Real numbers are numbers that can be found on the number line.
This includes both the rational and 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 $\sqrt{2}=1.41421356.....$ does not end.
The same can be said for any irrational number.

Let's see to how express any irrational number on number line
For example:- Take $\sqrt{2}$ as irrational number
1. Draw a number line, mark the origin and other integers.
2. Now, find distance between $0$ and $1$ using compass.
3. Draw a perpendicular to $1$ of the same length as between $0$ and $1$.
4. Join origin and other end of the new line i.e. perpendicular. You will get a figure as shown.
5. Name origin as $A$, $1$ on the number line as $B$ and end of perpendicular as $C$.
6. Distance between $A$ and $C$ is $\sqrt{2}$.
7. Use compass to find distance between $A$ and $C$.
8. Place compass at $A$ and mark the same length as between $A$ and $C$ on the number line.
9. And you get the point $\sqrt{2}$ at $D$.
Similarly, you can do it for other irrational numbers like $\sqrt{3},\sqrt{5},..$

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:-
$\frac{3}{8}=0.375$ has $3$ digits after decimal point
$\frac{5}{4}=1.25$ has $2$ digits after decimal point
Hence, $0.375$ and $1.25$ are terminating 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.
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:- 
$1.6666....,0.141414...$

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:- $1.41421356.....$

Let $a$ and $b$ be any two positive integers. Then there exist unique integers $q$ and $r$ such that 
$a=bq+r,$ where $0\le r<b$
If $b|a$, then $r=0$.
Otherwise, $r$ satisfies the stronger inequality $0\le r<b$

If $a$ and $b$ are positive integers such that $a=bq+r$, then every common divisor of $a$ and $b$ is a common divisor of $b$ and $r$, and vice-versa. 
Example: Find HCF of $420$ and $130$.
Since $420>130$ we apply the division lemma to $420$ and $130$ to get ,Since $300$ , we apply the division lemma to $130$ and $30$ to get 
$130=30\times 4+10$
$420=130\times 3+30$ 
Since $100$ , we apply the division lemma to $30$ and $10$ to get 
$30=10\times 3+0$
The remainder has now become zero, so our procedure stops. Since the divisor at this step is $10$, the HCF of $420$ and $130$ is $10$. 

Find the HCF  of $6$ and $20$ by the prime factorization method. 
We have : $6=2\times 3$
$20=2\times 2\times 5$ 
$20=2\times 2\times 5$.
Common factors of $6$ and $20$ are $2$ $1$ and $2$ $2$ 
So for HCF take the common number with lowest exponent.
HCF $=2\times 1=2$