Prove that for every rational number z and every irrational

Ayanna Trujillo

Ayanna Trujillo

Answered question

2022-06-16

Prove that for every rational number z and every irrational number x, there exists a unique irrational number such that x + y = z.

Answer & Explanation

scipionhi

scipionhi

Beginner2022-06-17Added 25 answers

Any z that is a rational number can be expressed as the ratio of two integers. (For strictness we require the denominator integer to be non-zero.)

Any x is an irrational number cannot be so expressed as the ratio of two integers.

For any real numbers, x and z, there is only a unique number y where x + y = z.

If this y were rational it could be expressed as the ratio of some two integers. For any sum x + y which can also be expressed as the ratio of two integers, it would then follow that x could be expressed as the sum of two integers. (By reason the product of any two integers is an integer.)

By contraposition: for any irrational x, and any rational z, the number y where x + y = z, must be both unique and irrational.

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