Final Answer:
The best description for the real number 4.6 repeating is a) Rational number.
Step-by-step explanation:
To determine the best description for the real number 4.6 repeating (which we can write as 4.\overline{6}), let's go through each option and consider what each term means:
a) Rational number: A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. Rational numbers include integers, fractions, and repeating or terminating decimals.
b) Irrational number: An irrational number is a number that cannot be expressed as a simple fraction. This means it cannot be written as a ratio of two integers. Irrational numbers have non-repeating, non-terminating decimal expansions. Examples include π and √2.
c) Whole number: Whole numbers are the set of non-negative integers, which include 0, 1, 2, 3, and so on. Whole numbers do not include any fractions or decimals.
d) Natural number: Natural numbers are a subset of whole numbers, generally defined as the set of positive integers (1, 2, 3, and so on). They do not include zero, fractions, or any decimal numbers.
Now, let's focus on the number in question: 4.6 repeating. Since this is a decimal number with a repeating pattern, we can indeed express it as a fraction, which makes it a rational number.
Here's how we can do that:
Let x = 4.overline6 (which means 4.6666...)
Multiplying both sides of this equation by 10 gives us a new equation since moving the decimal place one to the right essentially multiplies the number by 10.
So, 10x = 46.overline6.
Now let's subtract the original equation (x = 4.overline6) from this new equation (10x = 46.overline6):
10x - x = 46.overline6 - 4.overline6
9x = 42
To solve for x, divide both sides by 9:
x = 42 / 9
The division can be simplified, and we end up with:
x = 14 / 3
So, since 4.overline6 can be represented by the fraction 14/3, it is indeed a rational number.
The best description for the real number 4.6 repeating is: a) Rational number.