Describe the relationship between rational numbers and whole numbers. Are all
rational numbers also whole numbers? Are all whole numbers also rational
numbers? Give examples to support your answer. (4 points)
The relationship between rational numbers and whole numbers is that every whole number is also a rational number, but not every rational number is a whole number.
Whole numbers include all positive integers (numbers greater than zero) and zero itself. Rational numbers, on the other hand, include all numbers that can be expressed as a fraction, where the numerator and denominator are both integers and the denominator is not zero.
Since whole numbers can be expressed as a fraction with a denominator of 1, they can also be considered rational numbers. For example, the whole number 5 can be expressed as the fraction 5/1, making it a rational number.
However, not all rational numbers are whole numbers. For example, the fraction 4/3 is a rational number but not a whole number. This is because the numerator (4) and the denominator (3) are both integers, but the fraction is not equal to a whole number.
In summary, every whole number is a rational number, but not every rational number is a whole number. Whole numbers are a subset of rational numbers, and they can be expressed as a fraction with a denominator of 1.
The relationship between rational numbers and whole numbers is that rational numbers include whole numbers as a subset. Not all rational numbers are whole numbers, but all whole numbers are rational numbers.
A rational number is any number that can be expressed as a ratio of two integers, where the denominator is not zero. Whole numbers are a subset of rational numbers where the denominator is always 1.
For example:
- The rational number 2 can also be written as 2/1, where the denominator is 1. So, 2 is both a whole number and a rational number.
- The rational number 3/2 cannot be expressed as a whole number because the denominator is not 1. Therefore, 3/2 is a rational number but not a whole number.
- On the other hand, the whole number 5 can be expressed as 5/1, where the denominator is 1. So, 5 is both a whole number and a rational number.
In summary, while every whole number is a rational number, not every rational number is a whole number.
The relationship between rational numbers and whole numbers can be described as follows:
1. Are all rational numbers also whole numbers?
No, not all rational numbers are whole numbers. Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not zero. This includes fractions and decimals with terminating or repeating patterns. Whole numbers, on the other hand, are non-negative integers (0, 1, 2, 3, ...). So, while whole numbers can also be expressed as fractions or decimals (e.g., 2 can be written as 2/1 or 2.0), not all rational numbers fall under the definition of whole numbers.
Example:
The rational number 3/2 can be expressed as 1.5 in decimal form, so it is a rational number. However, it is not a whole number because it is not an integer.
2. Are all whole numbers also rational numbers?
Yes, all whole numbers are also rational numbers. Since whole numbers are integers, they can be expressed as a ratio of an integer (the numerator) and 1 (the denominator). In other words, every whole number can be written with a denominator of 1.
Example:
The whole number 10 can be expressed as 10/1, making it a rational number as well.
In summary, while all whole numbers are rational numbers, not all rational numbers are whole numbers.