Rational And Irrational Numbers


Rational Numbers

In mathematics, a rational number is a number which can be expressed as a ratio of two integers. Non-integer rational numbers (commonly called fractions) are usually written as the vulgar fraction a / b, where b is not zero. a is called the numerator, and b the denominator.

Each rational number can be written in infinitely many forms, such as 3 / 6 = 2 / 4 = 1 / 2, but it is said to be in simplest form when a and b have no common divisors except 1 (i.e., they are coprime). Every non-zero rational number has exactly one simplest form of this type with a positive denominator. A fraction in this simplest form is said to be an irreducible fraction, or a fraction in reduced form.

The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part). The same is true for any other integral base above one, and is also true when rational numbers are considered to be p-adic numbers rather than real numbers. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational. A real number that is not a rational number is called an irrational number.

The set of all rational numbers, which constitutes a field, is denoted \mathbb{Q}. Using the set-builder notation, \mathbb{Q} is defined as

\mathbb{Q} = \left\{\frac{m}{n} : m \in \mathbb{Z}, n \in \mathbb{Z}, n \ne 0 \right\},

where \mathbb{Z} denotes the set of integers.


Irrational Numbers

In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Informally, this means numbers that cannot be represented as simple fractions. It can be deduced that they also cannot be represented as terminating or repeating decimals, but the idea is more profound than that. While it may seem strange at first hearing, almost all real numbers are irrational, in a sense which is defined more precisely below. Perhaps[1][2] the most well known irrational numbers are π and \scriptstyle\sqrt{2}.

 \Bbb{R} \ \mathbb{Q} denotes the irrational numbers.

Having read the above recap on rational and irrational numbers, try answering the following multiple choice question which I had in a test today:

Which of the following is the first false statement?

(a) Between two different rational numbers, we can always find another rational number.
(b) Between two different rational numbers, we can always find an irrational number.
(c) Between two different irrational numbers, we can always find a rational number.
(d) Between two different irrational numbers, we can always find another irrational number.
(e) 0 = 1.

The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones.

This means that (a) is true.

It has been shown that there exists two irrational numbers a and b, such that ab is rational. Here is an example:

If √2√2 is rational, then take a = b = √2. Otherwise, take a to be the irrational number √2√2 and b = √2. Then ab = (√2√2)√2 = √2√2·√2 = √22 = 2 which is rational.

This means that (c) is true.

COROLLARY of the Archimedian property
a < b, a and b Є  \Bbb{R} Ǝ i Є  \Bbb{R} \ \mathbb{Q} : a < i < b.

a and b are two real numbers such as a < b. There exists an irrational number i such as a < i < b.
Since a and b are real numbers, we can choose them to be irrational. Therefore another irrational number will always sit between two irrational numbers. Similarly, we can take a and b as rational numbers. Then there is an irrational number in between them.

This means that (b) and (d) are true.

COROLLARY of the Archimedian property (Density property of rational numbers)
if a < b, a and b Є  \Bbb{R} then Ǝ r Є \mathbb{Q} : a < r < b.

if a and b are two real numbers such as a < b then there is another rational number that sits in between them.
If we take a and b as rational numbers, since \mathbb{Q} \Bbb{R} , then there is another rational number that is in between them.

This means that (b) is true.

Answer?

(e) is false.

Author: range

I'm mathematician/IT strategist/blogger from Canada living in Taipei.