# Irrational Numbers

#### Summary*arrow_upward*

**Natural Numbers**

Counting numbers 1, 2, 3, 4, 5, .......etc. are called Natural numbers. Set of natural numbers is generally denoted by N.

**Whole Numbers**

All the natural numbers together with zero are called Whole numbers. The numbers 0, 1, 2, 3, 4, 5, ....... etc. are called Whole numbers. Set of Whole numbers is generally denoted by W. Every Natural number is a Whole number.

**Integers**

All natural numbers, zero and negatives of the natural numbers are called Integers, i.e. ......– 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5,...........etc. are integers. Set of Integers is generally denoted by I or Z. Every Whole number is an Integer.

**Rational Numbers:**

Numbers that can be written in the form of $\frac{\text{p}}{\text{q}}$, where p and q are integers and q ≠ 0 are called Rational numbers. The collection of Rational numbers is denoted by Q. Between any two rational numbers there exists infinitely many rational numbers.

**Irrational Numbers**

Numbers which cannot be expressed in the form of $\frac{\text{p}}{\text{q}}$, where p and q are integers and q ≠ 0.

The set of irrational numbers is denoted by $\stackrel{\_}{\text{Q}}$. $\sqrt{\text{2}}$, $\sqrt{\text{3}}$, $\sqrt{\text{7}}$ are the examples of irrational numbers.

The ratio of the length of circumference of a circle to the length of its diameter is always constant. It is an irrational number and denoted by π. Decimal expansion of π is non-terminating and non-repeating. Value of π = 3.14159265......... Approximate value of π is $\frac{\text{22}}{\text{7}}$, but not equal to the exact value.

**Pythagoras Theorem**

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Irrational numbers can be represented on the number line using Pythagoras theorem.

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#### Questions & Answers*arrow_upward*

### 1 . represent root 17 on the number line

Sol :

We know that ?16 = 4 and ?17 = ?(4
^{2} +1
^{2} )

mark point A representing 4 on number lin...

### 2 . Represent root 11 on the number line

Representation of ?7 and ?11 on the number line:

Consider the number shown here

Mark point A at 0 and B at 1. Consider B

...### 3 . Arrange in ascending order: root 3, 3root 4, 4root 6

LCM of the denominators of exponents 2, 3 and 4 is 12

^{Hence the ascending order of the given numbers is}...

### 4 . Simplify: (64/125)^-2/3+(256/625)^-1/4+(3/7)^0

Can u prove 2/10=2

### 5 . Give exmple of two irrational numbers whose sum is rational

1) (2 + ?3) + (2 ? ?3) = 4

2) (2 + ?3) × (2 ? ?3) = 2
^{2} ? (?3)
^{2}= 4 ? 3 = 1

3) (?5/?5) = 1