Concept 1.4
Mixing Rational and Irrational Numbers
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
The sum or difference of a rational and an irrational number is irrational, and the product or quotient of a non-zero rational and an irrational number is irrational. For example, 5 − √3 and 3√2 are irrational.
The intuition
To show 5 − √3 is irrational, suppose it equals a rational number a/b. Then rearrange so √3 stands alone: √3 = 5 − a/b.
The right side is built only from rational numbers, so it is rational. But √3 is not. The rearrangement is the whole trick: isolate the irrational piece and the contradiction is plain to see.
The attention
Look at the side made only of rational numbers once the root stands alone.
5 − √3 = a/b ⇒ √3 = 5 − a/b
Look hereNot here
- Look here
- The side made only of rational numbers. 5 − a/b is built from rational numbers only, so it is rational. It can't equal √3, which isn't.
- Not here
- The 5 on its own. The 5 on its own proves nothing. The argument only works once √3 is alone on one side.
- What it teaches
- Rearrange until the thing you doubt stands alone. Then compare it with what the other side must be.