# 11A. Inverse

J provides a comprehensive calculus of inverses:

 a0=: I=: ^:_1 Inverse (adverb) m1 =: ^ I Natural log (^.); Inverse exponential m2 =: ^.I Exponential m3 =: 10&^. Base 10 log m4 =: m3 I Inverse base 10 log (10&^) m5 =: ] -: m4@m3 Tautology (test that m4 is left inverse) m6 =: ] -: m3@m4 Tautology (test that m4 is right inverse) m7 =: ssc=: +/\ Sum scan (subtotals) m8 =: ssc I Inverse sum scan; first differences m9 =: (]-:m8@m7)*.(]-:m7@m8) Tautology m10=: assc=: -/\ Alternating sum scan m11=: assc I m12=: ] -: m13=: * /\ I e.g. * /\ m13 3 1 4 15 9 26 5 3 m14=: % /\ I e.g. % /\ m14 3 1 4 15 9 26 5 3 m15=: ~:/\ I e.g. ~:/\ m15 1 1 0 1 1 0 1 1 m16=: = /\ I e.g. = /\ m16 1 1 0 1 1 0 1 1 m17=: + /\. I e.g. + /\. m17 3 1 4 15 9 26 5 3 m18=: - /\. I e.g. - /\. m18 3 1 4 15 9 26 5 3 m19=: * /\. I e.g. * /\. m19 3 1 4 15 9 26 5 3 m20=: % /\. I e.g. % /\. m20 3 1 4 15 9 26 5 3 m21=: ~:/\. I e.g. ~:/\. m21 1 1 0 1 1 0 1 1 m22=: = /\. I e.g. =/\. m22 1 1 0 1 1 0 1 1 d23=: # I Expand; 'ab' -: 1 0 1 # 1 0 1 d23 'ab' m24=: p: I p (n) the number of primes less than n m25=: x: I Floating point approx. of a rational. e.g. m25 3r7 m26=: 1&+ I Inverse increment; decrement m27=: +&1 I Inverse increment; decrement m28=: >: I Inverse increment; decrement m29=: _1&+ I Inverse decrement; increment m30=: +&_1 I Inverse decrement; increment m31=: -&1 I Inverse decrement; increment m32=: <: I Inverse decrement; increment m33=: 2&* I Inverse double; halve m34=: *&2 I Inverse double; halve m35=: +: I Inverse double; halve m36=: 0.5&* I Inverse halve; double m37=: *&0.5 I Inverse halve; double m38=: %&2 I Inverse halve; double m39=: -: I Inverse halve; double m40=: ^&2 I Inverse square m41=: ^&3 I Inverse cube m42=: ^&0.5 I Inverse square root m43=: ^&1r3 I Inverse cube root m44=: 2&^ I Inverse 2 with power; base 2 log m45=: 10&^ I Inverse 10 with power; base 10 log m46=: 2&! I Inverse triangular number. e.g. +/i.<.2&! I m m47=: +~ I Inverse double m48=: *~ I Inverse square m49=: ^~ I e.g. x^x=: ^~ I 12 m50=: (3&+)@(%&2)I -: (%&2 I)@(3&+ I) Inverse of composition is composition of inverses

These inverses may be illustrated as follows:

```   x=: 2 3 5 7
,.(] ; m1 ; m2 ; m1@m2) x
+---------------------------------+
&brvbar;2 3 5 7                          &brvbar;
+---------------------------------&brvbar;
&brvbar;0.6931472 1.09861 1.60944 1.94591&brvbar;
+---------------------------------&brvbar;
&brvbar;7.38906 20.0855 148.413 1096.63  &brvbar;
+---------------------------------&brvbar;
&brvbar;2 3 5 7                          &brvbar;
+---------------------------------+

(] ; m7 ; m8 ; m9) x
+---------------------------+
&brvbar;2 3 5 7&brvbar;2 5 10 17&brvbar;2 1 2 2&brvbar;1&brvbar;
+---------------------------+
(];m10;m11;m11@m10) x
+-----------------------------------+
&brvbar;2 3 5 7&brvbar;2 _1 4 _3&brvbar;2 _1 2 _2&brvbar;2 3 5 7&brvbar;
+-----------------------------------+```