pointwise properties of irrational numbers
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Properties of irrational numbers
1. Taking the sum of an irrational number and a rational number gives an irrational number. To see why this is true, suppose x is irrational y, is rational, and the sum X+y is a rational number z . Then we have X=z-y, and since the difference of two rational numbers is rational, this implies x is rational. This is a contradiction since x is irrational. Therefore, the sum X+y must be irrational.
2. Multiplying an irrational number with any nonzero rational number gives an irrational number. We argue as above to show that if xy=z is rational, then X=z/y is rational, contradicting the assumption that x is irrational. Therefore, the product xy must be irrational.
3. The lowest common multiple (LCM) of two irrational numbers may or may not exist.
4.The sum or the product of two irrational numbers may be rational; for example,
4=√2 - √2=2