Which set of numbers is always rational?(1 point) Responses (A)positive numbers, (B)decimals decimals, (C)integers (D)fractions
The set of numbers that is always rational is (D) fractions.
can you give me an explanation on why its D
Certainly!
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.
Fractions, by definition, are expressed as the quotient of two integers (numerator and denominator). For example, 1/2, 3/4, and -5/7 are all fractions.
On the other hand:
- Positive numbers (A) can be rational, such as 1 or 5/1, but they can also be irrational like √2 or π.
- Decimals (B) can also be rational, like 0.25 or 3.75, but they can also be irrational like √2 or π.
- Integers (C) include all positive and negative whole numbers and zero. While integers can be rational, such as 1 or -3, they can also be irrational if expressed as decimals like 2.348 or -0.837.
However, fractions (D) are always rational because they are explicitly defined as the quotient of two integers. As long as the denominator is not zero, fractions will always be rational numbers.
The set of numbers that is always rational is (C) integers. Integers are whole numbers, including both positive and negative numbers, as well as zero. All integers can be expressed as fractions with a denominator of 1, which makes them rational numbers.
To determine which set of numbers is always rational among the given options, let's understand the concept of rational numbers.
A rational number is any number that can be expressed as a fraction, where both the numerator and the denominator are integers. In other words, rational numbers can be written in the form a/b, where a and b are integers and b is not equal to 0.
Now let's evaluate each option to determine which set of numbers always satisfies this definition:
(A) Positive numbers: Positive numbers can be rational or irrational. For example, the number 4 can be expressed as 4/1, where both 4 and 1 are integers. However, numbers like √2 (√2/1) or π (π/1) are irrational. Therefore, the set of positive numbers is not always rational.
(B) Decimals: Decimals can represent both rational and irrational numbers. Some decimals, such as 0.5 (1/2) or 0.25 (1/4), can be expressed as fractions. However, decimals like √2 (approximately 1.414213) or π (approximately 3.14159) are irrational. Therefore, the set of decimals is not always rational.
(C) Integers: Integers are whole numbers, both positive, negative, and zero. All integers can be represented as fractions by placing them over 1, such as 3/1, (-4)/1, or 0/1. Hence, the set of integers is always rational.
(D) Fractions: Fractions are numbers expressed as a ratio of two integers. Since fractions are explicitly defined as rational numbers, the set of fractions is always rational.
Therefore, among the given options, the set of numbers that is always rational is (C) integers.