Mathematics with Python and Ruby/Real numbers in Ruby

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Decimal representation[edit]

Since the apparition of the digits, numbers are in some way more concrete than were the fractions: Seeing \frac{6}{5} one has the impression that there are two numbers instead of one, whereas 1.2 is a single number at first sight.

Decimal numbers[edit]

A number is decimal when it has a finite number of digits. Then the real numbers that are not decimal have an infinite decimal expansion. Such a number can be constructed on purpose like the Liouville numbers.

In Ruby, certain decimal numbers have an infinite representation yet: This apparent paradox is due to the fact that the internal representation of numbers in Ruby is binary and even such a simple number as 1.2 has an infinite binary expansion.

Fractions[edit]

A fraction is characterized by the fact that its (finite or infinite) expansion is eventually periodic. Examples:

puts(1.0/3)
puts(1.0/9.0)
puts(1/11.0)
puts(1.0/7)

Irrational numbers[edit]

The first numbers known as not being fractions were the square roots and the famous Golden mean. Here are some of them:

puts(2**0.5)
puts(Math.sqrt(2))
puts((1+5**0.5)/2)
puts((Math.sqrt(5)+1)/2)

Two famous transcendental numbers are e and π ; they are properties of the Math object:

puts(Math::E)
puts(Math::PI)

Champernowne's constant can be built from a string object:

c='0.'
(1..40).collect { |n| c=c+n.to_s }
puts(c.to_f)
puts(c.to_r)

At first, c is a string, and Ruby knows it because it is written inside a pair of quotes. It represents already the decimal expansion of the constant, which begins with a zero followed by the decimal period. Then the object (1..40) (a list of integers) is created on the fly, and its collect method is called upon, with a small Ruby block, which has only one variable called n and only one instruction, asking to add the string representation of n to c. As c is a string, + denotes the string addition, which is concatenation. At the end, c is converted, either in a float or in a fraction.

Functions[edit]

Operations[edit]

The four operations are denoted by the classical +, -, * and /. As soon as one of the operand includes a period, it is recognized as a real number, and the float operation is applied. Euclidian's division is available too, which allows Ruby to compute the principal value of an angle (in radians):

puts(100%Math::PI)

The minus sign can also be unary, in which case it represents the negation of the following number. To add a number h to an other number x, one can, instead of writing x=x+h, write x+=h which is shorter.

To get an integer approximation to a real number, floor, ceil and to_i can be used. To get the absolute value of the number x, use x.abs. And its square root can be obtained with any of the following methods:

r=2**0.5
puts(r)
r=Math.sqrt(2)
puts(r)


Logarithms and hyperbolic functions[edit]

Logarithms[edit]

To compute the natural logarithm, decimal logarithm, and the arguments of the hyperbolic circular functions, one writes

puts(Math.log(0.5))
puts(Math.log10(0.5))
puts(Math.acosh(0.5))
puts(Math.asinh(0.5))
puts(Math.atanh(0.5))

Hyperbolic functions[edit]

The following script computes and displays the exponential and hyperbolic circular functions of 2:

puts(Math.exp(2))
puts(Math.cosh(2))
puts(Math.sinh(2))
puts(Math.tanh(2))

Circular functions[edit]

To get the cosine, sine and tangent of an angle which measures one radian:

puts(Math.cos(1))
puts(Math.sin(1))
puts(Math.tan(1))

To get the angle from one of the above numbers, one just has to add an extra a before the name of the function:

puts(Math.acos(0.5))
puts(Math.asin(0.5))
puts(Math.atan(0.5))

To know an angle from the sides of a right triangle, on can use Math.atan(y/x) or Math.atan2(x,y). And to know the third side, one can use Math.hypot(x,y). For example, if one knows that the sides of a right triangle measure 5' and 12', one can know the angles and hypotenuse with this script:

cdr=180/Math::PI
a=12
b=5
puts(Math.atan2(a,b)*cdr)
puts(Math.atan2(b,a)*cdr)
puts(Math.hypot(a,b))