Is √2 a Rational Number Is √2 a Rational Number
To check if 21/2belongs to the set of rational numbers, let’s assume that 21/2is a rational number. Now,we can find any integer numbers a and b (≠ 0) such that 2= (a/b). Here,a & b are co-primes. b*(21/2)=a Squaring both sides: 2b2=a2. (1) =>2 divides a2. =>2 divides a. a=2c, let c be any integer. Substituting a we get- 2b2=4c2 =>2 divides b2 => 2 divides b =>a and b have atleast 2 as common factor≠1. This contradiction arises because our assumption 21/2 is rational number, is wrong. Hence, √2 is not a Rational Number. Definition of Rational Number In mathematics, Rational Numbers can be defined as a ratio of two integers. Rational numbers can be expressed in form of fraction 'a/b' in which 'a' and 'b' are integers Know More About Worksheets on Laws of Limit
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where denominator 'b' not equal to zero. We also have set of rational number and 'Q' is used to represent it. Examples:1/3 is a rational number (1 divided by 3, or the ratio of 1 to 3). 0.25 is a rational number (1/4). 3 is a rational number (3/1). 2.32 is a rational number (232/100). -6.8 is a rational number (-68/10). This is a brief gist of Rational Numbers. Introduction to Rational numbers Today, I will tell you a story. Once there was a family of Natural numbers where all counting numbers used to live. One day a guest named zero visited the house and requested for a permission to stay there. All were happy; they requested the eldest member of the family Mr. infinite (∞) to grant the permission for 0. The permission was granted and the name of the house was changed to house of Whole numbers. Now, after some time negative numbers also visited the house and requested for the permission to be the part of the family. They were permitted and now the family became the family of Integers i.e. -∞ . ……..-3,-2,-1,0,1,2,3,…….∞. Read More About Worksheet on Law of Limit
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On seeing the family living together, some numbers which were in form of p/q, where p and q are natural numbers also asked for a permission to stay there. They were called fractions. Some fractions are 4/7, 2/5 …etc. The family of fractions also told that if you all see the denominator with you, which is not usually visible, then you will also become the part of fractions family. All the numbers started trying it and realized that they all are the part of fractions. But this was not true for negative integers. The meeting was held, in which it was decided that a name Rational number will be given to the family. A family of Rational Numbers consists of all the numbers which can be expressed in form of p/q, where p and q are integers, but q≠ 0. Example of Rational Numbers Like addition and subtraction we don’t need to equalize the denominator we can multiply them as, numerator of 1st number *denominator of 2nd number/denominator of 1st number *numerator of 2nd In other words we can say dividing the number is just equivalent to multiplying the 1st number with the reciprocal of other. We also need to see one thing the sign convention that’s also plays a very important role in the division we need to know that in multiplication (+) *(+)= + , multiplication of (-) * (-) =+,multiplication of (-)* (+)= - and multiplication of (+) * (-) = - we can also remember these convention as if we have the same sign then the result will always be positive and if signs are negative then result will always be negative ,I think this is the simplest way to remember the sign convention.
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