Natural Numbers
	set of all positive integers		Natural Numbers
		also called whole numbers or counting numbers {1,2,3,4,5,...} closed under addition and multiplication

Integers
	set of all integers including the Natural Numbers and zero and the negative Integers		Integers Natural Numbers
		{...,-3,-2,-1,0,1,2,3,4,5,...} closed under addition, multiplication, subtraction

Rational Numbers
	set of all numbers that can be written as the ratio of two integers		Rational Numbers Integers Natural Numbers
		includes the Integers and, therefore, the Natural Numbers includes all fractions, proper (eg 2/3) and improper (eg 5/2) and integers closed under addition, multiplication, subtraction, and division note: an integer such as 3 can be written as 3/1; that is, as the ratio of two integers note: decimal representation is either a terminating decimal or a repeating one:       ______   1/2   =   0.5 3/7   =   0.428571 428571 428571 ...   =   0.428571 Decimal Representation of Rational Numbers

Real Numbers
	set of nonintersecting sets of Rational Numbers and Irrational Numbers (those that cannot be written as the ratio of two integers)		Real Numbers Rational Numbers Integers Natural Numbers
		since the Real Number Set includes the Rational Numbers, it therefore includes, the Integers and the Natural Numbers closed under addition, multiplication, subtraction, division, and square root of non-negatives Irrational Numbers are those that cannot be written as a terminating or repeating decimal in general, the square root of any natural number that is not a perfect square, cube roots of integers that are not perfect cubes, etc. plus many others such as e, the base of natural logarithms, and π, the ratio of the circumference to the diameter of a circle, are all irrational

5 = 5/1 = 5

13/4 = 3.25

1/17 =

_________________  0.05882 35294 11764 70588 23529 41176 47058 82352 94117 64705 88235 29411 76470 58823 52941 17647 05882 35294 11764 7...

1/31 =

________________  0.03225 80645 16129 03225 80645 16129 03225 80645 16129 03225 80645 16129 03225 80645 16129 03225 80645 16129...

By the way, any of the following is absolutely accurate and correct:

1/3 =

_  0.3

or

1/3 =

0.33333 33333 ...

or

1/3 ≈

0.33333 33333

The first two indicate that the decimal expansion goes on forever and are therefore exactly equal to 1/3. The last states that the truncated decimal expansion is only approximately equal to 1/3.

We often write 1/3 = 0.33333 which is only an approximation and not strictly true.

As long as this is understood, there is no problem. And, the concern is not the number of digits in the decimal expansion—more digits means we are closer— but whether there are a finite number of digits in the forever repeating decimal representation. Clearly writing 1/3 with one million 3's is closer than with five 3's, but it still in not exactly equal.



Examples of Irrational Numbers:



1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73799 07324 78462 10703 88503 87534 32764 15727
   35013 84623 09122 97024 92483 60558 50737 21264 41214 97099 93583 14132 22665 92750 55927 55799 95050 11527 82060 57147
   01095 59971 60597 02745 34596 86201 47285 17418 64088 91986 09552 32923 04843 08714 32145 08397 62603 62799 52514 07989
   68725 33965 46331 80882 96406 20615 25835 23950 54745 75028 77599 61729 83557 52203 37531 85701 13543 74603 40849 88471
   60386 89997 06990 04815 03054 40277 90316 45424 78230 68492 93691 86215 80578 46311 15966 68713 01301 56185 68987 23723
   52885 09264 86124 94977 15421 83342 04285 68606 01468 24720 77143 58548 74155 65706 96776 53720 22648 54470 15858 80162
   07584 74922 65722 60020 85584 46652 14583 98893 94437 09265 91800 31138 82464 68157 08263 01005 94858 70400 31864 80342
   19489 72782 90641 04507 26368 81313 73985 52561 17322 04024 50912 27700 22694 11275 73627 28049 57381 08967 50401 83698
   68368 45072 57993 64729 06076 29969 41380 47565 48237 28997 18032 68024 74420 62926 91248 59052 18100 44598 42150 59112
   02494 41341 72853 14781 05803 60337 10773 09182 86931 47101 71111 68391 65817 26889 41975 87165 82152 12822 95184 88472




2.23606 79774 99789 69640 91736 68731 27623 54406 18359 61152 57242 70897 24541 05209 25637 80489 94144 14408 37878 22749
   69508 17615 07737 83504 25326 77244 47073 86358 63601 21533 45270 88667 78173 19187 91658 11276 64532 26398 56580 53576
   13504 17533 78500 34233 92414 06444 20864 32539 09725 25926 27228 87629 95174 02440 68161 17759 08909 49849 23713 90729
   72889 84820 88641 54268 98940 99131 69357 70197 48678 88442 50897 54132 95618 31769 21499 97742 48015 30434 11503 59576
   68332 51249 88151 78139 40800 05624 20855 24354 22355 56106 30634 28202 34093 33198 29339 59746 35227 12013 41749 61420
   26359 04737 88550 43896 87061 13566 00457 57139 95659 55669 56917 56457 82219 52500 06053 92312 34005 00928 67648 75529
   72205 67662 53666 07448 58535 05262 33067 84946 33422 24231 76372 77026 63240 76801 04443 31582 57335 05893 09813 62263
   43198 68647 19469 89970 18081 89524 26445 96203 45221 41192 23291 25981 96325 81110 41704 95807 04812 04034 55994 94350
   68555 51855 57251 23886 41655 01026 24363 12571 02444 96187 89424 68290 34044 74716 11545 57232 01737 67659 04609 18529
   57560 35779 84398 05415 53807 79064 39363 97230 28756 06299 94822 13852 17734 85924 53515 12104 63455 55040 70722 78724




2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 45713 82178 52516 64274
   27466 39193 20030 59921 81741 35966 29043 57290 03342 95260 59563 07381 32328 62794 34907 63233 82988 07531 95251 01901
   15738 34187 93070 21540 89149 93488 41675 09244 76146 06680 82264 80016 84774 11853 74234 54424 37107 53907 77449 92069
   55170 27618 38606 26133 13845 83000 75204 49338 26560 29760 67371 13200 70932 87091 27443 74704 72306 96977 20931 01416
   92836 81902 55151 08657 46377 21112 52389 78442 50569 53696 77078 54499 69967 94686 44549 05987 93163 68892 30098 79312
   77361 78215 42499 92295 76351 48220 82698 95193 66803 31825 28869 39849 64651 05820 93923 98294 88793 32036 25094 43117
   30123 81970 68416 14039 70198 37679 32068 32823 76464 80429 53118 02328 78250 98194 55815 30175 67173 61332 06981 12509
   96181 88159 30416 90351 59888 85193 45807 27386 67385 89422 87922 84998 92086 80582 57492 79610 48419 84443 63463 24496
   84875 60233 62482 70419 78623 20900 21609 90235 30436 99418 49146 31409 34317 38143 64054 62531 52096 18369 08887 07016
   76839 64243 78140 59271 45635 49061 30310 72085 10383 75051 01157 47704 17189 86106 87396 96552 12671 54688 95703 50354




1.61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 28621 35448 62270 52604 62818 90244 97072 07204 18939 11374
   84754 08807 53868 91752 12663 38622 23536 93179 31800 60766 72635 44333 89086 59593 95829 05638 32266 13199 28290 26788
   06752 08766 89250 17116 96207 03222 10432 16269 54862 62963 13614 43814 97587 01220 34080 58879 54454 74924 61856 95364
   86444 92410 44320 77134 49470 49565 84678 85098 74339 44221 25448 77066 47809 15884 60749 98871 24007 65217 05751 79788
   34166 25624 94075 89069 70400 02812 10427 62177 11177 78053 15317 14101 17046 66599 14669 79873 17613 56006 70874 80710
   13179 52368 94275 21948 43530 56783 00228 78569 97829 77834 78458 78228 91109 76250 03026 96156 17002 50464 33824 37764
   86102 83831 26833 03724 29267 52631 16533 92473 16711 12115 88186 38513 31620 38400 52221 65791 28667 52946 54906 81131
   71599 34323 59734 94985 09040 94762 13222 98101 72610 70596 11645 62990 98162 90555 20852 47903 52406 02017 27997 47175
   34277 75927 78625 61943 20827 50513 12181 56285 51222 48093 94712 34145 17022 37358 05772 78616 00868 83829 52304 59264
   78780 17889 92199 02707 76903 89532 19681 98615 14378 03149 97411 06926 08867 42962 26757 56052 31727 77520 35361 39362




3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
   82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
   44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273
   72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094
   33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912
   98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 60943 70277 05392 17176 29317 67523 84674 81846 76694 05132
   00056 81271 45263 56082 77857 71342 75778 96091 73637 17872 14684 40901 22495 34301 46549 58537 10507 92279 68925 89235
   42019 95611 21290 21960 86403 44181 59813 62977 47713 09960 51870 72113 49999 99837 29780 49951 05973 17328 16096 31859
   50244 59455 34690 83026 42522 30825 33446 85035 26193 11881 71010 00313 78387 52886 58753 32083 81420 61717 76691 47303
   59825 34904 28755 46873 11595 62863 88235 37875 93751 95778 18577 80532 17122 68066 13001 92787 66111 95909 21642 01989

   38095 25720 10654 85863 27886 59361 53381 82796 82303 01952 03530 18529 68995 77362 25994 13891 24972 17752 83479 13151
   55748 57242 45415 06959 50829 53311 68617 27855 88907 50983 81754 63746 49393 19255 06040 09277 01671 13900 98488 24012
   85836 16035 63707 66010 47101 81942 95559 61989 46767 83744 94482 55379 77472 68471 04047 53464 62080 46684 25906 94912
   93313 67702 89891 52104 75216 20569 66024 05803 81501 93511 25338 24300 35587 64024 74964 73263 91419 92726 04269 92279
   67823 54781 63600 93417 21641 21992 45863 15030 28618 29745 55706 74983 85054 94588 58692 69956 90927 21079 75093 02955
   32116 53449 87202 75596 02364 80665 49911 98818 34797 75356 63698 07426 54252 78625 51818 41757 46728 90977 77279 38000
   81647 06001 61452 49192 17321 72147 72350 14144 19735 68548 16136 11573 52552 13347 57418 49468 43852 33239 07394 14333
   45477 62416 86251 89835 69485 56209 92192 22184 27255 02542 56887 67179 04946 01653 46680 49886 27232 79178 60857 84383
   82796 79766 81454 10095 38837 86360 95068 00642 25125 20511 73929 84896 08412 84886 26945 60424 19652 85022 21066 11863
   06744 27862 20391 94945 04712 37137 86960 95636 43719 17287 46776 46575 73962 41389 08658 32645 99581 33904 78027 59009

   94657 64078 95126 94683 98352 59570 98258 22620 52248 94077 26719 47826 84826 01476 99090 26401 36394 43745 53050 68203
   49625 24517 49399 65143 14298 09190 65925 09372 21696 46151 57098 58387 41059 78859 59772 97549 89301 61753 92846 81382
   68683 86894 27741 55991 85592 52459 53959 43104 99725 24680 84598 72736 44695 84865 38367 36222 62609 91246 08051 24388
   43904 51244 13654 97627 80797 71569 14359 97700 12961 60894 41694 86855 58484 06353 42207 22258 28488 64815 84560 28506
   01684 27394 52267 46767 88952 52138 52254 99546 66727 82398 64565 96116 35488 62305 77456 49803 55936 34568 17432 41125
   15076 06947 94510 96596 09402 52288 79710 89314 56691 36867 22874 89405 60101 50330 86179 28680 92087 47609 17824 93858
   90097 14909 67598 52613 65549 78189 31297 84821 68299 89487 22658 80485 75640 14270 47755 51323 79641 45152 37462 34364
   54285 84447 95265 86782 10511 41354 73573 95231 13427 16610 21359 69536 23144 29524 84937 18711 01457 65403 59027 99344
   03742 00731 05785 39062 19838 74478 08478 48968 33214 45713 86875 19435 06430 21845 31910 48481 00537 06146 80674 91927
   81911 97939 95206 14196 63428 75444 06437 45123 71819 21799 98391 01591 95618 14675 14269 12397 48940 90718 64942 31961

   56794 52080 95146 55022 52316 03881 93014 20937 62137 85595 66389 37787 08303 90697 92077 34672 21825 62599 66150 14215
   03068 03844 77345 49202 60541 46659 25201 49744 28507 32518 66600 21324 34088 19071 04863 31734 64965 14539 05796 26856
   10055 08106 65879 69981 63574 73638 40525 71459 10289 70641 40110 97120 62804 39039 75951 56771 57700 42033 78699 36007
   23055 87631 76359 42187 31251 47120 53292 81918 26186 12586 73215 79198 41484 88291 64470 60957 52706 95722 09175 67116
   72291 09816 90915 28017 35067 12748 58322 28718 35209 35396 57251 21083 57915 13698 82091 44421 00675 10334 67110 31412
   67111 36990 86585 16398 31501 97016 51511 68517 14376 57618 35155 65088 49099 89859 98238 73455 28331 63550 76479 18535
   89322 61854 89632 13293 30898 57064 20467 52590 70915 48141 65498 59461 63718 02709 81994 30992 44889 57571 28289 05923
   23326 09729 97120 84433 57326 54893 82391 19325 97463 66730 58360 41428 13883 03203 82490 37589 85243 74417 02913 27656
   18093 77344 40307 07469 21120 19130 20330 38019 76211 01100 44929 32151 60842 44485 96376 69838 95228 68478 31235 52658
   21314 49576 85726 24334 41893 03968 64262 43410 77322 69780 28073 18915 44110 10446 82325 27162 01052 65227 21116 60396

   66557 30925 47110 55785 37634 66820 65310 98965 26918 62056 47693 12570 58635 66201 85581 00729 36065 98764 86117 91045
   33488 50346 11365 76867 53249 44166 80396 26579 78771 85560 84552 96541 26654 08530 61434 44318 58676 97514 56614 06800
   70023 78776 59134 40171 27494 70420 56223 05389 94561 31407 11270 00407 85473 32699 39081 45466 46458 80797 27082 66830
   63432 85878 56983 05235 80893 30657 57406 79545 71637 75254 20211 49557 61581 40025 01262 28594 13021 64715 50979 25923
   09907 96547 37612 55176 56751 35751 78296 66454 77917 45011 29961 48903 04639 94713 29621 07340 43751 89573 59614 58901
   93897 13111 79042 97828 56475 03203 19869 15140 28708 08599 04801 09412 14722 13179 47647 77262 24142 54854 54033 21571
   85306 14228 81375 85043 06332 17518 29798 66223 71721 59160 77166 92547 48738 98665 49494 50114 65406 28433 66393 79003
   97692 65672 14638 53067 36096 57120 91807 63832 71664 16274 88880 07869 25602 90228 47210 40317 21186 08204 19000 42296
   61711 96377 92133 75751 14959 50156 60496 31862 94726 54736 42523 08177 03675 15906 73502 35072 83540 56704 03867 43513
   62222 47715 89150 49530 98444 89333 09634 08780 76932 59939 78054 19341 44737 74418 42631 29860 80998 88687 41326 04721