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Author: S.LAL
Created: 18 Jan, 2010; Last Modified: 29 Apr, 2018

Number Theory - 01

Rational & Irrational Numbers

Let us first recap the simpler numbers systems that we are familiar with.

Natural numbers
natural-numbers-number-line.png
Fig 1: The Natural Numbers on the Number Line

Also known as the counting numbers, the natural numbers are represented by the numerals:

{ 1 , 2 , 3 , 4 , 6 , }

The set of natural numbers is denoted by the symbol N. The numbers can be depicted on a number line as in Fig. , where the arrowhead indicates that the list of numbers is infinite.

Whole numbers
whole-numbers-number-line.png
Fig 2: The Whole Numbers on the Number Line

Adding zero (0) to the list of natural numbers gives the set of whole numbers:

{ 0 , 1 , 2 , 3 , 4 , 6 , }

The set of whole numbers is symbolised by the symbol W.

Integers
integers-number-line.png
Fig 3: The Integers on the Number Line

Mirroring the set of natural numbers (which can be thought of as positives) across zero (0) on the number line to create their negatives forms the complete set of integers. The set comprises:

{ , 3 , 2 , 1 , 0 , 1 , 2 , 3 , }

The set of integers is denoted by Z. The set of natural numbers and the set of whole numbers are subsets of the set of integers. Thus,

N Z and W Z

Rational Numbers

All numbers which can be expressed in the form p q , where p and q are integers having no common factors and q 0 , are called rational numbers (derived from the word ratio), symbolised by Q. Thus, Q = { p q : p , q Z    and    b 0 }

Note that:

The set of integers is actually a subset of the set of rational numbers. Each integer n can be considered a fraction with the denominator of 1 and the integer itself as the numerator, i.e. n 1 , which is a quotient of two integers. Thus, Z Q .

  • p and q are integers having no common factors (except 1). This implies that even though both 1 2 and 6 8 can qualify for being rational numbers, the numerator and denominator of 6 8 have an HCF of 2 which can be factored out to yield 3 4 , which is the actual representation of that rational number.
  • The fact that p and q are integers implies that just like integers have positive and negative counterparts, so do rational numbers; viz, for every p q there is a negative counterpart of p q .
Common fraction and Decimal fraction

Every rational number, which is not an integer, can be written as a fraction. A common fraction is a fraction having a numerator and denominator.

Performing division of a common fraction (whose numerator is less than the denominator) gives a decimal fraction, or simply decimal, which is a rational number with a decimal point, having only 0 on the left side of the decimal point. For example, the common fraction 3 4 can be reduced to the decimal fraction 0.75.

Terminating and Repeating decimals

In the case of rational numbers like 3 4 , 1 2 or 1 16 , the process of division reaches a point where the remainder becomes 0, thereby terminating the process. Such decimals are called terminating decimals. For the above cases, we have 3 4 = 0.75 , 1 2 = 0.5 and 1 16 = 0.0625 .

A repeating decimal is written in short form by placing a bar over the group of digits which repeat, or a dot over each of the digit(s) in the group which repeat. e.g.: 0.333333 = 0. 3 ¯ = 0. 3 ˙ 0.161616 = 0. 16 ¯ = 0. 1 ˙ 6 ˙

However, not all rational numbers can be expressed as terminating decimals. For example, 1 3 results in a decimal fraction 0.33333..., where the digit 3 repeats continuously. Similarly, for 1 6 , the decimal fraction turns out to be 0.16161616..., where the pair of digits 16 keeps repeating. Such decimals are called repeating decimals, periodic decimals or circulating decimals.

The digit or the group of digit which repeats is known as the period of the repeating decimal.

Note that the terminating decimal is actually a special case of the repeating decimal, whereupon it is the number 0 which repeats continuously after the point where the remainder becomes 0 and the division process is discontinued.

TIP: If the denominator of a rational number can be expressed as a power of 2 and/or 5 (either alone or multiplied together), the rational number can convert to a terminating decimal. e.g.

(i) 16 25 = 16 5 2 = 0.64 (ii) 13 250 = 13 5 3 × 2 = 0.052 (iii) 5 8 = 5 2 3 = 0.625
In all other cases, the rational numbers results in recurring decimals.

Properties of rational numbers

  1. The set of rational numbers is everywhere dense: Every rational number can be associated with a point on the number line. There are infinitely many rational numbers between any two rational numbers on the number line.
  2. Every rational number can be expressed as either a terminating decimal or a repeating decimal. Moreover, a number whose decimal expansion is either terminating or is repeating is a rational number.
  3. Rational numbers are 'closed' with respect to addition, subtraction, multiplication and division (except by 0). This means that any of these operations between two rational numbers always results in another rational number.
  4. Between any two rational numbers a and b, there exists another rational number a + b 2 . This means that:
    a < b a < a + b 2 < b a > b a > a + b 2 > b

Irrational Numbers

Irrational numbers are in the form of infinite non-repeating decimals. These numbers cannot be expressed in the form p q . The symbol for irrational numbers is S.

A rational approximation of an irrational number is a rational number which is close to, but not equal to, the value of the irrational number.

The most famous example of an irrational number is π, which is the circumference of a circle divided by its diameter, or π = circumference diameter . Two common rational approximations of π are π 3.142 and π = 22 7 .

A few other irrational numbers are 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15 and 17.

Properties of irrational numbers

  1. The set of irrational numbers is everywhere dense: Every irrational number can be associated with a point on the number line. There are infinitely many irrational numbers between any two irrational numbers on the number line.
  2. Every irrational number can be expressed as an infinite non-terminating decimal. Moreover, a number whose decimal expansion is infinite non-terminating is an irrational number.
  3. Irrational numbers are not 'closed' with respect to addition, subtraction, multiplication and division. This means that any of the above operations between two irrational numbers may not always result in another irrational number. For e.g., ( 6 ) + ( 6 ) and 3 × 3 result in rational numbers.

Real Numbers

Taken together, rational and irrational numbers comprise all numbers which can be written as decimals. The set of real numbers is the set that consists of all rational numbers and all irrational numbers. The symbol for real numbers is R. Thus, Q R and S R .

Also, since a real number is either rational or irrational, R = Q ∪︀ S .

Note that the rational and irrational numbers are completely disjoint. So, S ∩︀ Q = .

For every real number, there is a corresponding point on the number line. All these points taken together make up the real number line.

real-line.png
Fig 4: The Real Line (Source: Thomas et al. Calculus second-order, p. 1)

The different subsets of real numbers and their relationships are as depicted in Fig.

sets-of-real-numbers-and-relationships.png
Fig 5: Sets of real numbers and their relationships. (Source: Beecher, Penna & Bittinger, Algebra and Trignometry, p. 2)

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List of References

Beecher, JA, Penna, JA & Bittinger, ML, Algebra and Trigonometry, 3rd edn, USA: Addison Wesley, 2006.
Thomas, GB, Weir, MD, Hass, J & Giordano, FR, Thomas' Calculus – including second-order differential equations, 11th edn, USA: Pearson Addison-Wesley, 2004.

Bibliography

Bansal, RK, Concise Mathematics I.C.S.E., Part I – Class IX, New Delhi: Selina Publishers, 2005.
Gantert, AX, AMSCO's Integrated Algebra I, NY: AMSCO School Publication, 2007.
NCERT, Mathematics – Textbook for Class IX, viewed 25 February, 2008, <http://www.ncert.nic.in/textbooks/testing/index.htm>, (n.d.).