1) How do we prove that sum of Rational and Irrational no. is Irrational ?

2) Also how do we prove that sum of two Irrational nos. is Irrational ?

P.J.LAKHAPATE
pjlakhap@bechtel.com

When is your homework due?

1) Contradiction

2) You don't

2x pi is still irrational, and c plus pi is still irrational. However, what is the difference betweeen an irrational number and a transcendental number?

A transcendental number is not the solution of any polynomial equation.
http://en.wikipedia.org/wiki/Transcendental_number

I gather that all irrational numbers are transcendentasl?

Not at all, any square root, or sum of a square root and an integer can be a solution to a polynomial, eg. x^2-2=0.

polynomial equation

With rational coefficients :)

For example, pi is a root of x - pi = 0, but it's transcendental.

Any real number that can be expressed with a finite number of additions, multiplications, and extractions of (rational) roots of rational numbers is algebraic. Any real number that isn't algebraic is transcendental.

Oops, thanks Data!
That's what I meant to type.

It should be mentioned that all transcendental numbers are rational.

OOps I meant all transcendental numbers are irrational, sorry.

Can anybody give an example of Transcedental No.

Pi and e are transcendental, aren't they?

Er, how exactly do you express the sum of two irrational numbers in a way to allow you to work a proof?

2^0.5 and 3^0.5 both are irational.
Prove that (2^0.5) + (3^0.5) is an irrational no.

2) Also how do we prove that sum of two Irrational nos. is Irrational ?
You disprove the generalisation by counter-example:
√2 and (2-√2) are both irrational, but their sum is rational.

1) How do we prove that sum of Rational and Irrational no. is Irrational ?

a, b, c, and d are integers, e is an irrational number.

Assume:
a/b + e = c/d

a/b - c/d = e
(ad - bc)/bd = e

therefore e is rational. This contradicts our premises assumptions, which means that if e is irrational, a/b + e must also be irrational.


(insert rigour where necessary!)