Chapter 20: Scalar Numerical Functions

In this chapter we look at built-in scalar functions for computing numbers from numbers. This chapter is a straight catalog of functions, with links to the sections as follows:

Ceiling	Conjugate	cos	cos-1	cosh	cosh-1
Decrement	divide	Double	Exponential	Factorial	Floor
GCD	Halve	Increment	LCM	Logarithm	Log, Natural
Magnitude	Minus	multiply	Negate	OutOf	PiTimes
Plus	power	Pythagorean	Reciprocal	Residue	Root
Signum	sin	sin-1	sinh	sinh-1	Square
SquareRoot	tan	tan-1	tanh	tanh-1

20.1 Numbers from Numbers

20.1.1 Plus and Conjugate

Dyadic + is arithmetic addition.

2 + 2	3j4 + 5j4	2r3 + 1r6
4	8j8	5r6

Monadic + is "Conjugate". For a real number y, the conjugate is y. For a complex number xjy (that is, x + 0jy), the conjugate is x - 0jy.

+ 2	+ 3j4
2	3j_4

20.1.2 Minus and Negate

Dyadic - is arithmetic subtraction.

2 - 2	3 - 0j4	2r3 - 1r6
0	3j_4	1r2

Monadic - is "Negate".

- 2	- 3j4
_2	_3j_4

20.1.3 Increment and Decrement

Monadic >: is called "Increment". It adds 1 to its argument.

>: 2	>: 2.5	>: 2r3	>: 2j3
3	3.5	5r3	3j3

Monadic <: is called "Decrement". It subtracts 1 from its argument.

<: 3	<: 2.5	<: 2r3	<: 2j3
2	1.5	_1r3	1j3

20.1.4 Times and Signum

Dyadic * is multiplication.

2 * 3	3j1 * 2j2
6	4j8

Monadic * is called "Signum". For a real number y, the value of (* y) is _1 or 0 or 1 as y is negative, zero or positive.

* _2	* 0	* 2
_1	0	1

More generally, y may be real or complex, and the signum is equivalent to y % | y. Hence the signum of a complex number has magnitude 1 and the same angle as the argument.

y =: 3j4	| y	y % | y	* y	| * y
3j4	5	0.6j0.8	0.6j0.8	1

20.1.5 Division and Reciprocal

Dyadic % is division.

2 % 3	3j4 % 2j1	12x % 5x
0.666667	2j1	12r5

1 % 0 is "infinity" but 0 % 0 is 0

1 % 0	0 % 0
_	0

Monadic % is the "reciprocal" function.

% 2	% 0j1
0.5	0j_1

20.1.6 Double and Halve

Monadic +: is the "double" verb.

+: 2.5	+: 3j4	+: 3x
5	6j8	6

Monadic -: is the "halve" verb:

-: 6	-: 6.5	-: 3j4	-: 3x
3	3.25	1.5j2	3r2

20.1.7 Floor and Ceiling

Monadic <. (left-angle-bracket dot) is called "Floor". For real y the floor of y is y rounded downwards to an integer, that is, the largest integer not exceeding y.

<. 2	<. 3.2	<. _3.2
2	3	_4

For complex y, the floor lies within a unit circle center y, that is, the magnitude of (y - <. y) is less than 1.

y =: 3.4j3.4	z =: <. y	y - z	| y-z
3.4j3.4	3j3	0.4j0.4	0.565685

This condition (magnitude less than 1) means that the floor of say 3.8j3.8 is not 3j3 but 4j3 because 3j3 does not satisfy the condition.

y =: 3.8j3.8	z =: <. y	| y-z	| y - 3j3
3.8j3.8	4j3	0.824621	1.13137

Monadic >. is called "Ceiling". For real y the ceiling of y is y rounded upwards to an integer, that is, the smallest integer greater than or equal to y. For example:

>. 3.0	>. 3.1	>. _2.5
3	4	_2

Ceiling applies to complex y

>. 3.4j3.4	>. 3.8j3.8
3j4	4j4

20.1.8 Power and Exponentiation

Dyadic ^ is the "power" verb: (x^y) is x raised-to-the-power y

10 ^ 2	10 ^ _2	100 ^ 1%2
100	0.01	10

Monadic ^ is exponentiation (or antilogarithm): ^y means (e^y) where e is Euler's constant, 2.71828...

^ 1	^ 0j1
2.71828	0.540302j0.841471

Euler's equation, supposedly engraved on his tombstone is: e ^{i pi} +1 = 0

   (^ 0j1p1) + 1
0

20.1.9 Square

Monadic *: is "Square".

*: 4	*: 2j1
16	3j4

20.1.10 Square Root

Monadic %: is "Square Root".

%: 9	%: 3j4	2j1 * 2j1
3	2j1	3j4

20.1.11 Root

If x is integral, then x %: y is the "x'th root" of y:

3 %: 8	_3 %: 8
2	0.5

More generally, (x %: y) is an abbreviation for (y ^ % x)

x =: 3 3.1	x %: 8	8 ^ % x
3 3.1	2 1.95578	2 1.95578

20.1.12 Logarithm and Natural Logarithm

Dyadic ^. is the base-x logarithm function, that is, (x ^. y) is the logarithm of y to base x :

10 ^. 1000	2 ^. 8
3	3

Monadic ^. is the "natural logarithm" function.

e =: ^ 1	^. e
2.71828	1

20.1.13 Factorial and OutOf

The factorial function is monadic !.

! 0 1 2 3 4	! 5x 6x 7x 8x
1 1 2 6 24	120 720 5040 40320

The number of combinations of x objects selected out of y objects is given by the expression x ! y

1 ! 4	2 ! 4	3 ! 4
4	6	4

20.1.14 Magnitude and Residue

Monadic | is called "Magnitude". For a real number y the magnitude of y is the absolute value:

| 2	| _2
2	2

More generally, y may be real or complex, and the magnitude is equivalent to (%: y * + y).

y =: 3j4	y * + y	%: y * + y	| y
3j4	25	5	5

The dyadic verb | is called "residue". the remainder when y is divided by x is given by (x | y).

10 | 12	3 | _2 _1 0 1 2 3 4 5	1.5 | 3.7
2	1 2 0 1 2 0 1 2	0.7

If x | y is zero, then x is a divisor of y:

4 | 12	12 % 4
0	3

The residue function applies to complex numbers:

a =: 1j2	b=: 2j3	a | b	a | (a*b)	(b-1j1) % a
1j2	2j3	1j1	0	1

20.1.15 GCD and LCM

The greatest common divisor (GCD) of x and y is given by (x +. y). Reals and rationals in the domain of +..

6 +. 15	_6 +. _15	2.5 +. 3.5	6r7 +. 15r7
3	3	0.5	3r7

Complex numbers are also in the domain of +..

a=: 1j2	b=:2j3	c=:3j5	(a*b) +. (b*c)
1j2	2j3	3j5	2j3

If x and y are complex, then x +. y may differ from y +. x.

1 +. 0j1	0j1 +. 1
1	0j1

We can see that the same result is produced by Euclid's algorithm for (x GCD y), which is: if y=0 then x, otherwise (x|y) GCD x. Here is a verb E, to model the algorithm.

   E  =:  (| E [) ` [ @. (]=0:)
   

6 E 15	1 +. 0j1	1 E 0j1	0j1 +. 1	0j1 E 1
3	1	1	0j1	0j1

The Least Common Multiple of x and y is given by (x *. y).

(2 * 3) *. (3 * 5)	2*3*5
30	30

20.2 Circle Functions

20.2.1 Pi Times

There is a built-in verb o. (lower-case o dot). Monadic o. is called "Pi Times"; it multiplies its argument by 3.14159...

o. 1	o. 2	o. 1r6
3.14159	6.28319	0.523599

20.2.2 Trigonometric and Other Functions

If y is an angle in radians, then the sine of y is given by the expression 1 o. y. The sine of (pi over 6) is 0.5

y =: o. 1r6	1 o. y
0.523599	0.5

The general scheme for dyadic o. is that (k o. y) means: apply to y a function selected by k. Giving conventional names to the available functions, we have:

   sin   =:  1 & o.  NB.  sine
   cos   =:  2 & o.  NB.  cosine 
   tan   =:  3 & o.  NB.  tangent
   
   sinh  =:  5 & o.  NB.  hyperbolic sine 
   cosh  =:  6 & o.  NB.  hyperbolic cosine  
   tanh  =:  7 & o.  NB.  hyperbolic tangent 
   
   asin  =: _1 & o.  NB.  inverse sine 
   acos  =: _2 & o.  NB.  inverse cosine 
   atan  =: _3 & o.  NB.  inverse tangent 
   
   asinh =: _5 & o.  NB.  inverse hyperbolic sine
   acosh =: _6 & o.  NB.  inverse hyperbolic cosine 
   atanh =: _7 & o.  NB.  inverse hyperbolic tangent
   
   

y	sin y	asin sin y
0.523599	0.5	0.523599

20.2.3 Pythagorean Functions

There are also the "pythagorean"functions:

      0 o. y  means   %:   1 - y^2
      4 o. y  means   %:   1 + y^2
      8 o. y  means   %: - 1 + y^2
     _4 o. y  means   %:  _1 + y^2
     _8 o. y  means - %: - 1 + y^2 
   

y =: 0.6	0 o. y	%: 1 - y^2
0.6	0.8	0.8

This is the end of chapter 20