Prove square root of 3 is Irrational Number Prove square root of 3 is Irrational Number
Here, I am going to tell you the best way of understanding that root of 3 is an Irrational Number. 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. The values of our assumptions a and b cannot be found so we can say that root of 3 is an Irrational Number. Know More About Constant Law of Limit Worksheets
Page No. : 1/4
What is a Rational Number? The Rational numbers are those numbers which can either be whole numbers or fractions or decimals. Rational numbers can be written as a ratio of two integers in the form 'p/q' where 'p' and 'q' are integers and 'q' is nonzero. A rational number is simply a ratio of two integers, for example1/5 is a rational number (1 divided by 5, or the ratio of 1 to 5). 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. 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 where denominator 'b' not equal to zero. We also have set of rational number and 'Q' is used to represent it. Positive Rational Numbers Positive rational numbers can be expressed as the ratio 'p/q' where, 'p' and 'q' are both positive integers. Any positive rational number can be expressed as a sum of distinct reciprocals of positive integers like: 3/5 = 1/3+2/6+1/18. A positive rational number has infinitely many different such representations called Egyptian. Read More About Constant Laws of Limit Worksheet
Page No. : 2/4
Negative Rational Numbers The numbers which are written in form of a/b where a and b are integers such that b≠0 all comes in the family of Rational Numbers. Positive Rational Numbers are those which have both numerators and denominators as positive or negative. Example: 5/7, 6/5 or (-3/-4) are positive rational numbers. Properties of Rational Numbers The rational numbers are closed under addition, subtraction, multiplication and division by nonzero rational numbers. The properties are called closure properties of rational numbers. Operations on Rational Numbers In mathematics, we generally deal with four types of basic operations called as addition, subtraction, multiplication, and division. We can easily perform these four kinds of operations on different type of Rational Numbers. Rational Numbers on a Number Line A number line is graphical representation of numbers from negative infinity to positive infinity in a single line. Rational numbers can be expressed in fraction (a/b) form, where 'b' is not equation.
Page No. : 2/3 Page No. : 3/4