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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.
If X is irrational and Y is rational then X plus Y is irrational.?
Yes.
If x is a rational number and y is an irrational number what can you say about x plus y?
It is irrational.
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.
Are Irrational Numbers Closed Under Addition counter example?
No. Here is a counter-example: x = 1 + sqrt(2) y = 2 - sqrt(2) x and y are irrational. x + y = 3 is rational.
What is an irrational number 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.
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.
If you add a rational and irrational number what is the sum?
an irrational number PROOF : Let x be any rational number and y be any irrational number. let us assume that their sum is rational which is ( z ) x + y = z if x is a rational number then ( -x ) will also be a rational number. Therefore, x + y + (-x) = a rational number this implies that y is also rational BUT HERE IS THE CONTRADICTION as we assumed y an irrational number. Hence, our assumption is wrong. This states that x + y is not rational. HENCE PROVEDit will always be irrational.
What is an irrational and a rational number?
Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
Is the sum of two irrational numbers irrational?
not always. nothing can be generalized about the sum of two irrational number. counter example. x=(sqrt(2) + 1), y=(1 - sqrt20) then x + y = 1, rational.
What is an irrational number and why is pi an irrational number?
An irrational number is a real number that cannot be expressed as a ratio of two integers, x and y, where y>0. In 1761, Johann Heinrich Lambert proved that pi is irrational. His proof and alternatives by other mathematicians can be found at the attached link.
Is a rational number divided by an irrational number always irrational?
No. If we let x be irrational, then 0/x = 0 is a counterexample. However, if we consider nonzero rational numbers, then our conjecture is true. We shall prove this by contradiction. Suppose we have nonzero rational numbers x and y, and an irrational number z, such that x/z = y. Since z is not equal to 0, x = yz. Since y is not equal to 0, x/y = z. Since x/y is a quotient of rational numbers, x/y is rational. Therefore, z is rational, contradicting our assumption that z was irrational. QED.
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.
