Is 70.300722 a rational number?
The Correct answer and Explanation is:
Yes, 70.300722 is a rational number.
A rational number is defined as any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In mathematical terms, a rational number can be represented as pq\frac{p}{q}qp, where ppp and qqq are integers and q≠0q \neq 0q=0. This definition encompasses both integers (e.g., -2, 0, 1) and fractions (e.g., 12\frac{1}{2}21, −34\frac{-3}{4}4−3).
The number 70.300722 can be expressed as a fraction. To illustrate this, we can rewrite it in the form pq\frac{p}{q}qp. Since 70.300722 has six decimal places, we can multiply it by 10610^6106 (or 1,000,000) to eliminate the decimal. This gives us:70.300722=70.300722×10000001000000=70300722100000070.300722 = \frac{70.300722 \times 1000000}{1000000} = \frac{70300722}{1000000}70.300722=100000070.300722×1000000=100000070300722
Here, 70300722 is an integer (the numerator), and 1000000 is also an integer (the denominator), fulfilling the requirements for rational numbers. The denominator is not zero, which confirms that this expression is valid.
Furthermore, it’s essential to note that all terminating decimals (like 70.300722) are classified as rational numbers because they can be expressed as fractions. In contrast, non-terminating decimals may still be rational if they repeat a sequence (e.g., 0.333… is rational since it can be expressed as 13\frac{1}{3}31).
In conclusion, since 70.300722 can be expressed as a fraction of two integers, it is indeed a rational number. This characteristic is a fundamental aspect of numbers in mathematics, allowing us to categorize and understand different types of numerical values systematically.