Is 5.3 repeating rational
The Correct Answer and Explanation is :
Yes, 5.3 repeating (denoted as ( 5.\overline{3} )) is a rational number.
Explanation:
A rational number is defined as any number that can be expressed as the ratio of two integers. Specifically, if a number can be written in the form ( \frac{a}{b} ), where ( a ) and ( b ) are integers and ( b \neq 0 ), then the number is rational.
Now, let’s examine ( 5.\overline{3} ), which means the digit 3 repeats indefinitely:
[
5.\overline{3} = 5.33333\ldots
]
To show that this number is rational, we can convert it into a fraction. Here’s a step-by-step process:
- Let ( x = 5.\overline{3} ).
- Multiply both sides of the equation by 10 to shift the decimal point over:
[
10x = 53.\overline{3}
] - Now subtract the original equation ( x = 5.\overline{3} ) from this new equation ( 10x = 53.\overline{3} ):
[
10x – x = 53.\overline{3} – 5.\overline{3}
]
This simplifies to:
[
9x = 48
] - Solve for ( x ):
[
x = \frac{48}{9}
] - Simplify the fraction:
[
x = \frac{16}{3}
]
Thus, ( 5.\overline{3} = \frac{16}{3} ), which is a rational number because it can be expressed as the ratio of two integers.
Why is this important?
The key takeaway is that repeating decimals are always rational numbers. If a decimal repeats or terminates, it can be written as a fraction. This is in contrast to irrational numbers, which cannot be expressed as a fraction (e.g., ( \pi ), ( \sqrt{2} ), etc.), and have non-repeating, non-terminating decimals.
In conclusion, ( 5.\overline{3} ) is a rational number because it can be expressed as ( \frac{16}{3} ), which is a ratio of two integers.