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If X is irrational and Y is rational then X plus Y is irrational.?
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∙ 13y ago
Yes.
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∙ 13y ago
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If x is a rational number and y is an irrational number what can you say about x plus y?
It is irrational.
If x is a rational number and y is an irrational number then can you say about x plus y?
please rephrase or grammar-check your question.
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.
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.
Are imaginary numbers rational or irrational?
Imaginary numbers are not intrinsically rational or irrational.Of course, all real numbers are either rational or irrational numbers.Imaginary numbers are not real numbers.Imaginary numbers have a real part and an imaginary part, sometimes written like z=x+i y.The two parts, i.e. the x and the y, are real numbers. As real numbers, they are either rational or irrational. Its just that the two parts of a complex number may both be either rational or irrational or one may be rational and the other irrational. One could always make up a new name for these cases, but right now there is no such classification.
If x is a rational number and y is an irrational number what can you say about x plus y?
It is irrational.
Show that the sum of rational no with an irrational no is always irrational?
Suppose x is a rational number and y is an irrational number.Let x + y = z, and assume that z is a rational number.The set of rational number is a group.This implies that since x is rational, -x is rational [invertibility].Then, since z and -x are rational, z - x must be rational [closure].But z - x = y which implies that y is rational.That contradicts the fact that y is an irrational number. The contradiction implies that the assumption [that z is rational] is incorrect.Thus, the sum of a rational number x and an irrational number y cannot be rational.
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.
If x is a rational number and y is an irrational number then can you say about x plus y?
please rephrase or grammar-check your question.
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.
If x is a rational number and y is a rational number than is x plus y rational?
If both numbers are rational then x plus y is a rational number
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.
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.
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.
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 the different between a rational and irrational number?
Rational numbers can be represented in the form x/y but irrational numbers cannot.
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).
