**Real Numbers**

**Study Points**1. A

**is a proven statement used for proving another statement.**

*lemma*2. An

**is a series of well defined steps which gives a procedure for solving a type of problem.**

*Algorithm*3.

*Muḥammad ibn Mūsā al-Khwārizmī*, a persian mathematician coined the term 'algorithm'.

4.

**: Given positive integers**

*Euclid's division lemma**and*

**a***there exist whole numbers*

**b***and*

**q****satisfying a = bq +r, 0 ≤ r < b.**

*r*5.

*Euclid's division algorithm*: In order to compute the HCF of two positive integers say

*and*

**a***, with a > b by using Euclid's algorithm we follow the given below steps:*

**b**STEP-❶: | Apply Euclid's division lemma to a and b and obtain whole numbers q_{1} and r_{1} such that a = bq _{1} + r_{1}, 0 ≤ r < b |

STEP-❷: | If r_{1} = 0, b is the HCF of a and b. |

STEP-❸: | If r_{1}
≠ 0, apply Euclid's division lemma to b and r_{1} and obtain two whole numbers q_{1} and r_{2} such that b = q_{1}r_{1} + r_{2} |

STEP-❹: | If r_{2} = 0, r_{1} is the HCF of a and b |

STEP-❺: | If r_{2}
≠ 0, apply Euclid's division lemma to r_{1} and r_{2} and continue the above process till the remainder r_{n} is zero. The divisor at this stage i.e. r_{n-1}or the non-zero remainder at the previous stage is the HCF of a and b._{} |

6. Euclid’s Division Algorithm is stated for only positive integers but it can be extended for all integers except zero, i.e,

*≠ 0.*

**b**7.

*The Fundamental Theorem of Arithmetic*: Every composite number can be expressed (factorized) as product of primes, and this factorization is unique except for the order in which the prime factors occur.

8. Every composite number can be uniquely expressed as the product of powers of primes in ascending or descending order.

9. Let

**be a positive integer and p be a prime number such that p|a**

*a*^{2}(p divides a

^{2}), then p | a (p divides a).

10. There are infinitely many positive primes.

11. Every positive integer different from 1 can be expressed as product of non-negative power of 2 and an odd number.

12. A positive integer

*is a prime number, if it is not divisible by any prime less than or equal to √n.*

**n**13. If p is a positive prime, then √p is an irrational number. e.g. √2, √3, √5, √7, √11 etc. are irrational numbers.

14. The numbers which can be represented in the form of p/q where q ≠ 0 and p and q are integers are called

**.**

*Rational numbers*15. Irrational numbers are the numbers which are

*non-terminating*and

*non-repeating*.

16. An Irrational Number is a real number that cannot be written as a simple fraction.

17. Practically Irrational numbers are used in

- Finding the length of diagonal of a square whose sides are given.
- Finding the hypotenuse of a right triangle.
- Computing the circumference of a circle whose radius is known.
- Finding the diagonal of a square whose sides are given.

18. Rational and irrational numbers together form

*.*

**Real numbers**19. The sum, difference, product and quotient of two irrational numbers need not always be irrational number. e.g. √2 ✕ √2 = 2 is a rational but π ✕ π = π

^{2}which is irrational.

21. Terminating fractions are the numbers which leaves remainder 0 on normal division.

22. Recurring fractions are the numbers which never leave a remainder 0 on normal division.

23. There are more irrational numbers than rational numbers between two consecutive numbers.

24. Sum and product of a rational number and an irrational number is an irrational number.

25. Let

*x*be a rational number whose decimal expansion terminates. Then

*x*can be expressed in the form

*p*/

*q*, where

*p*and

*q*are co-prime and the prime factorization of q is of the form of 2

^{m}✕ 5

^{m}, where

*m*,

*n*are non-negative integers.

26. Let

*x*=

*p*/

*q*be a rational number, such that the prime factorization of q is of the form 2

^{m}✕ 5

^{m}, where

*m*,

*n*are non-negative integers. Then

*x*has a terminating decimal expansion.

27. Let x = p/q be a rational number, such that the prime factorization of q is not of the form 2

^{m}✕ 5

^{m}, where

*m*,

*n*are non-negative integers. Then

*x*has a terminating repeating decimal expansion.

28. HCF (a,b) ✕ LCM (a,b) = a ✕ b where

*a*and

*b*are positive integers.

See also ☛ CH 1: Real Numbers (MCQ)

Nice questions BT need more tough questions

ReplyDeleteNice questions BT need more tough questions

ReplyDeleteNice questions BT need more tough questions

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