy to see that the claim is correct. Too bad about the missing 998.


5 Responses to “Fun with Python’s decimal library”

  1. Matthew McVickar says:

    Hmmm… I’m not sure how to take the ‘musical superstar’ comment, but I appreciate the diacritical marks in ‘Kamakawiwoʻole’.

    Really, though, my tweet was poorly worded. By ‘up your alley of amateur math naïveté’ I meant ‘up your alley of looking into the veracity of trendy math facts’.

    I’m curious about the background in that embedded tweet of mine. My Twitter profile background is solid blue, and I don’t have a background image set. Looking up one of my tweets with the API, though, I see that my profile_background_image_url variable points to that pixelated background in the embedded tweet. I recognize it as a background image that I used a while ago. It would appear that hitting ‘Remove background’ on the Twitter ‘Design Settings’ page just flips a switch to show the color and not a background image, while the image itself remains attached to the profile. And since background images trump colors in CSS, that’s why my embedded tweet features that old image! Strange.

  2. pmcarlton says:

    The easiest way to check the fraction is to enter it in Wolfram Alpha.

    bc works pretty well too:

    echo "scale=3000;1/998001" | bc -l

  3. Seth Brown says:

    One way to get the formatting you desire is to use the wrap method in Python’s textwrap module. It’s fast and eliminates the conditional loop:

    #!/usr/bin/python
    
    import textwrap
    from decimal import *
    
    getcontext().prec = 3000
    num = Decimal(1)
    denom = Decimal(998001)
    dec = str(num/denom)[2:3002]
    num = ' '.join(textwrap.wrap(dec,3))
    line = '\n'.join(textwrap.wrap(num,40))
    print line
    
  4. Dr. Drang says:

    Matthew, it looks like I’ll have to check the Twitter API to see if there’s a public function that returns the value of the background switch.

    pm, nice use of bc! I’d forgotten that it was an arbitrary precision calculator.

    Seth, thanks for the pointers on textwrap. A much cleaner solution than mine. The formatting of your code got messed up by WordPress, so I went in and fixed it.

  5. james says:

    Using a similar method to this Hacker News comment, it is easy to see what is going on.

    0.000001+ 0.000000002+ 0.000000000003+ 0.000000000000004+… = Sum[(k)/10^(3 k+3), {k, 1, Infinity}]

    Wolfram Alpha