• Rational Numbers : Rational Numbers are numbers , which can be expressed in the form p/q, q≠0; where p and q are integers . The collection of all rationals are represented by Q.
∴ Q = {p/q ; p,q are integers and q ≠ 0}
A rational number is either a terminating or non-terminating repeating decimal number.
• Irrational Number: A number, which is not rational is called an irrational number.
In other words, numbers which cannot be expressed in the form of
Decimal Representation of a Rational Number: A rational number is either a terminating decimal or a non-terminating but recurring (repeated) decimal. In other words, a terminating decimal or a non-terminating but recurring decimal is a rational number. Recurring decimals are also expressed as below:
Note:
A rational number when expressed in lowest terms having factors 2 or 5 or both in the denominator can be expressed as a terminating decimal otherwise a non- terminating recurring decimal. Decimal Representation of Irrational Number: The decimal expansion of an irrational number is non-terminating, non- recurring. Moreover, a number whose decimal expansion is non- terminating, non-recurring is irrational number.
Insertion of rational numbers between two rationals: There lies infinite rational numbers between any two rationals.
Illustration:
A rational number between a and b is given by a+b/2.
i.e., a < (a+b/2) < 2
Writing (a+b/2) as x_{1}.
A rational number between a and x_{1} is a+x_{1}/2 .
Therefore, we can insert a rational number a+x_{1}+/2 = x_{2} between a and x_{1}.
Again between x_{1} and b we can insert a rational number x_{1 }+ b/2 = x_{3}.
Thus , a < x_{2} < x_{1} < x_{3}< b.
Repeating the process again and again we can show that there exists an infinite number of rational numbers x_{2} , x_{1}, x_{3},... between any two rational numbers a and b so that a <…< x_{2} < x_{1} < x_{3} < …<b.
This is called denseness property of rational numbers
Real Numbers: The collection of all rational numbers and irrational numbers taken together form a collection of real numbers. It can be understood through figure shown below:
• Every point on number line corresponds to a real number and vice- versa.
• If r_{1} and r_{2} are any two rational numbers then r_{1} + r_{2} , r_{1} – r_{2 }, r_{1} × r_{2},_{}
r_{1}/r_{2} (provided r_{2} ≠ 0) are rational numbers .
• If r is a rational number and s is an irrational number, then r + s, r – s, r × s and r/s (s ≠ 0) are irrational numbers .
(i) Draw a ray AX.
(ii) Mark another point Q such that AQ = x units.
(iii) Mark point R such that QR = 1 unit.
(iv) Find the mid – point of AR and mark it as O.
(v) Draw semicircle of radius OR centered O
(vi) Draw a line perpendicular to AX passing through Q and intersecting the semicircle at S.
(vii) With Q as centre and QS as radius draw an arc cutting AX at T. Then , QS = QT = √x .
• Another way of representing real numbers on real number line is through process of successive magnification. In this method we successively decrease the lengths of the intervals in which given number lines.
• For positive real numbers a and b.
• If m and n are rational numbers and a is a positive real number, then
• Rationalising the denominator : A number is easy to handle if its denominator is a rational number. We generally remove an irrational number the denominator by certain methods which are explained in the examples below :
• Rationalising Factor (RF) : When product of two irrational numbers is a rational number then each of them is called rationalizing factor of the other .
• If and b are two rational numbers which are not perfect squares, then irrational numbers √a + √b and √a - √b are said to be conjugate to each other .
• The product of two conjugate irrational numbers is always a rational number as
∴ Rationalising factor of : (√a + √b) is (√a - √b)
Similarly, rationalizing factor of (√a -√b) is (√a + √b).