Prove Square Root of 3 is Irrational Prove Square Root of 3 is Irrational
Here I am telling you the best way of understanding that root of 3 is an irrational number. Prove square root of 3 is Irrational can be solved in the below mention manner. So we are assuming √3 is a rational number i.e √3=a/b equation (1) Where a and b are integers having no common factor (b≠0) on squaring both side (√3)2= (a/b) 2 3= a2/b2 equation (2) 3b2=a2 equation (3) Where a and b are both odd number and a/b reduce to smallest possible terms . It is not possible that a and b are even because if a and b are even one can always be divided by 2 as we assume a/b is an rational numbers a=2m+1 Assuming a and b are odd b=2n+1 By putting the value of a and b in equation 3: 3(2n+1)2= (2m+1)2 3(4n2+1+4n)= (2m2+1+4m) 12n2+3+12n=4m2+1+4m 12n2+12n+2=4m2+4m 6n2+6n+1=2m2+2m 6n2+6n+1=2(m2+n) In this equation all the value of m is always odd and the value of n is always n for all values so this equation has no solution. Know More About Worksheets of Rational Numbers in Expressions
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The values of our assumptions a and b cannot be found so we can say that root of 3 is an irrational number. What are irrational Numbers Irrational numbers are the numbers which are not rational numbers. In other words we can say that any number that cannot be expressed in the form of p/q are termed as irrational numbers. If any floating point number (that is a number that has an integer part as well as an decimal part is termed as floating point number.) cannot expressed as the ratio of two integers that floating point number is termed as irrational numbers. Let us take some of the examples of Irrational numbers Now if we take the value of “ pi (π ) “ that is π = 3.1415926535897932384626433832795 This value of π is impossible to express as the simple ratio of two numbers or two integers instead. Thus the value of π is an irrational number. Let us take some more examples to clearly get an image about the irrational numbers Let us take a value 3.2. Now 3.2 is not an irrational number, it is a rational number as 3.2 can be expressed as a ratio of two integers that is A square root of every non perfect square is an irrational number and similarly, a cube root of non-perfect cube is also an example of the irrational number.
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When we multiply any two irrational numbers and the result is rational number, then each of these irrational numbers is called rationalizing factor of the other one. What are Rational and Irrational Numbers ? When we deal Rational and Irrational numbers, the first question arise in our mind is that what are rational and irrational numbers? Rational numbers are those numbers which can be represented as fraction means having numerator and denominator and both in integer form. Let’s take some examples of rational numbers: 1. 5 is a rational number because it has 1 in its denominator and can be written as 5/1. 2. 2/3 is also a rational number. Now, the next part of the same question i.e. what are irrational numbers? Irrational numbers are those which can be represented as a fraction i.e. numbers except rational numbers. They can only be represented as decimal number.
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Published on Apr 6, 2012
Here I am telling you the best way of understanding that root of 3 is an irrational number. Prove square root of 3 is Irrational can be solv...