Irrational Numbers

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Various types of numbers we have learned so far from the previous topics under Number System:

➣ Counting numbers ↬ 1, 2, 3, …

➣ Zero ↬ 0

➣ Negative numbers ↬ …, -3, -2, -1

➣ Fractions ↬ Number representing part of a whole.

➣ Decimals ↬ Numbers representing both integral and fractional part .

Different sets found to group these numbers

➢ Natural Numbers, N = {Counting numbers}

➢ Whole numbers, W = {0, N}

➢ Integers, Z = {…, 3, 2, 1, 0, 1, 2, 3, …}

➢ Rational Numbers, Q = {\frac{p}{q},\, q\neq 0},

→ Rational number is the set of numbers that includes both integers and fractions.

→ Every integers can be expressed as p/q with q = 1.

→ The rational numbers do not have a unique representation in the form p/q , where p and q are integers and q ≠ 0.

For example, 1/2 = 2/4 = 10/20 = 25/50 = 47/94, and so on. These are equivalent rational numbers (or equivalent fractions). They represent the same number.

→ On the number line, among the infinitely many fractions equivalent to 1/2, we will choose 1/2 i.e., the simplest form to represent all of them.

→ There are infinitely many rational numbers between any two given rational numbers.

Is Rational number the complete set of numbers ?

Are there numbers which can’t be represented as rational number ?

ℹ↬ Rational number is not a complete set of numbers.

↬ There are infinitely many numbers that can not be represented as a ratio of two integers. These numbers are called Irrational numbers.

e.g. √2, π (22/7 is its approximate value) etc.

“An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q.
Irrational numbers have decimal expansions that neither terminate nor reoccurring.”

Hippassus of Metapontum (5th century BC), a Greek Pythagorean (followers of Pythagoras) philosopher , found while working on square of unit side that the length of its diagonal (√2 unit) can’t be expressed as ratio of two integers.

Myths suggests that the other Pythagoreans who believed adamantly that only rational number could exist, threw Hippassus overboard on a sea voyage, and vowed to keep the existence of irrational numbers an official secret of their sect.

● In 425 BC, Theodorus of Cyrene showed that √3, √5, √6, √7 , √10 , √11, √12, √13, √14, √15 and √17 are also irrationals.

π was known to various cultures for thousands of years, it was proved to be irrational by Lambert and Legendre only in the late 1700s.

Square root of Prime numbers are irrational.

Irrational numbers on a Number line

Irrational numbers like √2, √3, √5 etc can be shown on number line using Pythagoras theorem.

√2 on number line

→ First, we consider a unit square OABC, with each side 1 unit in length. Then we can see by the Pythagoras theorem that, length of the diagonal

OB\, = \sqrt {(1^2 + 1^2)} = \sqrt{2} .

→ We transfer this square onto the number line making sure that the vertex O coincides with zero.

→ Using a compass with centre O and radius OB, draw an arc intersecting the number line at the point P. Then P corresponds to √2 on the number line.

√3 on number line

→ We draw number line with ∆OAB (OA = AB = 1unit & OB = √2 unit), O at origin as in previous case.

→ Construct BD of unit length perpendicular to OB.

→ Then using the Pythagoras theorem, we see that OD\, =\sqrt {\sqrt{2}^2 + 1^2}= \sqrt{3} .

→ Using a compass, with centre O and radius OD, draw an arc which intersects the number line at the point Q. Then Q corresponds to √3 .

● In the same way, we can locate √n for any positive integer n, after \sqrt{n - 1} has been located.

Difference Between Rational & Irrational numbers

Decimal expansions of rationals and irrationals numbers can be used to distinguish between them.

Decimal expansions of Rational Numbers

On division of p by q, two main things happen – either the remainder becomes zero or never becomes zero and we get a repeating string of remainders.

➢ When the remainder becomes zero, the decimal expansion terminates or ends after a finite number of steps. We call the decimal expansion of such numbers terminating.

e.g, 3/4 = 0.75, 7/8 = 0.875.

➢ When the remainder never becomes zero, we get a repeating string of remainders and we have a repeating block of digits in the quotient. We say that this expansion is non-terminating recurring.

Such numbers are represented by bar above the block of digits that repeats.

e.g, 4/3 = 1.333… = 1. \overline{3},

1/7 = 0.142857142857…= 0. \overline{142857}

The number of entries in the repeating string of quotient is generally less than the divisor.

❔ How can we express terminating & non-terminating reoccurring decimals in the form p/q, where p and q are integers and q ≠ 0.

Terminating Decimal to p/q form

Terminating decimals can be expressed in the form of p/q by first putting 1 in denominator and adding same number of zeros after it as number of digits in numerator after the decimal point and then removing the decimal point.

2.\overline{1423} = \frac{21423}{10000}

Non-terminating recurring decimal to p/q form

Non-terminating reccurring decimals can be expressed in the form of p/q by using algebraic equations.

e.g., Expressing 0.7777…= 0. \overline{7} in the form p/q; where p & q are integers & q ≠ 0.

Since we do not know what 0. \overline{7} is , let us call it ‘x’

So, x = 0.7777… (i)

Now here is where the trick comes in.

Multiplying both side with 10, we get

10 x = 10 × (0.777…) = 7.777… (ii)

Subtracting (i) from (ii), we get

10x – x = 7.777… – 0.777…

⇒ 9x = 7

⇒ x = 7/9

∴ 0.7= 7/9

Decimal expansion of an Irrational number

Decimal expansion of an irrational number is non-terminating non-recurring i.e., Irrational numbers niether terminate nor they have repeating string of digits in their decimal expansion.

→ √2 = 1.41421356237309504880168…

→ π = 3.141592653589793238462643…

→ We often take 22/7 as an approximate value for π, but π ≠ 22/7.

❕ The Greek genius Archimedes (287 BCE – 212 BCE) was the first to compute digits in the decimal expansion of π. He showed 3.140845 < π < 3.142857. Aryabhatta (476 – 550 C.E.), the great Indian mathematician and astronomer, found the value of π correct to four decimal places (3.1416). Using high speed computers and advanced algorithms, π has been computed to over 1.24 trillion decimal places!

ℹ The decimal expansion of a rational number is either terminating or non-terminating recurring whereas the decimal expansion of an irrational number is non-terminating non-recurring.

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