real numbers class 10 maths concepts

01

 

DEFINITION

Fundamental Theorem of Arithmetic

Any integer greater than  is either a prime number, or can be written as a unique product of prime numbers

02

 

EXAMPLE

Real number by Magnification

Find the number  by magnification.
Sol: We will initially look at , after magnifying we are see at various sub-divisions like , which later on breaks into 

03

 

DEFINITION

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.

04

 

DEFINITION

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:-

05

 

DEFINITION

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:- 

06

 

DEFINITION

Euclid's Division Lemma

Let  and  be any two positive integers. Then there exist unique integers  and  such that

If , then .
Otherwise,  satisfies the stronger inequality 

07

 

EXAMPLE

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: Find 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 . 

08

 

RESULT

Multiplicative inverse

Multiplicative inverse of any irrational number  is equal to  denoted by  such that : 
. Example:  is the multiplicative inverse of