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Are the products of irrational numbers always irrational?
No. The product of conjugate pairs is always rational.So suppose sqrt(y) is the irrational square root of the rational number y. ThenThus [x + sqrt(y)]*[x - sqrt(y)] = x^2 + x*sqrt(y) - x*sqrt(y) - sqrt(y)*sqrt(y)= x^2 + y^2 which is rational.
What is an irrational number and examples?
An irrational number is a number that cannot be written as a ratio of two whole numbers. That is, there are no two integers, X and Y (with Y>0) such that the number can be written as X/Y. Sqrt(2), pi, log(3) are examples of irrational numbers.
Why is the sum of a rational numbers and an irrational number is irrational?
Let x be a rational number and y be an irrational number.Suppose their sum = z, is rational.That is x + y = zThen y = z - xThe set of rational number is closed under addition (and subtraction). Therefore, z - x is rational.Thus you have left hand side (irrational) = right hand side (rational) which is a contradiction.Therefore, by reducio ad absurdum, the supposition that z is rational is false, ie the sum of a rational and an irrational must be irrational.
Is the product of rational no and irrational no is rational?
Unless you multiply 0 with some irrational number, it is impossible. Here's why: Let x,y be rational with x = a/b, z = c/d and y be the irrational number. If we presume xy = z then we have y = z/x. However, this is equal to (c/d)/(a/b) = (bc)/(ad), which is rational. Since y is assumed to be irrational, this cannot occur (unless one of b,c is zero).
Is 4 Over 4 An Irrational Number?
No. 4 over 4, or 1 whole, is not an irrational number. It can be written as a simple fraction, 4/4, so it is not an irrational number.
