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REAL NUMBERS
Real numbers - Real numbers are the numbers which can be obtained on the number line. Real numbers are denoted by 'R'. They includes Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers.
Natural numbers - Natural numbers are the counting and ordering numbers (excluding 0) i.e. N = { 1, 2, 3, 4, 5, .......... }
Whole numbers- Whole numbere are collection of natural numbers including zero (0) i.e. W = {0, 1, 2, 3, 4, 5, ........ }
Integers - Integers consist of natural numbers, their negatives and zero i.e. Z = { ....... -3, -2, -1, 0, 1, 2, 3, 4, 5, ........... }
Prime numbers - A prime number is defined as a number that has no factor other than 1 and itself.
Co-prime numbers- co-primes are considered in pairs and two numbers are co-prime if they have no common factors other than 1.
Twin prime numbers - Twin prime numbers are the prime numbers whose difference is always equal to 2. For example, the difference between 5 and 7 is 2, and hence 5 and 7 are twin prime numbers.
Composite numbers -
All the natural numbers which are not prime numbers are composite numbers as they can be divided by more than two numbers.
Rational Numbers - Rational numbers are the numbers which can be expressed as p/q, where p and q are integers and q ≠ 0. i.e. 3/4, 5/9, ...... Etc.
Irrational numbers - All real numbers which are not rational numbers are known as irrational numbers. A non- terminating and non- repeating (recurring) decimals are an example of irrational numbers. i.e. √5, √11, ........... Etc.
NOTE- 1 is neither a prime nor a composite number.
Important Points :-
1. An irrational number is a non terminating and non recurring decimal and cannot be expressed in the form P/q .
2. Sum of two irrational numbers does not always given irrational number.
3. Product of two irrational numbers does not always given irrational number.
4. Terminating or non terminating and recurring/ repeating decimals are examples of rational number.
5. Product of a non zero rational number with an irrational number always gives an irrational number.
6. Sum of a rational number and irrational number always gives irrational number.
7. If √ab is irrational, then √a + √b will definitely be irrational.
8. If c is a positive prime number, then √c is an irrational number.
9. Euclid's division Lemma - for any two positive integers a and b , a>b their exits unique integers q and r such that a = bq+ r (0≤r<b) .
10. Euclid's division Algorithm- Euclid's algorithm provides step wise procedure for computing the HCF of natural numbers.
11. Fundamental theorem of arithmetic- every composite number can be expressed as a product of primes and their factorization is unique.
Keep in Mind.....
1. L.C.M. × H.C.F = Product of two numbers.
2. H.C.F of two or more prime numbers is always 1.
3. L.C.M. of two or more prime numbers is equal to their product.
🎯For fractions x/y and c/d,
1.L. C. M. Of fraction = L. C. M. Of numerators / H. C. F. Of denominators
2. H. C. F. Of fraction = H. C. F. Of numerators /L.C.M. of denominators
👩🏫For three positive integers (x, y and z)
1. H. C. F. (x, y, z) × L. C. M. (x, y, z) ≠ x × y × z, where x, y and z are positive integers.
2. L. C. M. (x, y, z) = x. y. z. H. C. F. (a, b, c) /H.C.F (x, y). H. C. F(y, z) H.C.F(x, z)
3. H. C. F. (x, y, z) = x. y. z. L. C. M(x, y, z) /L.C.M(x, y) L. C. M. (y, z) L. C. M. (x, z)
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