Examples
Examples get quite complicated quickly, here are small examples:
n
- = n (by rule 1)
p→q
- = pq (by rule 2)
- Thus 3→4 = 34 = 81
1→(any arrowed expression)
- = 1 since the entire expression eventually reduces to 1number = 1. (Indeed, any chain containing a 1 can be truncated just before that 1; e.g. X→1→Y=X for any (embedded) chains X,Y.)
4→3→2
- = 4→(4→(4)→1)→1 (by 4) and then, working from the inner parentheses outwards,
- = 4→(4→4→1)→1 (remove redundant parentheses )
- = 4→(4→4)→1 (3)
- = 4→(256)→1 (2)
- = 4→256→1 (rrp)
- = 4→256 (3)
- = 4256 (2)
- = 13 407 807 929 942 597 099 574 024 998 205 846 127 479 365 820 592 393 377 723 561 443 721 764 030 073 546 976 801 874 298 166 903 427 690 031 858 186 486 050 853 753 882 811 946 569 946 433 649 006 084 096 exactly ≈ 1.34078079299 × 10154
With Knuth's arrows:
2→2→4
- = 2→(2)→3 (by 4)
- = 2→2→3 (rrp)
- = 2→2→2 (4, rrp)
- = 2→2→1 (4, rrp)
- = 2→2 (3)
- = 4 (2) (In fact any chain beginning with two 2s stands for 4.)
2→4→3
- = 2→(2→(2→(2)→2)→2)→2 (by 4) The four copies of X (which is 2 here) are in bold to distinguish them from the three copies of q (which is also 2)
- = 2→(2→(2→2→2)→2)→2 (rrp)
- = 2→(2→(4)→2)→2 (previous example)
- = 2→(2→4→2)→2 (rrp) (expression expanded in next equation shown in bold on both lines)
- = 2→(2→(2→(2→(2)→1)→1)→1)→2 (4)
- = 2→(2→(2→(2→2→1)→1)→1)→2 (rrp)
- = 2→(2→(2→(2→2)))→2 (3 repeatedly)
- = 2→(2→(2→(4)))→2 (2)
- = 2→(2→(16))→2 (2)
- = 2→65536→2 (2,rrp)
- = 2→(2→(2→(...2→(2→(2)→1)→1...)→1)→1)→1 (4) with 65535 sets of parentheses
- = 2→(2→(2→(...2→(2→(2))...)))) (3 repeatedly)
- = 2→(2→(2→(...2→(4))...)))) (2)
- = 2→(2→(2→(...16...)))) (2)
- = (a tower with 216 = 65536 stories) = 655362 (See Tetration)
With Knuth's arrows: .
2→3→2→2
- = 2→3→(2→3)→1 (by 4)
- = 2→3→8 (2 and 3) With Knuth's arrows: 2 ↑8 3 (prop1)
- = 2→(2→2→7)→7 (1)
- = 2→4→7 (two initial 2's give 4 ) With Knuth's arrows: 2 ↑7 4 (prop1)
- = 2→(2→(2→2→6)→6)→6 (4)
- = 2→(2→4→6)→6 (prop6)
- = 2→(2→(2→(2→2→5)→5)→5)→6 (4)
- = 2→(2→(2→4→5)→5)→6 (prop6)
- = 2→(2→(2→(2→(2→2→4)→4)→4)→5)→6 (4)
- = 2→(2→(2→(2→4→4)→4)→5)→6 (prop6)
- = 2→(2→(2→(2→(2→(2→2→3)→3)→3)→4) →5)→6 (4)
- = 2→(2→(2→(2→(2→4→3)→3)→4)→5)→6 (prop6)
- = 2→(2→(2→(2→(2→65536→2)→3)→4)→5)→6 (previous example)
- = much larger than previous number
With Knuth's arrows:
3→2→2→2
- = 3→2→(3→2)→1 (4)
- = 3→2→9 (2 and 3)
- = 3→3→8 (4)
With Knuth's arrows: .
Read more about this topic: Conway Chained Arrow Notation
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)