All numbers can be classified into two categories -- Real and Imaginary. A real number can be actually perceived and can be represented in a number line. It includes all integers -- positive and negative, all natural numbers 1, 2, 3, �..
An imaginary number on the other hand cannot be perceived. Square root of all negative numbers is imaginary. A real number can be represented as an infinite decimal expansion.
A real number can be rational or irrational. All rational numbers and all irrational numbers together make a group of real numbers. A rational number an be written in the form p q , both p and q being integers q 0 while an irrational number can not be impressed in such a way. The decimal expansion of a rational number say x can terminate after a finite number of digits.
On the other hand in case of an irrational number, the decimal expansion continues without repeating. Ex: 1. First, let us understand the meaning of Lemma and Algorithms.
A Lemma is a minor proven statement which is used to prove other statements. An algorithm means a series of well defined steps, which provides a procedure of calculations repeatedly successfully on the results of earlier steps. With the help of defined procedure the desired result can be obtained. F Highest Common Factor of the given positive integers. Ordinarily, if we want to find the HCF of two positive integers, we first find the factors of the two integers.
For ex: For two integers 24 and 42, the common factors are 1, 2, 3 and 6. As 6 is the highest of other numbers, 6 is the HCF of 24 and Next the smaller integer is divided by the above remainder and the process is repeated till zero is obtained. The remainder in the last but one division is the required HCF.
According to this theorem every positive integer is either a prime number or it can be expressed as the product of primes. It is also known as Unique Factorization Theorem. Thus, any integer that is greater than 1 can be either a prime number or can be expressed as a product of prime factors. Note :. It is the only number that is both prime and even.
A chain of factors that is demonstrated in the form of a tree, is called a factor tree. HCF is the product of the smallest power of each common prime factor present in the numbers. The product of the largest power of each prime factor present in the numbers is the LCM.
Important Note:. We use the method of contradiction in order to prove irrationality of the numbers. If a prime number p divides a 2 , then p divides a, where a is a positive integer. Let us understand how to write the decimal expansion of those rational numbers with terminating decimal expansion. The important questions for this chapter is basically a compilation of higher and more advanced techniques a student can expect in Class 10 examinations.
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