Real Numbers

Important Terms
Natural numbers - The number's starting from 1 to ∞ are known an natural numbers for eg. 2, 89, 119
Whole numbers - The numbers in 0 to all Natural numbers is known as whole numbers
Integers - The natural numbers in which the negative numbers are also included with 0
Rational numbers - A number that can be presented in the form of p/q where q ≠ 0
@@rational no. are terminating and non-terminating and recurring.@@

Terminating numbers are those numbers that ends with a finite number.

eg - 38879

2.Non-Terminating numbers are those numbers that has infinite digits.

eg - 0.54444443453

3.Recurring number is used when the terms have a pattern that is being repeated

eg - 0.01001000100001…

@@Numbers can be termed as-@@

1.Factor - When a no. is a factor of another no. - for eg 2 is a factor of 12. 2.Multiple - It is a no. that is exactly divisible by another number. for eg. 3 is a multiple of 27.

3.A prime no. which cannot be divided by any other than itself or 1 is a prime number."

4.Numbers which are not prime are composite.

%%1 is not a prime number.%%

"Real Numbers is a set of rational and irrational numbers.

%%divisibility - when a no is completely divided by another no. is called divisibility.%%

^^Euclid's division Lemme / Algorithm^^

a = dividend

b = divisor

q = quotient

r = remainder

$dividend = divisor X quotient + remainder a = b x q + r 0 < r < b$

-> HCF by Euclid's division lemma can be found when the last divisor with remainder 0 is found.

When remainder is not zero, then the remainder is the divisor and the old divisor is the new dividend

Example-

By Euclid’s Division Algorithm

a = 38220

b = 196

q = 195

r = 0 \n 38220 = 196 x 195 + 0 \n 196 = 196 x 1 + 0 \n ∴ HCF (38220, 196) = 196.

%%IMPORTANT POINTS FOR SOLVING QUESTIONS%%

When we are given 3 no. to find HCF, first we find HCF of no. a , b and after getting that we will find HCF of c and then we have to do the algorithm to find HCF of the obtained number and c.

Whenever it is told we have to find greatest ,longest, highest number the question is basically asking for finding its HCF.

$if we have to find remainders of the numbers given then-$

step 1 - subtract the remainders from the respective no's. if the numbers are 442, 589, 697. then, 442-1, 569-2, 441, 567, 693 697-4

step 2: find HCF of no. of obtained in step 1

$IMPORTANT MEANINGS$

Coprime no's only have 1 as common factor
Coprime is not necessarily prime number.
HCF of coprime no's is always 1 and LCM of Coprime no's is always their product

@@Application of Euclid’s division lemma is "a= bq+r@@ .

Consecutive number Euclid's division lemma means one after another number.
For any integer of 2n is even and 2n+1 is odd as 2 is multiple of all even numbers.

%%Fundamental Theorem of arithmetic%%

A positive number is either prime or composite number

prime number factors - 1 and number itself as factors
composite number factors - more factors than 2 numbers

^^Any composite number can be represented in prime factors as it is unique.^^

==Carl Friedrich gauss found HCF, LCM - Fundamental theorem of Arithmetic.==

HCF of Product multiplied by LCM of product = Product of 2 numbers

To find LCM after prime factorization

multiply all prime factors. -factors that also don't match → HCF multiplied by all the factors and divided by HCF.

==numbers that are least and exactly divisible and the numbers that are greatest and divisible is also both are LCM==

THEOREM -

@@if we bring any rational number to its standard form then it is coprime.@@

Decimal expansion just by observing the denominator

denominators factor (prime factors) are only 2 and 5 then they are terminating and if their is another no accept 2, 5 then it is non terminating.

if denominator of a rational number has its prime factors of the form

then that rational number has terminating decimal or else non-terminating, recurring decimal expansion.

Real Numbers

Important Terms

^^Euclid's division Lemme / Algorithm^^

Example-

`Important` `Terms`