3.2 Numbers

Numbers in Guide: PLT Scheme introduces numbers.

All numbers are complex numbers. Some of them are real numbers, and all of the real numbers that can be represented are also rational numbers, except for +inf.0 (positive infinity), -inf.0 (negative infinity), and +nan.0 (not-a-number). Among the rational numbers, some are integers, because round applied to the number produces the same number.

Orthogonal to those categories, each number is also either an exact number or an inexact number. Unless otherwise specified, computations that involve an inexact number produce inexact results. Certain operations on inexact numbers, however, produce an exact number, such as multiplying an inexact number with an exact 0. Some operations, which can produce an irrational number for rational arguments (e.g., sqrt), may produce inexact results even for exact arguments.

In the case of complex numbers, either the real and imaginary parts are both exact or inexact, or the number has an exact zero real part and an inexact imaginary part; a complex number with an exact zero imaginary part is a real number.

Inexact real numbers are implemented as either single- or double-precision IEEE floating-point numbers – the latter by default, and the former only when support for 32-bit inexact numbers is specifically enabled when the run-time system is built, and when computation starts with numerical constants specified as single-precision numbers.

The precision and size of exact numbers is limited only by available memory (and the precision of operations that can produce irrational numbers). In particular, adding, multiplying, subtracting, and dividing exact numbers always produces an exact result.

Inexact numbers can be coerced to exact form, except for the inexact numbers +inf.0, -inf.0, and +nan.0, which have no exact form. Dividing a number by exact zero raises an exception; dividing a non-zero number other than +nan.0 by an inexact zero returns +inf.0 or -inf.0, depending on the sign of the dividend. The +nan.0 value is not = to itself, but +nan.0 is eqv? to itself. Conversely, (= 0.0 -0.0) is #t, but (eqv? 0.0 -0.0) is #f. The datum -nan.0 refers to the same constant as +nan.0.

Calculations with infinites produce results consistent with IEEE double-precision floating point where IEEE specifies the result; in cases where IEEE provides no specification (e.g., (angle +inf.0+inf.0i)), the result corresponds to the limit approaching infinity, or +nan.0 if no such limit exists.

A fixnum is an exact integer whose two’s complement representation fit into 31 bits on a 32-bit platform or 63 bits on a 64-bit platform. Two fixnums that are = are also the same according to eq?. Otherwise, the result of eq? applied to two numbers is undefined.

Two numbers are eqv? when they are both inexact or both exact, and when they are = (except for +nan.0, 0.0, and -0.0, as noted above). Two numbers are equal? when they are eqv?.

3.2.1 Number Types

Examples:

> (number? 1)

#t

> (number? 2+3i)

#t

> (number? "hello")

#f

Examples:

> (real? 1)

#t

> (real? +inf.0)

#t

> (real? 2+3i)

#f

> (real? 2.0+0.0i)

#f

> (real? "hello")

#f

Examples:

> (rational? 1)

#t

> (rational? +inf.0)

#f

> (real? "hello")

#f

Examples:

> (integer? 1)

#t

> (integer? 2.3)

#f

> (integer? 4.0)

#t

> (integer? +inf.0)

#f

> (integer? 2+3i)

#f

> (integer? "hello")

#f

Examples:

> (exact-integer? 1)

#t

> (exact-integer? 4.0)

#f

Examples:

> (exact-nonnegative-integer? 0)

#t

> (exact-nonnegative-integer? -1)

#f

Examples:

> (exact-positive-integer? 1)

#t

> (exact-positive-integer? 0)

#f

Examples:

> (zero? 0)

#t

> (zero? -0.0)

#t

Examples:

> (positive? 10)

#t

> (positive? -10)

#f

> (positive? 0.0)

#f

Examples:

> (negative? 10)

#f

> (negative? -10)

#t

> (negative? -0.0)

#f

Examples:

> (even? 10.0)

#t

> (even? 11)

#f

> (even? +inf.0)

even?: expects argument of type <integer>; given +inf.0

Examples:

> (odd? 10.0)

#f

> (odd? 11)

#t

> (odd? +inf.0)

odd?: expects argument of type <integer>; given +inf.0

Examples:

> (exact? 1)

#t

> (exact? 1.0)

#f

Examples:

> (inexact? 1)

#f

> (inexact? 1.0)

#t

Examples:

> (inexact->exact 1)

1

> (inexact->exact 1.0)

1

Examples:

> (exact->inexact 1)

1.0

> (exact->inexact 1.0)

1.0

3.2.2 Arithmetic

Examples:

> (+ 1 2)

3

> (+ 1.0 2+3i 5)

8.0+3.0i

> (+)

0

Examples:

> (- 5 3.0)

2.0

> (- 1)

-1

> (- 2+7i 1 3)

-2+7i

Examples:

> (* 2 3)

6

> (* 8.0 9)

72.0

> (* 1+2i 3+4i)

-5+10i

Examples:

> (/ 3 4)

3/4

> (/ 81 3 3)

9

> (/ 10.0)

0.1

> (/ 1+2i 3+4i)

11/25+2/25i

Examples:

> (quotient 10 3)

3

> (quotient -10.0 3)

-3.0

> (quotient +inf.0 3)

quotient: expects type <integer> as 1st argument, given:

+inf.0; other arguments were: 3

• (abs q) is between 0 (inclusive) and (abs m) (exclusive), and

• (+ q (* m (quotient n m))) equals n.

Examples:

> (remainder 10 3)

1

> (remainder -10.0 3)

-1.0

> (remainder 10.0 -3)

1.0

> (remainder -10 -3)

-1

> (remainder +inf.0 3)

remainder: expects type <integer> as 1st argument, given:

+inf.0; other arguments were: 3

Example:

> (quotient/remainder 10 3)

3

1

• (abs q) is between 0 (inclusive) and (abs m) (exclusive), and

• the difference between q and (- n (* m (quotient n m))) is a multiple of m.

Examples:

> (modulo 10 3)

1

> (modulo -10.0 3)

2.0

> (modulo 10.0 -3)

-2.0

> (modulo -10 -3)

-1

> (modulo +inf.0 3)

modulo: expects type <integer> as 1st argument, given:

+inf.0; other arguments were: 3

Examples:

> (abs 1.0)

1.0

> (abs -1)

1

Examples:

> (max 1 3 2)

3

> (max 1 3 2.0)

3.0

Examples:

> (min 1 3 2)

1

> (min 1 3 2.0)

1.0

Examples:

> (gcd 10)

10

> (gcd 12 81.0)

3.0

Examples:

> (lcm 10)

10

> (lcm 3 4.0)

12.0

Examples:
> (round 17/4)
4
> (round -17/4)
-4
> (round 2.5)
2.0
> (round -2.5)
-2.0

Examples:
> (floor 17/4)
4
> (floor -17/4)
-5
> (floor 2.5)
2.0
> (floor -2.5)
-3.0

Examples:
> (ceiling 17/4)
5
> (ceiling -17/4)
-4
> (ceiling 2.5)
3.0
> (ceiling -2.5)
-2.0

Examples:
> (truncate 17/4)
4
> (truncate -17/4)
-4
> (truncate 2.5)
2.0
> (truncate -2.5)
-2.0

Examples:

> (numerator 5)

5

> (numerator 17/4)

17

> (numerator 2.3)

2589569785738035.0

Examples:

> (denominator 5)

1

> (denominator 17/4)

4

> (denominator 2.3)

1125899906842624.0

Examples:

> (rationalize 1/4 1/10)

1/3

> (rationalize -1/4 1/10)

-1/3

> (rationalize 1/4 1/4)

0

> (rationalize 11/40 1/4)

1/2

3.2.3 Number Comparison

Examples:

> (= 1 1.0)

#t

> (= 1 2)

#f

> (= 2+3i 2+3i 2+3i)

#t

Examples:

> (< 1 1)

#f

> (< 1 2 3)

#t

> (< 1 +inf.0)

#t

> (< 1 +nan.0)

#f

Examples:

> (<= 1 1)

#t

> (<= 1 2 1)

#f

Examples:

> (> 1 1)

#f

> (> 3 2 1)

#t

> (> +inf.0 1)

#t

> (< +nan.0 1)

#f

Examples:

> (>= 1 1)

#t

> (>= 1 2 1)

#f

3.2.4 Powers and Roots

Examples:

> (sqrt 4/9)

2/3

> (sqrt 2)

1.4142135623730951

> (sqrt -1)

0+1i

Examples:

> (integer-sqrt 4.0)

2.0

> (integer-sqrt 5)

2

Examples:

> (integer-sqrt/remainder 4.0)

2.0

0.0

> (integer-sqrt/remainder 5)

2

1

Examples:

> (expt 2 3)

8

> (expt 4 0.5)

2.0

> (expt +inf.0 0)

1

Examples:

> (exp 1)

2.718281828459045

> (exp 2+3i)

-7.315110094901103+1.0427436562359045i

> (exp 0)

1

Examples:

> (log (exp 1))

1.0

> (log 2+3i)

1.2824746787307684+0.982793723247329i

> (log 1)

0

3.2.5 Trignometric Functions

Examples:

> (sin 3.14159)

2.65358979335273e-06

> (sin 1.0+5.0i)

62.44551846769653+40.0921657779984i

Examples:

> (cos 3.14159)

-0.9999999999964793

> (cos 1.0+5.0i)

40.095806306298826-62.43984868079963i

Examples:

> (tan 0.7854)

1.0000036732118496

> (tan 1.0+5.0i)

8.256719834227411e-05+1.0000377833796008i

Examples:

> (asin 0.25)

0.25268025514207865

> (asin 1.0+5.0i)

0.1937931365549321+2.3309746530493123i

Examples:

> (acos 0.25)

1.318116071652818

> (acos 1.0+5.0i)

1.3770031902399644-2.3309746530493123i

In the two-argument case, the result is roughly the same as (/ (exact->inexact y) (exact->inexact x)), but the signs of y and x determine the quadrant of the result. Moreover, a suitable angle is returned when y divided by x produces +nan.0 in the case that neither y nor x is +nan.0.

Examples:

> (atan 0.5)

0.4636476090008061

> (atan 2 1)

1.1071487177940904

> (atan -2 -1)

-2.0344439357957027

> (atan 1.0+5.0i)

1.530881333938778+0.19442614214700213i

> (atan +inf.0 -inf.0)

2.356194490192345

3.2.6 Complex Numbers

Example:

> (make-rectangular 3 4.0)

3.0+4.0i

Example:

> (make-polar 2 3.14159)

-1.9999999999929585+5.30717958670546e-06i

Examples:

> (real-part 3+4i)

3

> (real-part 5.0)

5.0

Examples:

> (imag-part 3+4i)

4

> (imag-part 5.0)

0

> (imag-part 5.0+0.0i)

0.0

Examples:

> (magnitude -3)

3

> (magnitude 3.0)

3.0

> (magnitude 3+4i)

5

Examples:
> (angle -3)
3.141592653589793
> (angle 3.0)
0
> (angle 3+4i)
0.9272952180016122
> (angle +inf.0+inf.0i)
0.7853981633974483

3.2.7 Bitwise Operations

Examples:

> (bitwise-ior 1 2)

3

> (bitwise-ior -32 1)

-31

Examples:

> (bitwise-and 1 2)

0

> (bitwise-and -32 -1)

-32

Examples:

> (bitwise-xor 1 5)

4

> (bitwise-xor -32 -1)

31

Examples:

> (bitwise-not 5)

-6

> (bitwise-not -1)

0

Examples:

> (bitwise-bit-set? 5 0)

#t

> (bitwise-bit-set? 5 2)

#t

> (bitwise-bit-set? -5 (expt 2 700))

#t

(bitwise-and (sub1 (arithmetic-shift 1 (- end start)))

(arithmetic-shift n (- start)))

but in constant time when n is positive, start and end are fixnums, and (- end start) is no more than the maximum width of a fixnum.

Examples:

> (bitwise-bit-field 13 1 1)

0

> (bitwise-bit-field 13 1 3)

2

> (bitwise-bit-field 13 1 4)

6

Examples:

> (arithmetic-shift 1 10)

1024

> (arithmetic-shift 255 -3)

31

Examples:

> (integer-length 8)

4

> (integer-length -8)

3

3.2.8 Random Numbers

In each case, the number is provided by the given pseudo-random number generator (which defaults to the current one, as produced by current-pseudo-random-generator). The generator maintains an internal state for generating numbers. The random number generator uses a 54-bit version of L’Ecuyer’s MRG32k3a algorithm [L'Ecuyer02].