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You said:
Why is 70.300722 a rational number
The Correct Answer and Explanation is:
The number 70.300722 is a rational number because it can be expressed as a ratio of two integers, which is the defining characteristic of rational numbers.
Understanding Rational Numbers:
A rational number is any number that can be written in the form of a fraction or ratio, where both the numerator and the denominator are integers, and the denominator is not zero. In other words, a rational number can be represented as ( \frac{p}{q} ), where ( p ) and ( q ) are integers and ( q \neq 0 ).
Decimal Representation:
The given number, 70.300722, is a terminating decimal because it has a finite number of digits after the decimal point. Rational numbers can either have:
- Terminating decimals: These decimals end after a finite number of digits (like 0.75 or 2.5).
- Repeating decimals: These decimals have a repeating block of digits that go on infinitely (like 0.333… or 1.666…).
In the case of 70.300722, the decimal part is 300722, which ends after six digits. Since it terminates, we know that it is a rational number. Any terminating decimal can be converted into a fraction, which proves its rationality.
Converting 70.300722 into a Fraction:
To express 70.300722 as a fraction:
- Consider the number as 70300722/1000000 by moving the decimal point six places to the right.
- Simplify the fraction if possible. The number 70300722 and 1000000 can be divided by their greatest common divisor (GCD), which is 2:
[
\frac{70300722}{1000000} = \frac{35150361}{500000}
]
Thus, 70.300722 is equivalent to the fraction ( \frac{35150361}{500000} ), which is the ratio of two integers.
Conclusion:
Since 70.300722 can be expressed as the ratio of two integers, it is a rational number. Rational numbers include both integers and fractions, and their decimal representations can either terminate or repeat. Therefore, 70.300722 qualifies as a rational number because it has a finite decimal expansion and can be expressed as a fraction.