product of rational number and irrational number is
irrational
an irrational number.
To understand this, let’s first define what rational and irrational numbers are:
– Rational numbers are numbers that can be expressed as a ratio of two integers. For example, 1/3, 2/5, 7/2, etc. are all rational numbers.
– Irrational numbers, on the other hand, cannot be expressed as a ratio of two integers. They are non-repeating and non-terminating decimals. For example, π (pi), √2 (square root of 2), √3 (square root of 3), etc. are all irrational numbers.
Now, let’s take the product of a rational number and an irrational number, say 2/3 and √2.
2/3 x √2 = (2/3) x (√2)
To simplify this expression, we can first multiply the numerators and denominators separately:
(2/3) x (√2) = (2 x √2) / (3 x 1)
= (2√2) / 3
Since √2 is irrational and cannot be expressed as a ratio of two integers, the product (2√2)/3 is also irrational. Therefore, the product of a rational number and an irrational number is always an irrational number.
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