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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.
