Real Numbers
The part of speech is a category to which a word is assigned according to its syntactic functions. In English the main parts of speech are noun, pronoun, adjective, determiner, verb, adverb, preposition, conjunction, and interjection.
6.Revisiting Rational
Numbers and their
Decimal Expansion
An interjection is used to express emotion in a sentence.
Think of other interjections!
So by those example with get to know that rational number of the form of p by q where p and q are coprime and prime factorization of denominator that is q has only powers of 2 or 5 or both
Theorem 1.6: Let x=p/q be a rational number such that prime factorization of q is not in the form of 2^n5^m where m and n are non negative integers. Then X has a decimal representation which terminates
For ex: division of 1/7, here reminders are 3,2, 6,4,5,1,3 ,2,6,4,5,1 and so on
Notice that the denominator hair is clearly not from 2n and 5m therefore from theorem 1.5 and 1.6 1/7will not have a terminating decimal expansion
Theorem 1.7 : Let x=p/q be a rational number, such that the prime factorization of q is not of the form 2^n 5^m, where n, m are non negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).So by this theorem we get to know that decimal expansion of every rational number is even terminating or non terminating repeating
13/3125
3125=5x5x5x5x5
=13/5^5
So as denominator is having power of 5 so decimal expansion is terminating
64/455
455=5x7x13
So as denominator is having both 5 and other no except 2 so decimal expansion is recurring
Let us consider a rational number say P/q where q ≠ zero, for example
0.375 = 375/1000 = 375/10^3
-> It can also be written as
3 x 5^3 / 2^3 x 5^3 = 3 / 2^3
Theorem 1.5 : let x be a rational number whose decimal expansion terminates. Then x can be expressed in form of p/q ,where p and q are coprime and Prime factorisation of q is of the form 2^n 5^m where n and m are non negative integers so we will prove by some examples
For ex: 3/8 =
3 / 2^3 = 3 x 5^3 / 2^3 x 5^3 = 0.375
4.Fundamental Theorem
of Arithmetic
A numeral is a word or phrase that describes a numerical quantity.
Some theories of grammar use the word 'numeral' to refer to cardinal numbers that act as a determiner to specify the quantity of a noun, for example the 'two' in 'two hats'.
HCF(a,b) x LCM(a,b) = a x b
HCF is found by taking each prime factor in all the terms the least number of times it appear in any of the terms.
LCM is found by taking each prime factor in all the terms the most number of times it appears in any of the terms
Every composite number can be expressed
as a product of primes, and this factorization is unique apart from the order in which the prime factors occur. Ex- 36=2x2x3x3
5.Irrationality
An adverb is used to describe a verb, but it can also describe an adjective or another adverb.
Adverbs normally help paint a fuller picture by describing how something happens.
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Theorem 1.3: If p is a prime no. and p divides a^2 then p divides a where p is positive integer.
For example:
Root 5 is irrational number
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When we add or multiply Rational number with irrational then we get irrational number.
The numbers which are neither terminating nor repeating are known as irrational numbers.They cannot be written in the form of p/q where p & q are integers and are not equal zero.
3.Euclid division Algorithm
A pronoun is a word that can be used in place of a noun, typically after the noun itself has already been stated.
If "a" and "b" are positive integers
such that "a" = "b" q + r, then,
every common divisor of "a" and "b"
is a common divisor of b and r and
vice versa.
It is used to find HCF.
Euclid Division algorithm is based on the principle that he greatest common divisor of two numbers does not change is the smaller number is subtracted from the longer number.
A reflexive pronoun ends with ...self or ...selves and refers to another noun or pronoun in the sentence (usually the subject of the sentence). The reflexive pronouns are myself, yourself, herself, himself, itself, ourselves, yourselves, and themselves.
2.Euclid's Division Lemma
An adjective is a word that's used to describe a specific noun and to provide more detail to the listener.
Dividend = divisor x quotient + remainder
Given positive integers a and b, there exists unique integers q and r satisfying the above equation.
Superlative adjectives demonstrate a higher level of comparison between entities.
a = p q + r, 0 ≤ r < b
Expresses a comparison between two entities or groups of entities in quality or degree.
1.Introduction
A verb is an action word or 'doing' word that signifies movement in some way.
Examples- √2 , √3 , 1.010110111, 12/22, 25.5, 100 etc.
A linking verb connects the subject with a word that gives information about the subject, such as a condition or relationship.
Real number consist of both rational and irrational numbers.
A verb with its own meaning: a verb that is not an auxiliary verb.
