## Pre-Requisires

Test & Enrich

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**Introduction to Natural Numbers**

Non-negative counting numbers excluding zero are called **Natural Numbers**.

N = 1, 2, 3, 4, 5, ……….

**Whole Numbers**

All natural numbers including zero are called **Whole Numbers**.

W = 0, 1, 2, 3, 4, 5, ……………. **(Scroll down to continue …)**

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**Integers**

All natural numbers, negative numbers and 0, together are called **Integers**.

Z = – 3, – 2, – 1, 0, 1, 2, 3, 4, …………..

**Rational Numbers**

The number ‘a’ is called **Rational** if it can be written in the form of r/s where ‘r’ and ‘s’ are integers and s ≠ 0,

Q = 2/3, 3/5, etc. all are rational numbers.

**How to find a rational number between two given numbers?**

To find the rational number between two given numbers ‘a’ and ‘b’.

**Example:**

Find 2 rational numbers between 4 and 5.

**Solution:**

To find the rational number between 4 and 5

To find another number we will follow the same process again.

Hence the two rational numbers between 4 and 5 are 9/2 and 17/4.

**Remark: **There could be unlimited rational numbers between any two rational numbers.

**Irrational Numbers**

The number ‘a’ which cannot be written in the form of p/q is called irrational, where p and q are integers and q ≠ 0 or you can say that the numbers which are not rational are called **Irrational Numbers**.

**Example –** √7, √11 etc.

**Real Numbers**

All numbers including both rational and irrational numbers are called **Real Numbers**.

R = – 2, – (2/3), 0, 3 and √2

**Real Numbers and their Decimal Expansions**

**1. Rational Numbers**

If the rational number is in the form of a/b then by dividing a by b we can get two situations.

**a. If the remainder becomes zero**

While dividing if we get zero as the remainder after some steps then the decimal expansion of such a number is called terminating.

**Example:**

7/8 = 0.875

**b. If the remainder does not become zero**

While dividing if the decimal expansion continues and not becomes zero then it is called non-terminating or repeating expansion.

**Example:**

1/3 = 0.3333….

Hence, the decimal expansion of rational numbers could be terminating or non-terminating recurring and vice-versa.

**2. Irrational Numbers**

If we do the decimal expansion of an irrational number then it would be **non –terminating non-recurring **and vice-versa. i. e. the remainder does not become zero and also not repeated.

**Example:**

π = 3.141592653589793238……

**Representing Real Numbers on the Number Line**

To represent the real numbers on the number line we use the process of successive magnification in which we visualise the numbers through a magnifying glass on the number line.

**Example:**

**Step 1:** The number lies between 4 and 5, so we divide it into 10 equal parts. Now for the first decimal place, we will mark the number between 4.2 and 4.3.

**Step 2:** Now we will divide it into 10 equal parts again. The second decimal place will be between 4.26 and 4.27.

**Step 3:** Now we will again divide it into 10 equal parts. The third decimal place will be between 4.262 and 4.263.

**Step 4:** By doing the same process again we will mark the point at 4.2626.

**Operations on Real Numbers**

1. The sum, difference, product and quotient of two rational numbers will be rational.

**Example:**

2. If we add or subtract a rational number with an irrational number then the outcome will be irrational.

**Example:**

If 5 is a rational number and √7 is an irrational number then 5 + √7 and 5 – √7 are irrational numbers.

3. If we multiply or divide a non-zero rational number with an irrational number then also the outcome will be irrational.

**Example:**

If 7 is a rational number and √5 is an irrational number then 7√7 and 7/√5 are irrational numbers.

4. The sum, difference, product and quotient of two irrational numbers could be rational or irrational.

**Example:**

**Finding Roots of a Positive Real Number ‘x’ geometrically and mark it on the Number Line**

To find √x geometrically

1. First of all, mark the distance x unit from point A on the line so that AB = x unit.

2. From B mark a point C with the distance of 1 unit, so that BC = 1 unit.

3. Take the midpoint of AC and mark it as O. Then take OC as the radius and draw a semicircle.

4. From the point B draw a perpendicular BD which intersects the semicircle at point D.

**The length of BD** **= √x.**

To mark the position of √x on the number line, we will take AC as the number line, with B as zero. So C is point 1 on the number line.

Now we will take B as the centre and BD as the radius, and draw the arc on the number line at point E.

Now E is √x on the number line.

**Identities Related to Square Roots**

If p and q are two positive real numbers

**Examples:**

1. Simplify

We will use the identity

2. Simplify

We will use the identity

**Rationalising the Denominator**

Rationalize the denominator means to convert the denominator containing square root term into a rational number by finding the equivalent fraction of the given fraction.

For which we can use the identities of the real numbers.

**Example:**

Rationalise the denominator of 7/(7- √3).

**Solution:**

We will use the identityhere.

**Laws of Exponents for Real Numbers**

If we have a and b as the base and m and n as the exponents, then

1. a^{m} × a^{n} =a^{m+n}

^{ }2. (a^{m})^{n} = a^{mn}

4. a^{m }b^{m} = (ab)^{m}

5.^{ }a^{0} = 1

6. a^{1} = a

7. 1/a^{n} = a^{-n}

- Let a > 0 be a real number and n a positive integer.

- 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,

**Example:**

Simplify the expression (2x^{3}y^{4}) (3xy^{5})^{2}.

**Solution:**

Here we will use the law of exponents

a^{m} × a^{n} =a^{m+n} and (a^{m})^{n} = a^{mn}

(2x^{3}y^{4})(3xy^{5})^{2}

(2x^{3}y^{4})(3 ^{2} x ^{2} y^{10})

18. x^{3}. x^{2}. y^{4}. y^{10}

18. x^{3+2}. y^{4+10}

18x^{5}y^{14}

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