Concepts

## Real Numbers

- Let's dig deep into the concepts of the chapter
1

## Real Numbers

Real numbers are numbers that can be found on the number line.
This includes both the rational and irrational numbers.
2

## Irrational Numbers

An irrational number is number that cannot be expressed as a ratio of integers, i.e. as a fraction.
Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat.
For example:- The number  does not end.
The same can be said for any irrational number.
3

## Express rational numbers on number line

Let's see to how express any irrational number on number line
For example:- Take as irrational number
1. Draw a number line, mark the origin and other integers.
2. Now, find distance between and using compass.
3. Draw a perpendicular to of the same length as between and .
4. Join origin and other end of the new line i.e. perpendicular. You will get a figure as shown.
5. Name origin as , on the number line as and end of perpendicular as .
6. Distance between and is .
7. Use compass to find distance between and .
8. Place compass at and mark the same length as between and on the number line.
9. And you get the point at .
Similarly, you can do it for other irrational numbers like
4

## Define and identify examples of terminating decimals

A terminating decimal is a decimal that ends i.e. it has finite number of digits.
For a fraction in decimal form, while performing division after a certain number of steps, we get the remainder zero.
The quotient obtained as decimal is called the terminating decimal.
For eg:-
has digits after decimal point
has digits after decimal point
Hence, and are terminating decimals.
5

## Non-Terminating Recurring Decimals

While expressing a fraction in the decimal form, when we perform division we get some remainder.
If the division process does not end we do not get the remainder equal to zero; then such decimal is known as non-terminating decimal.
In some cases, a digit or a block of digits repeats itself in the decimal part, then the decimal is non-terminating recurring decimal.
For eg:-

6

## Non-terminating and non-recurring decimals

While expressing a fraction in the decimal form, when we perform division we get some remainder.
If the division process does not end i.e. we do not get the remainder equal to zero; then such decimal is known as non-terminating decimal.
And if a digit or a block of digits does not repeats itself in the decimal part, such decimals are called non-terminating and non-recurring decimals.
For eg:-
7

## Euclid's Division Lemma

Let and be any two positive integers. Then there exist unique integers and such that
where
If , then .
Otherwise, satisfies the stronger inequality
8

## HCF using Euclid's Divison

If and are positive integers such that , then every common divisor of and is a common divisor of and , and vice-versa.
Example: F
ind HCF of and .
Since we apply the division lemma to and to get ,Since , we apply the division lemma to and to get

Since , we apply the division lemma to and to get

The remainder has now become zero, so our procedure stops. Since the divisor at this step is , the HCF of and is
9

## HCF using Fundamenal Theorem

Find the HCF  of and by the prime factorization method.
We have :

.
Common factors of and are   and
So for HCF take the common number with lowest exponent.
HCF