Real Numbers

Definition:

· Real Numbers are just numbers, nearly any number you can think of is a Real Number.

· So, what is NOT a Real Number,

Euclid’s Division Lemma:

· Lemma: A lemma is a statement which is already proved and is used for proving other statements.

· Euclid’s Division Lemma states, for each pair of given positive integers a and b, there exist unique whole numbers q and r which satisfies the relation,

· Where ‘a’ is a dividend, ‘b' is divisor, ‘q’ is quotient and ‘r’ is remainder.

Euclid’s Division Algorithm:

· Algorithm: An algorithm gives us some definite steps to solve a particular type of problem in a well-defined manner.

· Euclid’s Division Algorithm is a method used to find the H.C.F of two numbers, say a and b where a>b.

· Step I: We apply Euclid’s Division Lemma to find two integers q and r such that a=b*q + r and 0≤r<b.

· Step II: If r = 0, the H.C.F is b, else, we apply Euclid’s division Lemma to b (the divisor) and r (the remainder) to get another pair of quotient and remainder.

· Step III: The above method is repeated until a remainder of zero is obtained. The divisor in that step is the H.C.F of the given set of numbers.

· Example on finding HCF using Euclid’s Division Algorithm.

The Fundamental Theorem of Arithmetic:

Prime Numbers:

· A Prime Number is a number that cannot be evenly divided by any other number except 1 or itself.

· PRIME NUMBERS to 100

Composite Numbers:

· A whole number that can be divided evenly by numbers other than 1 or itself

Prime Factorisation:

· Prime Factorisation is the method of expressing a natural number as a product of prime numbers.

· The Fundamental Theorem of Arithmetic states, every composite number can be written as the product of power of primes and this factorization is unique and the order of the prime factors does not matter.

HCF/GCD by prime factorization method:

· HCF (Highest Common factor) or GCD(Greatest Common Divisior) = product of the smallest power of each common factor in the numbers

· Examples of HCF for 2 Numbers

· Examples of HCF for 3 Numbers

LCM by prime factorization method:

· LCM (Lowest Common Multiple) = Product of the greatest power of each prime factor involved in the number.

· Examples for finding LCM

Relation Between HCF & LCM:

Irrational Numbers:

· A number r is called irrational number if it cannot be expressed in the form p/q where p and q are integer and q≠ 0

· Theorem 1: If p is a prime number and p divides , then p is one of the prime factors of which divides a, where a is a positive integer.

· Theorem 2: If p is a positive number and not a perfect square, then √n is definitely an irrational number.

· Theorem 3: If p is a prime number, then √p is also an irrational number.

· Theorem 4: The sum or difference of a rational and an irrational number is irrational

· Theorem 5: The product and quotient of a non-zero rational and irrational number are irrational.

Rational numbers:

· A number r is called rational number if it can be expressed in the form p/q where p and q are integer and q≠ 0

· Decimal expressions of rational number are of two types

· Terminating decimal expression: Terminating decimals are decimals that end at a certain point. Example: 0.2, 2.56 and so on. Terminating decimal expression can be written in the form

· Non-terminating repeating decimal expression: Non-terminating decimals are decimals where the digits after the decimal point don’t terminate. Example: 0.333333….., 0.13135235343…

· Non-terminating decimals can be :

· Recurring – a part of the decimal repeats indefinitely (0.142857142857….)

· Non-recurring – no part of the decimal repeats indefinitely. Example: π=3.1415926535…

· Example on how to represent Recurring number in p/q form

· Example to check if a number is Terminating decimal expression or Non-terminating repeating decimal expression