Test integerness
Mathematically,
- the integers are included in the rational numbers ,
- which are included in the real numbers ,
- which can be generalized to the complex numbers .
This means that each of those larger sets, and the data types used to represent them, include some integers.
Given a rational, real, or complex number of any type, test whether it is mathematically an integer.
Your code should handle all numeric data types commonly used in your programming language. (If necessary, write a separate function for each input number type.)
For the purposes of this task, integerness means that a number could in theory be represented as an integer at no loss of precision. In other words:
| Set | Common representation | C++ examples | Considered an integer... |
|---|---|---|---|
| rational numbers | fraction | std::ratio
|
...if, in its reduced form, the denominator is 1 |
| real numbers (approximated) |
fixed-point floating-point |
float, double
|
...if, in decimal notation, it has no non-zero digits after the decimal point |
| complex numbers | pair of real numbers | std::complex
|
...if its real part is considered an integer and its imaginary part is zero |
(The table shows C++ types merely as an example – you should use the native data types of your programming language or standard library.)
| Input | Output | |
|---|---|---|
25.00000
|
(decimal) | true |
24.99999
|
(decimal) | false |
25.00001
|
(decimal) | false |
-2.1e120
|
(floating-point) | true |
-5e-2
|
(floating-point) | false |
5.0+0.0i
|
(complex) | true |
5-5i
|
(complex) | false |
(The notations shown in this table are merely examples – you should use the native number literals of your programming language. Feel free to use a different set of test-cases altogether.)
ALGOL 68
Uses LONG LONG values which in Algol 68 have a programmer specifiable number of digits. As with the C version, we need only handle the complex case directly, as Algol 68 will automatically coerce integer and real values to complex as required. <lang algol68># set the required precision of LONG LONG values using #
- "PR precision n PR" if required #
PR precision 24 PR
- returns TRUE if v has an integer value, FALSE otherwise #
OP ISINT = ( LONG LONG COMPL v )BOOL:
IF im OF v /= 0 THEN
# v has an imaginary part #
FALSE
ELSE
# v has a real part only #
ENTIER re OF v = v
FI; # ISINT #
- test ISINT #
PROC test is int = ( LONG LONG COMPLEX v )VOID:
print( ( re OF v, "_", im OF v, IF ISINT v THEN " is " ELSE " is not " FI, "integral", newline ) );
test is int( 1 );
test is int( 1.00000001 );
test is int( 4 I 3 );
test is int( 4.0 I 0 );
test is int( 123456789012345678901234 )
</lang>
- Output:
+1.0000000000000000000000000000000000e +0_+0.0000000000000000000000000000000000e +0 is integral +1.0000000100000000000000000000000000e +0_+0.0000000000000000000000000000000000e +0 is not integral +4.0000000000000000000000000000000000e +0_+3.0000000000000000000000000000000000e +0 is not integral +4.0000000000000000000000000000000000e +0_+0.0000000000000000000000000000000000e +0 is integral +1.2345678901234567890123400000000000e +23_+0.0000000000000000000000000000000000e +0 is integral
C
The main function that checks a numeric value is actually quite short. Because of C's weak types and implicit casting we can get away with making a function which checks long double complex types only.
<lang c>
- include <stdio.h>
- include <complex.h>
- include <math.h>
/* Testing macros */
- define FMTSPEC(arg) _Generic((arg), \
float: "%f", double: "%f", \ long double: "%Lf", unsigned int: "%u", \ unsigned long: "%lu", unsigned long long: "%llu", \ int: "%d", long: "%ld", long long: "%lld", \ default: "(invalid type (%p)")
- define CMPPARTS(x, y) ((long double complex)((long double)(x) + \
I * (long double)(y)))
- define TEST_CMPL(i, j)\
printf(FMTSPEC(i), i), printf(" + "), printf(FMTSPEC(j), j), \
printf("i = %s\n", (isint(CMPPARTS(i, j)) ? "true" : "false"))
- define TEST_REAL(i)\
printf(FMTSPEC(i), i), printf(" = %s\n", (isint(i) ? "true" : "false"))
/* Main code */ static inline int isint(long double complex n) {
return cimagl(n) == 0 && nearbyintl(creall(n)) == creall(n);
}
int main(void) {
TEST_REAL(0); TEST_REAL(-0); TEST_REAL(-2); TEST_REAL(-2.00000000000001); TEST_REAL(5); TEST_REAL(7.3333333333333); TEST_REAL(3.141592653589); TEST_REAL(-9.223372036854776e18); TEST_REAL(5e-324); TEST_REAL(NAN); TEST_CMPL(6, 0); TEST_CMPL(0, 1); TEST_CMPL(0, 0); TEST_CMPL(3.4, 0);
/* Demonstrating that we can use the same function for complex values
* constructed in the standard way */
double complex test1 = 5 + 0*I,
test2 = 3.4f,
test3 = 3,
test4 = 0 + 1.2*I;
printf("Test 1 (5+i) = %s\n", isint(test1) ? "true" : "false");
printf("Test 2 (3.4+0i) = %s\n", isint(test2) ? "true" : "false");
printf("Test 3 (3+0i) = %s\n", isint(test3) ? "true" : "false");
printf("Test 4 (0+1.2i) = %s\n", isint(test4) ? "true" : "false");
} </lang>
- Output:
Note: Some of the printed results are truncated and look incorrect. See the actual code if you wish to verify the actual value.
0 = true 0 = true -2 = true -2.000000 = false 5 = true 7.333333 = false 3.141593 = false -9223372036854775808.000000 = true 0.000000 = false nan = false 6 + 0i = true 0 + 1i = false 0 + 0i = true 3.400000 + 0i = false Test 1 (5+i) = true Test 2 (3.4+0i) = false Test 3 (3+0i) = true Test 4 (0+1.2i) = false
COBOL
COBOL likes to work with fixed-point decimal numbers. For the sake of argument, this program tests the "integerness" of values that have up to nine digits before the decimal point and up to nine digits after. It can therefore be "tricked", as in the third of the four tests below, by computing a result that differs from an integer by less than 0.000000001; if there is any likelihood of such results arising, it would be a good idea to allow more digits of precision after the decimal point. Support for complex numbers (in a sense) is included, because the specification calls for it—but it adds little of interest. <lang cobol>IDENTIFICATION DIVISION. PROGRAM-ID. INTEGERNESS-PROGRAM. DATA DIVISION. WORKING-STORAGE SECTION. 01 INTEGERS-OR-ARE-THEY.
05 POSSIBLE-INTEGER PIC S9(9)V9(9). 05 DEFINITE-INTEGER PIC S9(9).
01 COMPLEX-NUMBER.
05 REAL-PART PIC S9(9)V9(9). 05 IMAGINARY-PART PIC S9(9)V9(9).
PROCEDURE DIVISION. TEST-PARAGRAPH.
MOVE ZERO TO IMAGINARY-PART. DIVIDE -28 BY 7 GIVING POSSIBLE-INTEGER. PERFORM INTEGER-PARAGRAPH. DIVIDE 28 BY 18 GIVING POSSIBLE-INTEGER. PERFORM INTEGER-PARAGRAPH. DIVIDE 3 BY 10000000000 GIVING POSSIBLE-INTEGER. PERFORM INTEGER-PARAGRAPH.
TEST-COMPLEX-PARAGRAPH.
MOVE ZERO TO REAL-PART. MOVE 1 TO IMAGINARY-PART. MOVE REAL-PART TO POSSIBLE-INTEGER. PERFORM INTEGER-PARAGRAPH. STOP RUN.
INTEGER-PARAGRAPH.
IF IMAGINARY-PART IS EQUAL TO ZERO THEN PERFORM REAL-PARAGRAPH, ELSE PERFORM COMPLEX-PARAGRAPH.
REAL-PARAGRAPH.
MOVE POSSIBLE-INTEGER TO DEFINITE-INTEGER. IF DEFINITE-INTEGER IS EQUAL TO POSSIBLE-INTEGER THEN DISPLAY POSSIBLE-INTEGER ' IS AN INTEGER.', ELSE DISPLAY POSSIBLE-INTEGER ' IS NOT AN INTEGER.'.
COMPLEX-PARAGRAPH.
DISPLAY REAL-PART '+' IMAGINARY-PART 'i IS NOT AN INTEGER.'.</lang>
- Output:
-000000004.000000000 IS AN INTEGER. 000000001.555555555 IS NOT AN INTEGER. 000000000.000000000 IS AN INTEGER. 000000000.000000000+ 000000001.000000000i IS NOT AN INTEGER.
Elixir
<lang elixir>defmodule Test do
def integer?(n) when n == trunc(n), do: true def integer?(_), do: false
end
Enum.each([2, 2.0, 2.5, 2.000000000000001, 1.23e300, 1.0e-300, "123", '123', :"123"], fn n ->
IO.puts "#{inspect n} is integer?: #{Test.integer?(n)}"
end)</lang>
- Output:
2 is integer?: true 2.0 is integer?: true 2.5 is integer?: false 2.000000000000001 is integer?: false 1.23e300 is integer?: true 1.0e-300 is integer?: false "123" is integer?: false '123' is integer?: false :"123" is integer?: false
Fortran
Straightforward
The issue is a little delicate, because a number such as 3E120 is integral, but, cannot be represented in an ordinary integer variable. If the idea slides towards whether the value can be represented exactly by an integer variable then the task become easy. The following code takes advantage of the availability of INTEGER*8 variables, and an associated function KIDINT(x) that deals with double precision values and returns a 64-bit integer result. This function truncates - if it were to round then it might want to round up to one beyond the maximum possible integer of that sign. There is a bewildering variety of these truncation and rounding functions, some of which are generic and some not, and if only INTEGER*4 were available, different choices would have to be made.
The MODULE protocol of F90 is used, merely to save on the need to define the types of the function in each routine that uses them, since there is no default type for LOGICAL. Otherwise, this is F77 style. <lang Fortran> MODULE ZERMELO !Approach the foundations of mathematics.
CONTAINS
LOGICAL FUNCTION ISINTEGRAL(X) !A whole number?
REAL*8 X !Alas, this is not really a REAL number.
INTEGER*8 N !Largest available.
IF (ISNAN(X)) THEN !Avoid some sillyness.
ISINTEGRAL = .FALSE. !And possible error messages.
ELSE !But now it is safe to try.
N = KIDINT(X) !This one truncates.
ISINTEGRAL = N .EQ. X !Any difference?
END IF !A floating-point number may overflow an integer.
END FUNCTION ISINTEGRAL !And even if integral, it will not seem so.
LOGICAL FUNCTION ISINTEGRALZ(Z) !For complex numbers, two tests.
DOUBLE COMPLEX Z !Still not really REAL, though.
ISINTEGRALZ = ISINTEGRAL(DBLE(Z)) .AND. ISINTEGRAL(DIMAG(Z)) !Separate the parts.
END FUNCTION ISINTEGRALZ!No INTEGER COMPLEX type is offered.
END MODULE ZERMELO !Much more mathematics lie elsewhere.
PROGRAM TEST
USE ZERMELO
DOUBLE COMPLEX Z
WRITE (6,*) "See if some numbers are integral..."
WRITE (6,*) ISINTEGRAL(666D0),666D0
Z = DCMPLX(-3D0,4*ATAN(1D0))
WRITE (6,*) ISINTEGRALZ(Z),Z
END</lang>
See if some numbers are integral... T 666.000000000000 F (-3.00000000000000,3.14159265358979)
Tricky
If however large numbers are to be affirmed as integral even if there is no integer variable capable of holding such values, then a different approach is required. Given that a floating-point number has a finite precision, there will be some number above which no digits can be devoted to fractional parts and so the number represented by the floating-point value must be integral, while for smaller numbers the floating point value can be compared to its integer truncation, as above. Suppose a decimal computer (like the IBM1620!) for convenience, using eight decimal digits for the mantissa (and two for the exponent, as did the IBM1620). A (non-zero) normalised number must be of the form d·ddddddd and the largest number with a fractional digit would be 9999999·9 (represented as 9·9999999E+06 or possibly as ·99999999E+07 depending on the style of normalisation) and the next possible floating-point number would be 1·0000000E+07, then 1·0000001E+07, etc. advancing in steps of one. No fractional part is possible. Thus the boundary is clear and a test need merely involve a comparison: integral if greater than that, otherwise compare the floating-point number to its truncated integer form.
The argument is the same with binary (or base 4, 8 or 16), but, you have to know what base is used to prepare the proper boundary value, similarly you must ascertain just how many digits of precision are in use, remembering that in binary the leading one of normalised numbers may be represented implicitly, or it may be explicitly present. One would have to devise probing routines with delicate calculations that may be disrupted by various compiler optimisation tricks and unanticipated details of the arithmetic mill. For instance, The Intel 8087 floating-point co-processor and its descendants use an implicit leading-one bit for 32- and 64-bit floating-point numbers, but not for 80-bit floating-point numbers, and during a calculation (add, subtract, multiply, divide) a further three guard bits (with special meanings) are employed... Calculations are done with full 83-bit precision to yield an 80-bit result; it is only when values are stored that they are converted to single or double precision format in storage - the register retains full precision. Probe routines that worked for other designs will likely need tweaking. To determine the number of digits of precision, one probes somewhat as follows:<lang Fortran> X = 1
10 X = X*BASE
Y = X + 1
D = Y - X
IF (D .EQ. 1) GO TO 10</lang>
Or alternatively, compare 1 + eps to 1, successively dividing eps by BASE.
The difficulties include the risk that a compiler might wrongly apply the axia of mathematics to floating-point arithmetic and deduce that D was always one. Similarly, after assigning the result of X + 1 to Y, it may notice that the register could retain that value and so there would be no need to load Y's value to calculate Y - X: if the register was of greater precision than the variable, the probe will err. Producing output can help. As well as being interesting, otherwise a compiler might deduce that there is no need to calculate something because it is not used to produce output nor affects something that does lead to output.
Conveniently, F90 standardised the functions FRACTION(x) and EXPONENT(x) that reveal the parts of a floating point number and with the assistance of the RADIX(x) function that reports the base of the number system and DIGITS(x) the number of digits represented, a suitable boundary value can be constructed. For the decimal example above, 1E7 is the first value that has no space for fractional digits.
If the highest-precision floating-point number is 64-bit, and the largest integer is 64-bit, then, given that some of the 64 bits of the floating-point number are devoted to the exponent, floating-point values up to the threshold will never overflow a 64-bit integer range, and all will be well... A similar process would apply for 32-bit floating-point variables, and so on.
Since the special functions are only available for F90 and later, the example proceeds to activate the F90 protocol for making a function (or subroutine) generic. This requires a suitable function for each desired combination of parameter types and ringing the changes can soon become tedious as well as error-prone, though fortunately here, there is only one parameter and its types to work through. It would be helpful to have a decent pre-processor scheme (such as in pl/i) whereby the various routines would be generated, but there can be surprise obstacles also. Here, INTEGER*8 is not fully incorporated into the compiler as a possibility for integer constants, so it is necessary to pre-define INTEGER*8 BIG so that in its PARAMETER statement the compiler's calculation will have sufficient scope. The INTEGER*4 routine has no such difficulty.
Despite the attempt at generality, there will be difficulties on systems whose word sizes are not multiples of eight bits so that the REAL*4 scheme falters. Still, it is preferable to a blizzard of terms such as small int, int, long int, long long int. A decimal computer would be quite different in its size specifications, and there have been rumours of a Russian computer that worked in base three... <lang Fortran>
MODULE ZERMELO !Approach the foundations of mathematics.
INTERFACE ISINTEGRAL !And obscure them with computerese.
MODULE PROCEDURE ISINTEGRALF4, ISINTEGRALF8,
1 ISINTEGRALZ8, ISINTEGRALZ16
END INTERFACE !Selection is by parameter type and number.
CONTAINS !Sop, now for a grabbag of routines.
LOGICAL FUNCTION ISINTEGRALF8(X) !A whole number?
REAL*8 X !Alas, this is not really a REAL number.
INTEGER*8 N !Largest available.
INTEGER*8 BIG !The first number too big to have any fractional digits in floating-point.
PARAMETER (BIG = RADIX(X)**(DIGITS(X) - 1)) !These "functions" are in fact constants.
IF (ISNAN(X)) THEN !Avoid some sillyness.
ISINTEGRALF8 = .FALSE. !And possible error messages.
ELSE IF (ABS(X).GE.BIG) THEN !But now it is safe to try.
ISINTEGRALF8 = .TRUE. !Can't have fractional digits => integral.
ELSE !But smaller numbers can have fractional digits.
N = KIDINT(X) !So, truncate to an integral value.
ISINTEGRALF8 = N .EQ. X !Any difference?
END IF !So much for inspection.
END FUNCTION ISINTEGRALF8 !No need to look at digit sequences.
LOGICAL FUNCTION ISINTEGRALF4(X) !A whole number?
REAL*4 X !Alas, this is not really a REAL number.
INTEGER*4 N !Largest available.
IF (ISNAN(X)) THEN !Avoid some sillyness.
ISINTEGRALF4 = .FALSE. !And possible error messages.
ELSE IF (ABS(X) .GE. RADIX(X)**(DIGITS(X) - 1)) THEN !Constant results as appropriate for X.
ISINTEGRALF4 = .TRUE. !Can't have fractional digits => integral.
ELSE !But smaller numbers can have fractional digits.
N = INT(X) !So, truncate to an integral value.
ISINTEGRALF4 = N .EQ. X !Any difference?
END IF !A real*4 should not overflow INTEGER*4.
END FUNCTION ISINTEGRALF4 !Thanks to the size check.
LOGICAL FUNCTION ISINTEGRALZ8(Z) !For complex numbers, two tests.
COMPLEX Z !Still not really REAL, though.
ISINTEGRALZ8 = ISINTEGRAL(REAL(Z)) .AND. ISINTEGRAL(AIMAG(Z)) !Separate the parts.
END FUNCTION ISINTEGRALZ8 !No INTEGER COMPLEX type is offered.
LOGICAL FUNCTION ISINTEGRALZ16(Z) !And there are two sorts of complex numbers.
DOUBLE COMPLEX Z !Still not really REAL.
ISINTEGRALZ16 = ISINTEGRAL(DBLE(Z)) .AND. ISINTEGRAL(DIMAG(Z)) !Separate the parts.
END FUNCTION ISINTEGRALZ16 !No INTEGER COMPLEX type is offered.
END MODULE ZERMELO !Much more mathematics lie elsewhere.
PROGRAM TEST
USE ZERMELO
DOUBLE COMPLEX Z
DOUBLE PRECISION X
REAL U
Cast forth some pearls.
WRITE (6,1) 4,DIGITS(U),RADIX(U)
WRITE (6,1) 8,DIGITS(X),RADIX(X)
1 FORMAT ("REAL*",I1,":",I3," digits, in base",I2)
WRITE (6,*) "See if some numbers are integral..."
WRITE (6,*) ISINTEGRAL(666D0),666D0
WRITE (6,*) ISINTEGRAL(665.9),665.9
Z = DCMPLX(-3D0,4*ATAN(1D0))
WRITE (6,*) ISINTEGRAL(Z),Z
END
</lang>
REAL*4: 24 digits, in base 2 REAL*8: 53 digits, in base 2 See if some numbers are integral... T 666.000000000000 F 665.9000 F (-3.00000000000000,3.14159265358979)
Go
<lang go>package main
import ( "fmt" "math" "math/big" "reflect" "strings" "unsafe" )
// Go provides an integerness test only for the big.Rat type in the standard // library.
// The fundamental piece of code needed for built-in floating point types // is a test on the float64 type:
func Float64IsInt(f float64) bool { _, frac := math.Modf(f) return frac == 0 }
// Other built-in or stanadard library numeric types are either always // integer or can be easily tested using Float64IsInt.
func Float32IsInt(f float32) bool { return Float64IsInt(float64(f)) }
func Complex128IsInt(c complex128) bool { return imag(c) == 0 && Float64IsInt(real(c)) }
func Complex64IsInt(c complex64) bool { return imag(c) == 0 && Float64IsInt(float64(real(c))) }
// Usually just the above statically typed functions would be all that is used, // but if it is desired to have a single function that can test any arbitrary // type, including the standard math/big types, user defined types based on // an integer, float, or complex builtin types, or user defined types that // have an IsInt() method, then reflection can be used.
type hasIsInt interface { IsInt() bool }
var bigIntT = reflect.TypeOf((*big.Int)(nil))
func IsInt(i interface{}) bool { if ci, ok := i.(hasIsInt); ok { // Handles things like *big.Rat return ci.IsInt() } switch v := reflect.ValueOf(i); v.Kind() { case reflect.Int, reflect.Int8, reflect.Int16, reflect.Int32, reflect.Int64, reflect.Uint, reflect.Uint8, reflect.Uint16, reflect.Uint32, reflect.Uint64, reflect.Uintptr: // Built-in types and any custom type based on them return true case reflect.Float32, reflect.Float64: // Built-in floats and anything based on them return Float64IsInt(v.Float()) case reflect.Complex64, reflect.Complex128: // Built-in complexes and anything based on them return Complex128IsInt(v.Complex()) case reflect.String: // Could also do strconv.ParseFloat then FloatIsInt but // big.Rat handles everything ParseFloat can plus more. // Note, there is no strconv.ParseComplex. if r, ok := new(big.Rat).SetString(v.String()); ok { return r.IsInt() } case reflect.Ptr: // Special case for math/big.Int if v.Type() == bigIntT { return true } } return false }
// The rest is just demonstration and display
type intbased int16 type complexbased complex64 type customIntegerType struct { // Anything that stores or represents a sub-set // of integer values in any way desired. }
func (customIntegerType) IsInt() bool { return true } func (customIntegerType) String() string { return "<…>" }
func main() { hdr := fmt.Sprintf("%27s %-6s %s\n", "Input", "IsInt", "Type") show2 := func(t bool, i interface{}, args ...interface{}) { istr := fmt.Sprint(i) fmt.Printf("%27s %-6t %T ", istr, t, i) fmt.Println(args...) } show := func(i interface{}, args ...interface{}) { show2(IsInt(i), i, args...) }
fmt.Print("Using Float64IsInt with float64:\n", hdr) neg1 := -1. for _, f := range []float64{ 0, neg1 * 0, -2, -2.000000000000001, 10. / 2, 22. / 3, math.Pi, math.MinInt64, math.MaxUint64, math.SmallestNonzeroFloat64, math.MaxFloat64, math.NaN(), math.Inf(1), math.Inf(-1), } { show2(Float64IsInt(f), f) }
fmt.Print("\nUsing Complex128IsInt with complex128:\n", hdr) for _, c := range []complex128{ 3, 1i, 0i, 3.4, } { show2(Complex128IsInt(c), c) }
fmt.Println("\nUsing reflection:") fmt.Print(hdr) show("hello") show(math.MaxFloat64) show("9e100") show("(4+0i)", "(complex strings not parsed)") show(4 + 0i) show(rune('§'), "or rune") show(byte('A'), "or byte") var t1 intbased = 5200 var t2a, t2b complexbased = 5 + 0i, 5 + 1i show(t1) show(t2a) show(t2b) x := uintptr(unsafe.Pointer(&t2b)) show(x) show(math.MinInt32) show(uint64(math.MaxUint64)) b, _ := new(big.Int).SetString(strings.Repeat("9", 25), 0) show(b) r := new(big.Rat) show(r) r.SetString("2/3") show(r) show(r.SetFrac(b, new(big.Int).SetInt64(9))) show("12345/5") show(new(customIntegerType)) }</lang>
- Output:
Using Float64IsInt with float64:
Input IsInt Type
0 true float64
-0 true float64
-2 true float64
-2.000000000000001 false float64
5 true float64
7.333333333333333 false float64
3.141592653589793 false float64
-9.223372036854776e+18 true float64
1.8446744073709552e+19 true float64
5e-324 false float64
1.7976931348623157e+308 true float64
NaN false float64
+Inf false float64
-Inf false float64
Using Complex128IsInt with complex128:
Input IsInt Type
(3+0i) true complex128
(0+1i) false complex128
(0+0i) true complex128
(3.4+0i) false complex128
Using reflection:
Input IsInt Type
hello false string
1.7976931348623157e+308 true float64
9e100 true string
(4+0i) false string (complex strings not parsed)
(4+0i) true complex128
167 true int32 or rune
65 true uint8 or byte
5200 true main.intbased
(5+0i) true main.complexbased
(5+1i) false main.complexbased
833358060808 true uintptr
-2147483648 true int
18446744073709551615 true uint64
9999999999999999999999999 true *big.Int
0/1 true *big.Rat
2/3 false *big.Rat
1111111111111111111111111/1 true *big.Rat
12345/5 true string
<…> true *main.customIntegerType
J
Solution:<lang j> isInt =: (= <.) *. (= {.@+.)</lang> Alternative solution (remainder after diving by 1?): <lang j> isInt=: (0 = 1&|) *. (0 = {:@+.)</lang> Example:<lang j> isInt 3.14 7 1.4j0 4j0 5j3 0 1 0 1 0</lang>
jq
jq does not have builtin support for complex numbers or rationals, but in conformity with the Rosetta Code page Arithmetic/Complex#jq, we shall assume in the following that the complex number x+iy has been identified with the array [x,y]. To illustrate how the task can be solved for rationals, we shall also identify the rational numbers p/q with JSON objects that have the form: {"type": "rational", "p": p, "q": q}. <lang jq>def is_integral:
if type == "number" then . == floor
elif type == "array" then
length == 2 and .[1] == 0 and (.[0] | is_integral)
else type == "object"
and .type == "rational"
and .q != 0
and (.q | is_integral)
and ((.p / .q) | is_integral)
end ;</lang>
Example: <lang jq>(
0, -1, [3,0], {"p": 4, "q": 2, "type": "rational"},
1.1, -1.1, [3,1], {"p": 5, "q": 2, "type": "rational"}
) | "\(.) => \(if is_integral then "integral" else "" end)"</lang>
- Output:
<lang sh>$ jq -r -n -f is_integral.jq 0 => integral -1 => integral [3,0] => integral {"p":4,"q":2,"type":"rational"} => integral 1.1 => -1.1 => [3,1] => {"p":5,"q":2,"type":"rational"} => </lang>
Lua
<lang lua>function isInt (x) return type(x) == "number" and x == math.floor(x) end
print("Value\tInteger?") print("=====\t========") local testCases = {2, 0, -1, 3.5, "String!", true} for _, input in pairs(testCases) do print(input, isInt(input)) end</lang>
- Output:
Value Integer? ===== ======== 2 true 0 true -1 true 3.5 false String! false true false
ooRexx
<lang oorexx>/* REXX ---------------------------------------------------------------
- 22.06.2014 Walter Pachl using a complex data class
- ooRexx Distribution contains an elaborate complex class
- parts of which are used here
- --------------------------------------------------------------------*/
Numeric Digits 1000 Call test_integer .complex~new(1e+12,0e-3) Call test_integer .complex~new(3.14) Call test_integer .complex~new(1.00000) Call test_integer .complex~new(33) Call test_integer .complex~new(999999999) Call test_integer .complex~new(99999999999) Call test_integer .complex~new(1e272) Call test_integer .complex~new(0) Call test_integer .complex~new(1.000,-3) Call test_integer .complex~new(1.000,-3.3) Call test_integer .complex~new(,4) Call test_integer .complex~new(2.00000000,+0) Call test_integer .complex~new(,0) Call test_integer .complex~new(333) Call test_integer .complex~new(-1,-1) Call test_integer .complex~new(1,1) Call test_integer .complex~new(,.00) Call test_integer .complex~new(,1) Call test_integer .complex~new(0003,00.0) Exit
test_integer: Use Arg cpx cpxa=left(changestr('+-',cpx,'-'),13) -- beautify representation Select
When cpx~imaginary<>0 Then Say cpxa 'is not an integer' When datatype(cpx~real,'W') Then Say cpxa 'is an integer' Otherwise Say cpxa 'is not an integer' End
Return
- class complex
- method init /* initialize a complex number */
expose real imaginary /* expose the state data */ use Strict arg first=0, second=0 /* access the two numbers */ real = first + 0 /* force rounding */ imaginary = second + 0 /* force rounding on the second */
- method real /* return real part of a complex */
expose real /* access the state information */ return real /* return that value */
- method imaginary /* return imaginary part */
expose imaginary /* access the state information */ return imaginary /* return the value */
- method string /* format as a string value */
expose real imaginary /* get the state info */ return real'+'imaginary'i' /* format as real+imaginaryi */</lang> output
1E+12+0i is an integer 3.14+0i is not an integer 1.00000+0i is an integer 33+0i is an integer 999999999+0i is an integer 1.00000000E+1 is an integer 1E+272+0i is an integer 0+0i is an integer 1.000-3i is not an integer 1.000-3.3i is not an integer 0+4i is not an integer 2.00000000+0i is an integer 0+0i is an integer 333+0i is an integer -1-1i is not an integer 1+1i is not an integer 0+0i is an integer 0+1i is not an integer 3+0i is an integer
Mathematica / Wolfram Language
The built-in function IntegerQ performs the required test <lang Mathematica>IntegerQ /@ {E, 2.4, 7, 9/2}</lang>
- Output:
{False,False,True,False}
PARI/GP
The operator == does what we want here, comparing a number mathematically regardless of how it's stored. === checks literal equivalence instead.
<lang parigp>isInteger(z)=real(z)==real(z)\1 && imag(z)==imag(z)\1; apply(isInteger, [7, I, 1.7 + I, 10.0 + I, 1.0 - 7.0 * I])</lang>
- Output:
%1 = [1, 1, 0, 1, 1]
Perl
<lang perl6>use Math::Complex;
sub is_int {
my $number = shift;
if (ref $number eq 'Math::Complex') {
return 0 if $number->Im != 0;
$number = $number->Re;
}
return int($number) == $number;
}
for (5, 4.1, sqrt(2), sqrt(4), 1.1e10, 3.0-0.0*i, 4-3*i, 5.6+0*i) {
printf "%20s is%s an integer\n", $_, (is_int($_) ? "" : " NOT");
}</lang>
- Output:
5 is an integer
4.1 is NOT an integer
1.4142135623731 is NOT an integer
2 is an integer
11000000000 is an integer
3 is an integer
4-3i is NOT an integer
5.6 is NOT an integer
Perl 6
In Perl 6, classes that implement the Numeric role have a method called narrow which returns an object with the same value but with the most appropriate type. So we can just test the type of that object.
<lang perl6>for pi, 1e5, 1+0i {
say "$_ is{" NOT" if .narrow !~~ Int} an integer.";
}</lang>
- Output:
3.14159265358979 is NOT an integer. 100000 is an integer. 1+0i is an integer.
PicoLisp
Pico Lisp scaled fixed-point numbers. Every number is stored an an Integer and a Non-integer only relative to the scale applied. For this example we assume that all numbers are generated with the same scale. This is the common case. <lang PicoLisp> (de int? (N)
(= N (* 1.0 (/ N 1.0)))) #returns T or NIL
(de integer? (N)
(and (= N (* 1.0 (/ N 1.0))) N)) #returns value of N or NIL
(scl 4) #-> 4 # *Scl the global which holds 1.0 #-> 10000 (int? 1.0) #-> T (int? 1) #-> NIL # 1 with a scale of 4 is same as 0.0001 which is not an Integer (int? -1.0) #-> T (int? -0.0) #-> T (int? "RE") #-> "RE" -- Number expected (int? (*/ 2.0 1.0 3.0)) #-> NIL # 6667 is not an integer of the scale of 4, use of */ because of the scale </lang>
Python
<lang python>>>> def isint(f):
return complex(f).imag == 0 and complex(f).real.is_integer()
>>> [isint(f) for f in (1.0, 2, (3.0+0.0j), 4.1, (3+4j), (5.6+0j))] [True, True, True, False, False, False] >>> </lang>
Racket
The scheme/racket number pyramid is notoriously difficult to navigate. The following are integers representations that *I* know of, but I'm sure there are plenty more!
See documentation for integer?
<lang racket>#lang racket (require tests/eli-tester)
(test ;; known representations of integers:
;; - as exacts (integer? -1) => #t (integer? 0) => #t (integer? 1) => #t (integer? 1234879378539875943875937598379587539875498792424323432432343242423432432) => #t (integer? -1234879378539875943875937598379587539875498792424323432432343242423432432) => #t (integer? #xff) => #t ;; - as inexacts (integer? -1.) => #t (integer? 0.) => #t (integer? 1.) => #t (integer? 1234879378539875943875937598379587539875498792424323432432343242423432432.) => #t (integer? #xff.0) => #t ;; - but without a decimal fractional part (integer? -1.1) => #f ;; - fractional representation (integer? -42/3) => #t (integer? 0/1) => #t (integer? 27/9) => #t (integer? #xff/f) => #t (integer? #b11111111/1111) => #t ;; - but obviously not fractions (integer? 5/7) => #f ; - as scientific (integer? 1.23e2) => #t (integer? 1.23e120) => #t ; - but not with a small exponent (integer? 1.23e1) => #f ; - complex representations with 0 imaginary component ; ℤ is a subset of the sets of rational and /real/ numbers and (integer? 1+0i) => #t (integer? (sqr 0+1i)) => #t (integer? 0+1i) => #f ;; oh, there's so much else that isn't an integer: (integer? "woo") => #f (integer? "100") => #f (integer? (string->number "22/11")) => #t ; just cast it! (integer? +inf.0) => #f (integer? -inf.0) => #f (integer? +nan.0) => #f ; duh! it's not even a number! (integer? -NaN.0) => #f (integer? pi) => #f )
</lang> All tests pass.
REXX
version 1
<lang rexx>/* REXX ---------------------------------------------------------------
- 20.06.2014 Walter Pachl
- 22.06.2014 WP add complex numbers such as 13-12j etc.
- (using 13e-12 or so is not (yet) supported)
- --------------------------------------------------------------------*/
Call test_integer 3.14 Call test_integer 1.00000 Call test_integer 33 Call test_integer 999999999 Call test_integer 99999999999 Call test_integer 1e272 Call test_integer 'AA' Call test_integer '0' Call test_integer '1.000-3i' Call test_integer '1.000-3.3i' Call test_integer '4j' Call test_integer '2.00000000+0j' Call test_integer '0j' Call test_integer '333' Call test_integer '-1-i' Call test_integer '1+i' Call test_integer '.00i' Call test_integer 'j' Call test_integer '0003-00.0j' Exit
test_integer: Parse Arg xx Numeric Digits 1000 Parse Value parse_number(xx) With x imag If imag<>0 Then Do
Say left(xx,13) 'is not an integer (imaginary part is not zero)' Return End
Select
When datatype(x)<>'NUM' Then
Say left(xx,13) 'is not an integer (not even a number)'
Otherwise Do
If datatype(x,'W') Then
Say left(xx,13) 'is an integer'
Else
Say left(xx,13) 'isnt an integer'
End
End
Return parse_number: Procedure
Parse Upper Arg x
x=translate(x,'I','J')
If pos('I',x)>0 Then Do
pi=verify(x,'+-','M')
Select
When pi>1 Then Do
real=left(x,pi-1)
imag=substr(x,pi)
End
When pi=0 Then Do
real=0
imag=x
End
Otherwise /*pi=1*/Do
p2=verify(substr(x,2),'+-','M')
If p2>0 Then Do
real=left(x,p2)
imag=substr(x,p2+1)
End
Else Do
real=0
imag=x
End
End
End
End
Else Do
real=x
imag='0I'
End
pi=verify(imag,'+-','M')
If pi=0 Then Do
Parse Var imag imag_v 'I'
imag_sign='+'
End
Else
Parse Var imag imag_sign 2 imag_v 'I'
If imag_v= Then
imag_v=1
imag=imag_sign||imag_v
Return real imag</lang>
output
3.14 isn't an integer 1.00000 is an integer 33 is an integer 999999999 is an integer 99999999999 is an integer 1E272 is an integer AA is not an integer (not even a number) 0 is an integer 1.000-3i is not an integer (imaginary part is not zero) 1.000-3.3i is not an integer (imaginary part is not zero) 4j is not an integer (imaginary part is not zero) 2.00000000+0j is an integer 0j is an integer 333 is an integer -1-i is not an integer (imaginary part is not zero) 1+i is not an integer (imaginary part is not zero) .00i is an integer j is not an integer (imaginary part is not zero) 0003-00.0j is an integer
version 2
This REXX version handles an exponent indicator of E, D, or Q (either lower or uppercase), and it also supports a trailing I or J imaginary indicator.
This version also handles numbers larger than can be stored (within REXX) as simple integers within the limits of numeric digits. Also, most REXXes have a limit on the minimum/maximum value of the power in exponentiated numbers. <lang rexx>/*REXX pgm tests if a # (possibly complex) is equivalent to an integer.*/ numeric digits 3000 /*be able to handle big integers.*/ parse arg #s /*get optional #s list from C.L. */ if #s= then #s= '3.14 1.00000 33 999999999 99999999999 1e272 AA 0' ,
'1.000-3i 1.000-3.3i 4j 2.00000000+0j 0j 333 -1-i' ,
'1+i .00i j 0003-00.0j 1.2d1 2e55666 +0003-00.0j +0j' ,
'-.3q+2 -0i +03.0e+01+0.00e+20j -030.0e-001+0.0e-020j'
/* [↑] use these #s for defaults*/
do j=1 for words(#s); ox=word(#s,j) /*obtain a word from the #s list.*/
parse upper var ox x /*get an uppercase version of OX.*/
x=translate(x, 'EEI', "QDJ") /*alt. exponent & imag indicator.*/
if right(x,1)=='I' then call tImag /*has the X number an imag. part?*/
if isInt(x) then say right(ox,55) " is an integer." /*yup. */
else say right(ox,55) " isn't an integer." /*nope.*/
end /*j*/ /* [↑] process each # in list. */
exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────ISINT subroutine────────────────────*/ isInt: procedure; parse arg n /*obtain the number in question. */ if datatype(n, 'Whole') then return 1 /*it's a simple integer (small). */ parse var n m 'E' p /*separate base from the 10's pow*/ if \datatype(p, 'Numb') then return 0 /*Not an integer if P not a int.*/ return p>0 | m=0 /*Is power>0 or mantissa = zero? */ /*──────────────────────────────────ISSIGN subroutine───────────────────*/ isSign: arg ? 2; return ?=='+' |?=='-' /*concise method to test a sign. */ /*──────────────────────────────────TIMAG subroutine────────────────────*/ tImag: x=left(x, length(x)-1) /*strip the trailing I or J. */ if isInt(x) then do /*is what's remaining an integer?*/
if x\=0 then x=. /*what's remaining isn't = zero. */
return /*return to invoker either way. */
end /* [↑] handle simple imag. case.*/
if isSign(x) then x=substr(x,2) /*strip leading sign from X ? */ e=verify(x, .0123456789) /*find 1st char not a digit | dot*/ if e==0 then do; x=.; return; end /*Nothing? Then it's ¬ an integer*/ y=substr(x, e, 1) /*Y is the suspect character. */ if isSign(y) then do /*is suspect char a plus | minus?*/
z=substr(x, e+1) /*obtain the imaginary part of X.*/
x= left(x, e-1) /* " " real " " " */
if isInt(z) then if z=0 then return /*imag. is 0.*/
x=. /*imaginary part isn't zero. */
end /* [↑] end of imag. part of X. */
if y=='E' then do /*the real part of X has a power.*/
p=substr(x, e+1) /*obtain power of real part of X.*/
_= left(p, 1) /*obtain the possible sign of pow*/
if isSign(_) then p=substr(p, 2) /*strip the power sign*/
s=verify(p, '-+', "M") /*imaginary sep char. */
if s==0 then do; x=.; return; end /*No sign? Not integer*/
z=substr(p, s+1) /*get the imag. part. */
x= left(x, e+s) /* " " real " */
if isInt(z) then if z\=0 then x=. /*Not imag=0? Not int*/
end /* [↑] handle imag. */
return /*return to invoker. */</lang> output using the default input:
3.14 isn't an integer.
1.00000 is an integer.
33 is an integer.
999999999 is an integer.
99999999999 is an integer.
1e272 is an integer.
AA isn't an integer.
0 is an integer.
1.000-3i isn't an integer.
1.000-3.3i isn't an integer.
4j isn't an integer.
2.00000000+0j is an integer.
0j is an integer.
333 is an integer.
-1-i isn't an integer.
1+i isn't an integer.
.00i is an integer.
j isn't an integer.
0003-00.0j is an integer.
1.2d1 is an integer.
2e55666 is an integer.
+0003-00.0j is an integer.
+0j is an integer.
-.3q+2 is an integer.
-0i is an integer.
+03.0e+01+0.00e+20j is an integer.
-030.0e-001+0.0e-020j is an integer.
version 3
This REXX version is the same as the 2nd version, but it also supports multiple (abutted) unary operators.
I.E.: ++30e-1 - +0j
would be considered an integer (extra blanks were added to show the number with more clarity). <lang rexx>/*REXX pgm tests if a # (possibly complex) is equivalent to an integer.*/ numeric digits 3000 /*be able to handle big integers.*/ unaB='++ -- -+ +-' /*list of unary operators. */ unaA='+ + - -' /*list of unary operators trans. */ parse arg #s /*get optional #s list from C.L. */ if #s= then #s= '245+-00.0e-12i 245++++++0e+12j --3450d-1----0.0d-1j' ,
'4.5e11111222223333344444555556666677777888889999900'
/* [↑] use these #s for defaults*/
do j=1 for words(#s); ox=word(#s,j) /*obtain a word from the #s list.*/
parse upper var ox x /*get an uppercase version of OX.*/
x=translate(x, 'EEJ', "QDI") /*alt. exponent & imag indicator?*/
do k=1 for words(unaB) /*process every possible unary op*/ _=word(unaB,k) /*unary operator to be changed. */
do while pos(_,x)\==0 /*keep changing 'til no more left*/
x=changestr(_, x, word(unaA,k)) /*reduce all uniry operators. */
end /*while*/
end /*k*/
if right(x,1)=='J' then call tImag /*has the X number an imag. part?*/
if isInt(x) then say right(ox,55) " is an integer." /*yup. */
else say right(ox,55) " isn't an integer." /*nope.*/
end /*j*/
exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────ISINT subroutine────────────────────*/ isInt: procedure; parse arg n /*obtain the number in question. */ if datatype(n, 'Whole') then return 1 /*it's a simple integer (small). */ parse var n m 'E' p /*separate base from the 10's pow*/ if \datatype(p, 'Numb') then return 0 /*Not an integer if P not a int.*/ return p>0 | m=0 /*Is power>0 or mantissa = zero? */ /*──────────────────────────────────ISSIGN subroutine───────────────────*/ isSign: arg ? 2; return ?=='+' |?=='-' /*concise method to test a sign. */ /*──────────────────────────────────TIMAG subroutine────────────────────*/ tImag: x=left(x, length(x)-1) /*strip the trailing I or J. */ if isInt(x) then do /*is what's remaining an integer?*/
if x\=0 then x=. /*what's remaining isn't = zero. */
return /*return to invoker either way. */
end /* [↑] handle simple imag. case.*/
if isSign(x) then x=substr(x,2) /*strip leading sign from X ? */ e=verify(x, .0123456789) /*find 1st char not a digit | dot*/ if e==0 then do; x=.; return; end /*Nothing? Then it's ¬ an integer*/ y=substr(x, e, 1) /*Y is the suspect character. */ if isSign(y) then do /*is suspect char a plus | minus?*/
z=substr(x, e+1) /*obtain the imaginary part of X.*/
x= left(x, e-1) /* " " real " " " */
if isInt(z) then if z=0 then return /*imag. is 0.*/
x=. /*imaginary part isn't zero. */
end /* [↑] end of imag. part of X. */
if y=='E' then do /*the real part of X has a power.*/
p=substr(x, e+1) /*obtain power of real part of X.*/
_= left(p, 1) /*obtain the possible sign of pow*/
if isSign(_) then p=substr(p, 2) /*strip the power sign*/
s=verify(p, '-+', "M") /*imaginary sep char. */
if s==0 then do; x=.; return; end /*No sign? Not integer*/
z=substr(p, s+1) /*get the imag. part. */
x= left(x, e+s) /* " " real " */
if isInt(z) then if z\=0 then x=. /*Not imag=0? Not int*/
end /* [↑] handle imag. */
return /*return to invoker. */</lang> output using the default input:
245+-00.0e-12i is an integer.
245++++++0e+12j isn't an integer.
--3450d-1----0.0d-1j is an integer.
4.5e11111222223333344444555556666677777888889999900 is an integer.
Ruby
Testing for integerness of floats, rationals and complex numbers: <lang ruby> class Numeric
def integer? self == self.to_i rescue false end
end
- Demo
ar = [2.0, 2.5, # 2 floats
2.to_r, 2.5.to_r, # 2 rationals
2.to_c, 2+0.5i] # 2 complex numbers
ar.each{|num| puts "#{num} integer? #{num.integer?}" } </lang>
- Output:
2.0 integer? true 2.5 integer? false 2/1 integer? true 5/2 integer? false 2+0i integer? true 2+0.5i integer? false
Tcl
The simplest way is to test whether the value is (numerically) equal to itself cast as an integer. entier() performs this cast without imposing any word-size limits (as int() or wide() would).
<lang tcl>proc isNumberIntegral {x} {
expr {$x == entier($x)}
}
- test with various kinds of numbers:
foreach x {1e100 3.14 7 1.000000000000001 1000000000000000000000 -22.7 -123.000} {
puts [format "%s: %s" $x [expr {[isNumberIntegral $x] ? "yes" : "no"}]]
}</lang>
- Output:
1e100: yes 3.14: no 7: yes 1.000000000000001: no 1000000000000000000000: yes -22.7: no -123.000: yes
Note that 1.0000000000000001 will pass this integer test, because its difference from 1.0 is beyond the precision of an IEEE binary64 float. This discrepancy will be visible in other languages, but perhaps more obvious in Tcl as such a value's string representation will persist:
<lang Tcl>% set a 1.0000000000000001 1.0000000000000001 % expr $a 1.0 % IsNumberIntegral $a 1 % puts $a 1.0000000000000001</lang>
compare Python:
<lang Python>>>> a = 1.0000000000000001 >>> a 1.0 >>> 1.0 == 1.0000000000000001 True</lang>
.. this is a fairly benign illustration of why comparing floating point values with == is usually a bad idea.
zkl
No complex type. <lang zkl>T(1, 2.0,4.1,"nope",self).apply((1).isType)</lang>
- Output:
L(True,False,False,False,False)
All is not golden as BigInts (lib GMP) don't consider themselves to be integers so the above test would fail. For that case: <lang zkl>fcn isInt(x){ try{x==x.toInt()}catch{False}} var BN=Import("zklBigNum");
T(1, 2.0,4.1,"nope",self,BN(5)).apply(isInt);</lang>
- Output:
L(True,True,False,False,False,True)
Note that the first float is now considered to have an integer equivalent.