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 Floor <.  0 0 0 Lesser Of (Min)

 <.y gives the floor of y , that is, the largest integer less than or equal to y . Thus: ``` <. 4.6 4 _4 _4.6 4 4 _4 _5 ``` The implied comparison with integers is tolerant, as discussed under Equal (=), and is controlled by <.!.t . See below for complex arguments. x<.y is the lesser of x and y . For example: ``` 3 <. 4 _4 3 _4 <./7 8 5 9 2 2 <./\7 8 5 9 2 7 7 5 5 2 ```

For a complex argument, the definition of <. is modelled by:
```   floor=: j./@(ip+(c2>c1),c1+:c2)
'`c1 c2 fp ip'=:(1:>+/@fp)`(>:/@fp)`(+.-ip)`(<.@+.)
```
As developed by McDonnell [10], this function has the following properties:

 Convexity: If (<.z1)=(<.z2) and z3 lies on the line between z1 to z2, then (<.z3)=(<.z1) . Translatability: If z4 is a Gaussian integer, then (z4+<.z5)=(<.z4+z5) . Compatibility: (<.x j.0)=((<.x)j.0) and (<.0 j.x)=(0 j.(<.x))

The function <. can be viewed as a tiling by rectangles of unit area, all arguments within a rectangle sharing the same floor. One rectangle has vertices at 1j0 and 0j1, with the other side passing through the origin. Rectangles along successive diagonals are displaced by one-half the length.

The phrase j./@ip “floors” the individual parts of a complex argument. Moreover, the floor <.y is equivalent to ->.-y . In other words, it is the dual of ceiling with respect to (that is, under) arithmetic negation: <. >.&.- and >. <.&.- . Thus:
```   (>.&.- ; <.) 4.6 4 _4 _4.6
+---------+---------+
|4 4 _4 _5|4 4 _4 _5|
+---------+---------+
```
The expression <.x+0.5 gives the integer nearest to the real argument x . The number of digits needed to represent a positive integer is given by one plus the floor of its base ten logarithm:
```   a ,. (,. 1:+<.) 10^. a=: 9 10 11 99 100 101
9 0.954243 1
10        1 2
11  1.04139 2
99  1.99564 2
100        2 3
101  2.00432 3
```

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