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Question

Is the product of a rational and irrational number always irrational? Give an example.


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Solution

Find the product of rational and irrational numbers.

Both rational and irrational numbers are real numbers, but they have different characteristics. A rational number is a number that can be expressed in the form of the ratioPQ&Q0 and an irrational number cannot be expressed in the ratio PQ&Q0. However, both numbers are real and can be represented by a number line.

Example 1: a=23,b=3

Product ab=23

23 is an irrational number.

Example 2: a=0,b=3

Product ab=0

0 is a rational number.

Example 3: a=4,b=3

Product ab=43

43 is an irrational number.

Using the above example, we can conclude that if the rational number is 0, the product of the rational number and the non-rational number is always a rational number, but if the rational number is non-zero, the product of the rational number and irrational numbers are always irrational numbers.

Hence, the given statement is only true when the rational number is no zero.


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