Friday, February 26, 2010

Irrational Numbers

Of the infinite set of numbers, the most known are the counting numbers, 1,2,3... The negative numbers and zero are combined with the counting numbers to form the wider set of integers. Integers can be divided to form fractions, therefore expanding the system to rational numbers.

All of the above is common knowledge, but the number system can be further expanded by taking the infinite sum of a convergent sequence of rationals, or, more commonly, simply taking the square root of a non-perfect square. Either method results in a number that has an infinite decimal expansion that never repeats. Mathematically, this means that the number cannot be expressed as a fraction. Numbers of this type are called irrational numbers.

Irrational numbers can also be subdivided further into two groups: the algebraic irrationals, and the transcendental irrationals. The algebraic irrationals are simply the ones that solve a polynomial equation with integer coefficients. This simply means any equation like x^2+3x+5=0, x^5-97x^3+4=0, etc. Other numbers can also be algebraic, and all rational numbers are. (for equations like 3x-2=0, the solution is 2/3. All fractions, and therefore all rationals, can be expressed this way) It is also obvious why square roots are algebraic, they all fit equations of the form x^2=n, where the solution will always be the square root of n. Cube roots, and all other roots come about in the same way. Finally, all numbers of the form a+square root of b can be reached by means of the quadratic formula. Some other irrationals are algebraic as well, but it is a lot less obvious.

Transcendental numbers are all irrational, and these are simply the remaining numbers that do not fit the above category. Among these are π (pi), e, and any nonzero powers of either of them. There are still some numbers, like the square root of two to the power of the square root of two, π-e, and others that it is not known whether they are algebraic or transcendental.

Another method for representing the irrational numbers is the continued fraction. A continued fraction looks like this:

In the above picture, the first constant a0 must be an integer, and the remaining numbers must be positive integers. If this conditions are filled, the expression is a continued fraction. Continued fractions can represent all real numbers, rational and irrational (technically, complex numbers are the sum of two continued fractions, one real, one imaginary). Unlike decimal expansions, where rational numbers can go on forever (Ex. 1/3=.333333...) every continued fraction for a rational number is finite, meaning that there are a limited number of constants in the continued fraction. Conveniently, all irrational numbers have infinite continued fractions. The continued fraction for the square root of 2 has the constants 1,2,2,2,2,2... with the two repeating indefinitely. Another interesting property of continued fractions is that the constants of a algebraic irrational number repeat (Ex. square root of three has constants 1,1,2,1,2,1,2,1,2) while transcendental irrationals have a non-repeating sequence of constants (Ex. π has the constants 3, 7, 15, 1, 292, 1, 1, 1, 2, 1...). In a sense, the number is a irrational irrational!

There are infinite irrational numbers, but three notable ones are listed below.

π (pi)
A detailed article on pi in available elsewhere on this blog, see here.

Type: Transcendental irrational
Decimal Expansion: 2.71828182845904523536...
Continued Fraction Constants: 2,1,2,1,1,4,1,1,8,1,1,16...(this sequence of constants does have a pattern, but it still does not repeat, making e a transcendental irrational)
Description: The constant e is important in many aspects of mathematics, particularly calculus, and it appears in many functions. e is the limit of the function (1+1/x)^x as x approaches infinity and is the base of the function e^x, which is the only function of x that when differentiated stays the same. e is also the base of the natural logarithm, denoted ln(x) which has many properties. e is related to the notable constants π, i, 1, and 0, through Leonhard Euler's famous eqaution e^πi+1=0. Overall, the constant e shows up in many unlikely places in various regions of mathematics.

φ (phi)
Type: Algebraic irrational
Decimal Expansion: 1.6180339887
Continued fraction constants: 1,1,1,1,1,1,1...
Description: φ is an irrational number that is also commonly known as the golden ratio. This is because it is the ration between two line segments that makes it also the ratio between the longer segment and the sum of the two segments. The image below illustrates this property.

φ is also the only number whose reciprocal is itself minus one (1/φ+1=φ). Interestingly, φ is also related to the Fibonacci Sequence (the sequence that starts 1,1 and each new number is the sum of the previous two). The ratio of two consecutive terms of the Fibonacci sequence approximates φ, and the approximation becomes more accurate with higher Fibonacci numbers. φ has numerous properties, and is the only number to satisfy many formulae throughout mathematics.

There are many other irrationals but the above are the most notable.

Sources: (info and image), (info)

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