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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** =

→ 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

→ 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

→ 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 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… =

1/7 = 0.142857142857…=

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.

### 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…= in the form p/q; where p & q are integers & q ≠ 0.

Since we do not know what 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.*