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*The collection of numbers including all rational numbers and irrational numbers is called Real number.*

➧ It is denoted by **R**.

➧ Every Real number is either a Rational number or an Irrational number.

➧ Every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number. This is why we call the number line, the real number line.

➧ In 1870s two German mathematicians, Canton and Dedekind, showed that: Corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number.

## Representing Real numbers on Number line

➧ It is easy to represent integers on number line and we know how to do it including fractions and decimals up to few decimal places.

❔*How can we represent numbers with more than two digits after decimal point on the number line?*

*How can we represent non-terminating decimal numbers on the number line?*

### Successive magnification 🔍

The process of visualisation or representation of numbers on the number line, through a magnifying glass, is known as the process of successive magnification.

It is possible by sufficient successive magnifications to visualise the position (or representation) of a real number with a terminating and non-terminating decimal expansion on the number line.

➢ Let us locate 3.765 on the number line.

↬ Integral part of the number is positive 3, therefor the number lie between 3 and 4. Locate 3 and 4 on the number line.

↬ Suppose we are using magnifying glass which will give us the clear view of subdivisions as in figure.

↬ The number at first decimal place is 7, therefore we look at the portion between 3.7 & 3.8. Again suppose this portion is divided into ten equal parts then the first mark is 3.71, the second mark is 3.72 & so on.

↬ Now, the number at second decimal place is 6, so the number will lie between 3.76 & 3.77, therefore we look at the portion between 3.76 & 3.77. Again suppose this portion is divided into ten equal parts then the first mark is 3.761, the second is 3.762 & so on.

↬ Therefore, 3.765 is the 5th mark in these subdivisions.

## Properties of Real numbers

🔵 Rational numbers satisfy the *commutative*, *associative* and *distributive* laws for addition and multiplication.

🔵 Rational numbers are *‘closed’* with respect to addition**, **subtraction**, **multiplication and** **division (*except by zero*).

🔴 Irrational numbers also satisfy the *commutative*, *associative* and* distributive* laws for addition and multiplication.

🔴 The sum, difference, product and division of irrational numbers are not always irrational i.e., *not closed*.

If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.

2√3 + √3 = (2+1)√3 = 3√3, (= irrational )

√3 – √3 = 0, (= rational)

√3 × √3 = 3, (= rational)

⚫ (i) The sum and difference of a rational number and an irrational number is irrational.

(ii) The product and quotient of a non-zero rational number with an irrational number is irrational.

## Finding Roots of a Positive Real number *x* geometrically.

(Square root of a Negative real number does not exist as square of a real number is always positive).

→ Mark the distance** 3.5 units **from a fixed point A on a given line to obtain a point B such that **AB = 3.5 units**

→ From B, mark a distance of **1 unit **and mark the new point as C. **BC = 1 unit**

→ Find the **mid-point of AC** and mark that point as** O**. Draw a** semicircle with centre O** and **radius OC.**

→ Draw a line BD perpendicular to AC passing through B and intersecting the semicircle at D.

Then, **BD = .**

*x*, we mark B so that

**AB =**

*x***units**, and, as above, mark C so that

**BC = 1 unit.**Then, as we have done for the case x =

**√**3.5, we find

**BD=**.

➢ **Proof **– We can prove this result using the **Pythagoras Theorem**.

→ The radius of the circle is units.

OC = OD = OA = units.

OB =

→ In right-angled triangle ∆OBD,

BD^{2} = OD^{2} – OB^{2} (by the **Pythagoras Theorem**)

This construction gives us a visual, and geometric way of showing that √*x* exists for all real numbers > 0.

## Position of √x on the real number line.

**B as zero**, then

**C = 1**, and so on.

→ Draw an arc with centre B and radius BD, which intersects the number line at E.

Then, **E** represents on the number line.

➧ We can extend the idea of square roots to cube roots, fourth roots, and in general nth roots, where n is a positive integer.

→ We can define for a real number *a* > 0 and a positive integer *n*,

let *a* > 0 be a real number and *n* be a positive integer, then

and .

→ The symbol ‘’ used in , etc. is called the radical sign.

## Identities relating to square roots of Real numbers.

## Rationalising the denominator

The method of converting a fraction with irrational (roots) denominator to an equivalent fraction with rational denominator is called rationalisation.

→ Thus, rationalization is a process of eliminating radicals from the denominator of a real fraction.

→ When denominator has single term – Multiply the numerator and the denominator with the irrational factor of the denominator.

e.g.,

→ When denominator has two terms (binomial expression) – Multiply the numerator and the denominator with the conjugate of the expression in the denominator.

Conjugate of an expression is the expression with opposite sign in between the terms. e.g., a+b and a-b are conjugate of each other.

After multiplying with conjugate we use the following idenditity

e.g.,

(Rationalising)

## Laws of Exponents for Real numbers

➧ We have learned about the following laws of exponents from previous topics, (Here *a*, *n* and *m* are integers. *a* is the base and *m* and *n* are the exponents.)

(putting m= 0 in (iii))

➧ Let, a > 0 be a real number and n be a positive integer. Then and .

In the language of exponents, we define

➧ Let, **a** > 0 be a real number. Let, m and n be integers such that m and n have no common factors other than 1, and n > 0. Then,

➧ We now have the following extended laws of exponents for real numbers:

Let *a* > 0 be a real number and *p* and *q* be rational numbers. Then, we have