Which of the is a rational number but not an integer?
a.0
b. 9/10
c -18
d. 2.52943...
To determine which of the given options is a rational number but not an integer, we need to understand the definitions of rational numbers and integers.
Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. Integers, on the other hand, are whole numbers (positive, negative, or zero) without any fractional or decimal parts.
Let's analyze each option:
a. 0: Zero (0) is an integer because it is a whole number with no fractional or decimal parts.
b. 9/10: This number can be expressed as a fraction, where the numerator is 9 and the denominator is 10. It is not an integer because it has a fractional part (9/10).
c. -18: Negative 18 is an integer because it is a whole number with no fractional or decimal parts.
d. 2.52943...: This number is given as a decimal with an ellipsis, indicating that it has an infinite number of decimal places. However, it can still be expressed as a fraction. To convert it into a fraction, we need to write it as a repeating decimal. From the given information, we can see that the decimal part repeats after the decimal point, which suggests that it is a repeating decimal. We can denote the repeating part with an overline: 2.52943... = 2.52943¯. To express it as a fraction, we need to multiply it by a power of 10 that eliminates the repeating part:
Let x = 2.52943...
Then 1000x = 2529.43434...
Subtracting these two equations, we get:
1000x - x = 2529.43434... - 2.52943...
999x = 2526.90591...
Now, we can see that the repeating part is canceled out:
999x = 2526.90591 - 2.52943
999x = 2524.37648
Simplifying this equation:
x = 2524.37648 / 999
After performing the division, we find that x ≈ 2.52703003003...
Thus, the correct answer is d. 2.52943..., which is a rational number (as it can be expressed as a fraction) but not an integer, as it has a decimal part.