show that 2+√5 is an irrational number

Asked by poojithav2468 | 20th Apr, 2022, 08:22: PM

Expert Answer:

Suppose 2+√5 is a rational number
rightwards double arrow 2 plus square root of 5 equals straight p over straight q comma space space straight p comma straight q space are space integers space and space straight q not equal to 0
rightwards double arrow square root of 5 equals straight p over straight q minus 2
Since comma space difference space of space two space rational space numbers space is space rational
Therefore comma space straight p over straight q minus 2 space is space straight a space rational space number.
Thus comma space square root of 5 space is space rational.
This space is space straight a space contradiction space as space square root of 5 space is space irrational
Hence comma space 2 plus square root of 5 space is space an space irrational space number.

Answered by Renu Varma | 21st Apr, 2022, 07:49: PM