Question

Write three numbers whose decimal expansions are non-terminating and non-recurring?

Answer 1

Hint: - Irrational numbers are those numbers which when expressed in the form of a decimal number then the result is non-terminating (never ending) and non-recurring (digits not repeat after a fixed interval).

Complete step by step solution:
Decimal expansion of any number means that when we change the number to decimal number or decimal point is involved in the number. Like \[\dfrac{1}{2}\] is a fractional number but when we change it to decimal then we completely divide the numerator by denominator. So, decimal expansion of \[\dfrac{1}{2}\] will be 0.5
So, as we know that rational numbers are those numbers that either terminate or if they are non-terminating then they are recurring numbers or digits of the number after decimal points repeat after a fixed period.
But the irrational numbers are non-terminating and non-recurring numbers.
So, we had to write three irrational numbers.
So, examples of irrational numbers are\[\sqrt 2 ,\pi {\text{ }}and{\text{ }}\sqrt 3 \].
Now let us change them to decimal expansion.
So, decimal expansion of \[\sqrt 2 \] is 1.414213562373095……….
Decimal expansion of \[\sqrt 3 \] is 1.7320508075688…….
And decimal expansion of \[\pi \] is 3.141592653589…….
Hence, decimal expansion of \[\sqrt 2 ,\pi {\text{ }}and{\text{ }}\sqrt 3 \] are non-terminating and non-recurring.
Note: - Terminating numbers are those which when converted to decimal expansion then the number of digits after decimal point are finite, like 1.732 is a terminating number but 1.7325698357203803832………. is non-terminating number. Now recurring numbers are those which when converted to decimal number then the digits of the number on the right of the decimal point repeats after a regular interval. Like \[3.\bar 2\bar 3\] = 3.232323232323…….. is a recurring number because 23 repeats after the decimal point but 3.23436782443….. is a non-recurring number.