Is 90.790170 rational or irrational
The Correct answer and Explanation is:
To determine whether the number 90.79017090.79017090.790170 is rational or irrational, we first need to understand the definitions of rational and irrational numbers.
A rational number is any number that can be expressed as the quotient of two integers, where the denominator is not zero. In other words, if a number can be written in the form ab\frac{a}{b}ba, where aaa and bbb are integers and b≠0b \neq 0b=0, it is considered rational. Rational numbers include integers (like −2-2−2 or 555), fractions (like 13\frac{1}{3}31), and terminating or repeating decimals.
An irrational number, on the other hand, cannot be expressed as a simple fraction. Irrational numbers have decimal expansions that neither terminate nor repeat. Examples of irrational numbers include π\piπ (approximately 3.14159…3.14159…3.14159…) and the square root of 222 (approximately 1.414213…1.414213…1.414213…).
Now, let’s analyze the number 90.79017090.79017090.790170. This number is a decimal with a finite number of digits after the decimal point. Specifically, it has six digits following the decimal: 790170790170790170. Since it can be expressed as a fraction of two integers, for instance:90.790170=90790170100000090.790170 = \frac{90790170}{1000000}90.790170=100000090790170
Here, 907901709079017090790170 is an integer, and 100000010000001000000 (which is 10610^6106) is also a non-zero integer. This shows that 90.79017090.79017090.790170 can be written in the form ab\frac{a}{b}ba, thus confirming it meets the criteria for being a rational number.
To summarize, the number 90.79017090.79017090.790170 is a rational number because it has a terminating decimal representation and can be expressed as the quotient of two integers. This characteristic clearly distinguishes it from irrational numbers, which do not conform to these properties. Therefore, the correct answer is that 90.79017090.79017090.790170 is a rational number.